\documentclass{gtart} % to submit to AGT, uncomment %\agtart \newcommand{\pathtotrunk}{./} \input{text/article_preamble.tex} \input{text/top_matter.tex} \usepackage{xcolor} \definecolor{dark-red}{rgb}{0.7,0.25,0.25} \definecolor{dark-blue}{rgb}{0.15,0.15,0.55} \definecolor{medium-blue}{rgb}{0,0,0.65} \hypersetup{ colorlinks, linkcolor={black}, citecolor={medium-blue}, urlcolor={medium-blue} } \begin{document} \begin{abstract} We summarise the known obstructions to principal graphs of subfactors which begin with a triple point. One is based on Jones's quadratic tangles techniques, although we apply it in a novel way. The other two are based on connections techniques; one due to Ocneanu, and the other previously unpublished, although likely known to Haagerup. We then apply these obstructions to the classification of subfactors with index below $5$. In particular, we eliminate two of the five families of possible principal graphs called ``weeds" in the classication from \cite{1007.1730}. In an appendix we rule out a third of the five families; while this principal graph does begin with a triple point, none of the known triple point obstructions apply, and we use an argument based on connections and combinatorial properties of the graph norm. \end{abstract} \maketitle % remove table of contents for submitted version \tableofcontents \hypersetup{ colorlinks, linkcolor={purple}, citecolor={medium-blue}, urlcolor={medium-blue} } \newcommand{\MagicNumbersOneZero}{\ensuremath{\left(\bigraph{bwd1v1v1v1p1v1x0p0x1duals1v1v1x2},\bigraph{bwd1v1v1v1p1v1x0p1x0duals1v1v1x2}\right)}} \newcommand{\crab}{\ensuremath{\left(\bigraph{bwd1v1v1v1p1v1x0p0x1v1x0p0x1p0x1v1x0x0p0x0x1duals1v1v1x2v2x1x3}, \bigraph{bwd1v1v1v1p1v1x0p1x0v0x1v1p1duals1v1v1x2v1}\right)}} \newcommand{\BadSeed}{\ensuremath{\left(\bigraph{bwd1v1v1v1p1v1x0p0x1v1x0p1x0p0x1p0x1v0x1x0x0p0x0x1x0v1x0p0x1v1x0p0x1duals1v1v1x2v1x3x2x4v1x2}, \bigraph{bwd1v1v1v1p1v1x0p1x0v1x0p0x1v0x1p1x0v1x0p0x1v0x1p1x0duals1v1v1x2v1x2v2x1}\right) }} \newcommand{\FSMp}{\bigraph{bwd1v1v1v1p1v1x0p0x1v1x0p1x0p0x1p0x1v0x1x0x0p0x0x1x0p0x0x0x1v1x0x0p1x0x0p0x1x0p0x0x1v1x0x0x0p0x0x1x0p0x0x0x1duals1v1v1x2v1x3x2x4v1x4x3x2}} \newcommand{\FSMd}{\bigraph{bwd1v1v1v1p1v1x0p1x0v1x0p0x1v0x1p1x0p0x1v1x0x0p0x1x0v0x1p1x0p0x1duals1v1v1x2v1x2v2x1}} \newcommand{\FSM}{\ensuremath{\left(\FSMp, \FSMd\right)}} \newcommand{\FSMFp}{\bigraph{bwd1v1v1v1v1v1p1v1x0p0x1v1x0p1x0p0x1p0x1v0x1x0x0p0x0x1x0p0x0x0x1v1x0x0p1x0x0p0x1x0p0x0x1v1x0x0x0p0x0x1x0p0x0x0x1duals1v1v1v1x2v1x3x2x4v1x4x3x2}} \newcommand{\FSMFd}{\bigraph{bwd1v1v1v1v1v1p1v1x0p1x0v1x0p0x1v0x1p1x0p0x1v1x0x0p0x1x0v0x1p1x0p0x1duals1v1v1v1x2v1x2v2x1}} \newcommand{\FSMF}{\ensuremath{\left(\FSMFp, \FSMFd\right)}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Introduction} Jones' index theorem for subfactors \cite{MR696688} states that the index of a subfactor lies in the range $\{ 4 \cos^2(\frac{\pi}{n}) | n =3, 4, \ldots \} \cup [4,\infty]$. Any of these numbers can be realized as the index of a subfactor whose standard invariant is either a quotient of Temperley-Lieb (if the index is less than $4$) \cite{MR696688} or a Temperley-Lieb (if the index is at least $4$) \cite{MR1198815}. However, once you ignore the subfactors with TL standard invariant, the possible indices for irreducible subfactors are again quantized in an interval above $4$. Haagerup began the classification of subfactors with index `only a little larger' than four in \cite{MR1317352}. In that paper, he showed there are no extremal subfactors (other than those with TL standard invariant) with index in the range $(4,\frac{5+\sqrt{13}}{2})$. Furthermore, he gave a complete list of possible principal graphs of extremal subfactors whose index falls in the range $(4,3+\sqrt{3})$. (He states the result up to $3+\sqrt{3}$, and proves it up to $3+\sqrt{2}$.) Most of the graphs on this list were excluded by Bisch \cite{MR1625762} and Asaeda-Yasuda \cite{MR2472028}, while the remaining $3$ graphs were shown to come from (unique) subfactors by Asaeda-Haagerup \cite{MR1686551} and Bigelow-Morrison-Peters-Snyder \cite{0909.4099}. Haagerup's classification stops at index $3+\sqrt{3}$ for reasons of computational convenience, and because a Goodman-de la Harpe-Jones subfactor \cite{MR999799} was already known to exist at that index. However, recent theoretical progress \cite{math/1007.1158,1004.0665} and modern computer algebra systems make it possible to extend the classification of small-index subfactors further. This paper is the second in a series of papers which gives a classification of subfactors of index less than $5$. In the first paper \cite{1007.1730} we gave an initial classification result analogous to Haagerup's classification. The subsequent papers rule out the remaining families. Thus they are more closely analogous to the papers of Bisch \cite{MR1625762} (which applied fusion algebras to eliminate one family) and Asaeda--Yasuda \cite{MR2472028} (which applied number theory to eliminate one family). In this paper, we discuss several different ``triple point obstructions" and apply them to eliminate two families. These two families are more complicated in structure than the ones ruled out by Bisch and Asaeda--Yasuda. %{\color{blue} \todo{No longer true!} %Like Haagerup's classification, the first paper emphasizes ``local" techniques which eliminate certain small features (like certain kinds of triple points) without regard for the rest of the graph. The subsequent papers, including this one, use more global techniques. Thus these papers are more closely analogous to the papers of Bisch \cite{MR1625762} (which applied fusion algebras to eliminate one family) and Asaeda--Yasuda \cite{MR2472028} (which applied number theory to eliminate one family). In this paper we apply theoretical advances from Jones's quadratic tangles technique \cite{math/1007.1158} to eliminate two families. These two families are more complicated in structure than the ones ruled out by Bisch and Asaeda--Yasuda. %} In part $1$ of this series, we use the term \emph{translation of a graph pair} to indicate a graph pair obtained by increasing the supertransitivity by an even integer (the \emph{supertransitivity} is the number of edges between the initial vertex and the first vertex of degree more than two). An \emph{extension of a graph pair} is a graph pair obtained by extending the graphs in any way at greater depths (i.e. adding vertices and edges at the right), even infinitely. The main result of that paper was the following. \begin{thm}[From \protect{\cite{1007.1730}}] The principal graph of any subfactor of index between $4$ and $5$ is a translate of one of an explicit finite list of graph pairs (which we call the \emph{vines}), or is a translated extension of one of the following graph pairs (which we call the \emph{weeds}). \begin{align*} \cC &= \crab, \\ \cF &=\FSM, \\ \cB &=\BadSeed, \\ \cQ &= \left(\bigraph{bwd1v1v1v1p1p1v0x0x1duals1v1v2x1x3}, \bigraph{bwd1v1v1v1p1p1v0x0x1duals1v1v2x1x3} \right), \\ \cQ' &= \left(\bigraph{bwd1v1v1v1p1p1v0x0x1duals1v1v1x2x3}, \bigraph{bwd1v1v1v1p1p1v0x0x1duals1v1v1x2x3} \right). \end{align*} \end{thm} (As in \cite{1007.1730}, the trivial bimodule always appears as the leftmost vertex of a principal graph, and dual pairs of bimodules are indicated by red tags.) The main results of this paper are to show that three of the above weeds do not appear as principal graphs of subfactors. We do this by applying two stronger versions of the triple point obstruction, one using Jones's which uses ``quadratic tangles" planar algebra techniques \cite{math/1007.1158}, and one using more traditional connections arguments. These arguments also suffice to eliminate all but one graph in the ``Asaeda--Haagerup" family, which is an unpublished result of Haagerup's. \begin{thm}\label{thm:Crab} There are no subfactors, of any index, with principal graphs a translated extension of the pair $$\cC=\crab.$$ \end{thm} \begin{thm}\label{thm:FSM} There are no amenable subfactors, of any index, with principal graphs a translated extension of the pair $$\cF=\FSM$$ and such a non-amenable subfactor must have index (approximately) in the set $\{5.04468, 5.69758, 6.28995\}$ (see Proposition \ref{FSMprop} below for the exact values). \end{thm} \begin{thm}\label{thm:BadSeed} There are no subfactors, of any index \todo{true?}, with principal graphs a translated extension of the pair $$\cB =\BadSeed.$$ \end{thm} \begin{rem} The triple point obstructions in this paper are also capable of ruling out a large subset of the vines described in the first paper \cite{1007.1730}. However, it requires a certain amount of work for each vine, along the lines of the calculations we do here. Happily, there is a uniform approach arithmetic approach, which works for all vines, based on \cite{1004.0665}. A later paper in this series \cite{index5-part4} will use that technique to reduce the vines to a finite set of graphs. \end{rem} The structure of this paper is as follows. \todo{write this once the rest of the paper is written} Bundled with the arXiv sources of this article are two Mathematica notebooks, \code{Crab.nb} and \code{FSM.nb} \todo{and something for AH?}, which contain all relevant calculations for what follows. These make use of a package called FusionAtlas; see \cite{1007.1730} for a terse tutorial on its use. Note that in this paper, unlike in several of the other papers in the series, every calculation can be easily checked by hand and thus this paper does not use a computer in an essential way. A typical calculation in this paper involves solving a system of a dozen or so linear equations or multiplying several polynomials in a single variable. Nonetheless we have included notebooks which perform these calculations because computer calculations are easier to check and less prone to minor errors than calculations by hand. We would like to thank Vaughan Jones for helpful conversations and for hosting several ``Planar algebra programming camps'' where most of this work was done. Scott Morrison was at Microsoft Station Q at UC Santa Barbara and at the Miller Institute for Basic Research at UC Berkeley during this work, David Penneys was supported by UC Berkeley's Geometry, Topology, and Operator Algebras NSF grant EMSW21-RTG, Emily Peters was in part at the University of New Hampshire and in part supported by an NSF Postdoctoral Fellowship at MIT, and Noah Snyder was supported in part by RTG grant DMS-0354321 and in part by an NSF Postdoctoral Fellowship at Columbia University. We would also like to acknowledge support from the DARPA HR0011-11-1-0001 grant. \section{Background} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Annular Temperley-Lieb} The goal of this paper is to describe various triple-point obstructions, and apply these to graph pairs having annular multiplicities $*10$. We rapidly recall the language from \cite{MR1929335, math/1007.1158} to make sense of this statement and put it in context. A subfactor is called {\em $n$-supertransitive} if up to the $n$-box space its planar algebra is just Temperley-Lieb. Equivalently, a subfactor is $n$-supertransitive if and only if the principal graph up to depth $n$ is $A_{n+1}$. Any planar algebra is a module for the annular Temperley-Lieb algebra, and as such decomposes into irreducible modules. The theory of annular Temperley-Lieb modules is laid out in Graham-Lehrer \cite{MR1659204} (and in Jones \cite{MR1929335}, where the idea to apply annular Temperley-Lieb theory to planar algebras appears). Each such module is cyclic, generated by a `lowest weight vector' (that is, a submodule of a planar algebra $\cP$ is a direct sum of subspaces of $\cP_k$ closed under action by annular Temperley-Lieb tangles; the weight of a vector in $\cP_k$ is $k$, and for $n$ the lowest weight appearing in a submodule, the subspace of $\cP_n$ is one dimensional). Each such lowest weight vector with nonzero weight $n$ has a rotational eigenvalue which is an $n$-th root of unity. (Lowest weight vectors with weight $0$ have instead a `ring eigenvalue'.) The \emph{annular multiplicities} of a planar algebra are the sequence of multiplicities of lowest weight vectors. A theorem of Jones \cite{MR1929335} shows that the annular multiplicities are actually determined entirely by the principal graph. Thus we can discuss the annular multiplicities of a graph pair regardless of whether it comes from a subfactor. The $0$th annular multiplicity of a subfactor planar algebra is always $1$, corresponding to the empty diagram which generates Temperley-Lieb as an annular Temperley-Lieb module. If the planar algebra is $n$-supertransitive, then the next $n$ annular multiplicities are $0$, because the vector spaces $\cP_1$ through $\cP_n$ are each no larger than their Temperley-Lieb subalgebra. An $n$-supertransitive subfactor of annular multiplicities $*10$ means that the first two annular multiplicities after the long string of $n$ zeroes are $1$ and $0$. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Connections} In this subsection we rapidly recall the theory of paragroups and biunitary connections on graphs developed by Ocneanu in \cite{MR996454}. We have taken our conventions and normalizations from \cite{MR1642584}. This section is especially brief because we do not need the key notion of ``flatness" for a connection, because all the obstructions in this paper are obstructions even to the existence of non-flat biunitary connections. For simplicity, we assume all graphs are simply laced. This is not at all necessary for the theory of connections, but will make our notation cleaner; and all the graphs we consider in this paper are simply laced. Suppose we have a bigraph pair $(\Gamma, \Gamma')$ (recall from \cite{1007.1730} that this means $\Gamma$, $\Gamma'$ are bipartite graphs with dual data and specified root vertices, that $\Gamma$ and $\Gamma'$ have the same supertransitivity, and at each odd depth, there is a bijection called duality between vertices of $\Gamma$ and $\Gamma'$.) If further $\Gamma$ and $\Gamma'$ have the same Perron-Frobenius eigenvalue, and Perron-Frobenius dimensions which agree on odd vertices, then we can assemble a {\em 4-partite Ocneanu graph} from $\Gamma$ and $\Gamma'$: $$\begin{tikzpicture} \node (V00) at (0,0) {$V_{00}$}; \node (V11) at (2,-2) {$V_{11}$}; \node (V01) at (0,-2) {$V_{01}$}; \node (V10) at (2,0) {$V_{10}$}; \draw (V00)--(V01) node [left, midway] {$\Gamma$}; \draw (V01)--(V11) node [below, midway] {$\Gamma'$}; \draw (V00)--(V10) node [above, midway] {$\Gamma$}; \draw (V10)--(V11) node [right, midway] {$\Gamma'$}; \end{tikzpicture} $$ If we started with a subfactor $N \subset M$, the graph built from $N-N$, $N-M$, $M-M$ and $M-N$ bimodules under fusion with $X$ and $X^*$ is a 4-partite Ocneanu graph, with $$\begin{tikzpicture} \node (V00) at (0,0) {$V_{00}=\{ N-N \text{ bimodules}\}$}; \node (V11) at (6,-2) {$V_{11}=\{ M-M \text{ bimodules}\}$}; \node (V01) at (0,-2) {$V_{01}=\{ M-N \text{ bimodules}\}$}; \node (V10) at (6,0) {$V_{10}=\{N-M \text{ bimodules}\}$}; \draw (V00)--(V01) node [left, midway] {$\Gamma$}; \draw (V01)--(V11) node [below, midway] {$\Gamma'$}; \draw (V00)--(V10) node [above, midway] {$\Gamma$}; \draw (V10)--(V11) node [right, midway] {$\Gamma'$}; \end{tikzpicture} $$ \todo{Example of 4-partite graph. either $\cB$ or $\cF$} \begin{defn} Suppose we have a 4-partite Ocneanu graph $\cG$, with the additional property that the root vertices $*$ in $\Gamma$ and $\Gamma'$ have degree-one \eep{This is one of EK's axioms for a connection; I agree it's not needed but we're trying to stick to their conventions}\noah{I think this additional assumption is only needed for flatness not for defining a connection?}. Then a {\em connection} on $\cG$ is a map \begin{align*} W: \{ \text{based loops of length 4 around } W \} & \rightarrow \Complex \end{align*} where the based loops are in one of four orders: \begin{align*} V_{0,0} \rightarrow V_{1,0} \rightarrow V_{1,1} \rightarrow V_{0,1}, &\\ V_{1,0} \rightarrow V_{0,0} \rightarrow V_{0,1} \rightarrow V_{1,1},& \\ V_{0,1} \rightarrow V_{1,1} \rightarrow V_{1,0} \rightarrow V_{0,0},&\\ \text{ or } V_{1,1} \rightarrow V_{0,1} \rightarrow V_{0,0} \rightarrow V_{1,0}.& \end{align*} If there is no edge in $\cG$ between vertices $A_i$ and $A_{i+1\mod 4}$, then $W(A_1,A_2,A_3,A_4)=0$. A connection is {\em biunitary} if the following properties hold \begin{itemize} \item {\em{Unitarity}}: for all vertices $A$ and vertices $C$ diagonally opposite to $A$ in $\cG$, the matrix $W(A,-,C,-)$ is unitary; ie, $$\sum_{D} W(A,B,C,D) \overline{W(A,B',C,D)}=\delta_{B,B'}.$$ \item {\em{Renormalization}}: for all $A$, $B$, $C$, $D$, $$W(A,B,C,D)=\sqrt{\frac{\dim(B) \dim(D)}{\dim(A) \dim(C)}} \overline{ W(B,A,D,C) }$$ \end{itemize} \end{defn} Connections are important to us because \begin{thm}\label{thm:subfactorconn} A subfactor $N \subset M$ defines a (flat) connection on its 4-partite principal graph. \end{thm} and in particular, \begin{cor} A 4-partite graph which does not have a biunitary connection is not the principal graph of a subfactor. \end{cor} \begin{proof}[Proof of Theorem \ref{thm:subfactorconn}] See \cite[ch.10-11]{MR1642584} for a full proof; we give a brief outline only.\eep{I'm not convinced we need to include a proof here. However, if you don't want to include it, please comment out but don't delete!}\dave{i like it. it's way easier to read than EK...} Let $X= _{N}\!\!M_M$; by $X^\pm$, we mean whichever of $X$ or $X^*$ makes sense in context. $A$, $B$, $C$ and $D$ are irreducible bimodules; their type ($N-N$, $N-M$, etc.) is specified when necessary. For bimodules $A$ and $B$, define an inner product on $\sigma, \tau \in \Hom{}{A \otimes X^\pm}{B}$ by $\ip{\sigma, \tau}=\sigma \tau^* \in \Complex$. Here we interpret $\sigma \tau^*$ as a complex number via \begin{align*} \End{B} & \overset{\sim}{\longrightarrow} \Complex \\ \id & \longmapsto 1. \end{align*} Let $X= _{N}\!\!M_M$, and choose an orthonormal basis $\{ \sigma_{A,X}^B \}$ for each space $\Hom{N,M}{A \otimes X}{B}$ or $\Hom{M,N}{A \otimes X^*}{B}$. (Since we assume our graphs are simply laced, each of these spaces is one-dimensional and we are only making a choice of normalization.) These can be used to define orthonormal bases $\{ \sigma_{A,X}^B \} $ of $\Hom{N,N}{A \otimes X^*}{B}$ or $\Hom{M,M}{A \otimes X}{B}$ by requiring rotational invariance ($\sigma_{A,X}^B = \sigma_{X,B^*}^{A^*}$\footnote{Though these morphisms seem to live in different Hom-spaces, this equality makes sense because we have Frobenius reciprocity: $\Hom{}{A \otimes X^\pm }{B} \overset{\sim}{\longrightarrow} \Hom{}{X^\pm \otimes B^*}{A^*}$.}), and orthonormal bases $\{ \sigma_{X,A}^B \}$ of $\Hom{\_,\_}{X^\pm \otimes A}{B}$ by requiring duality ($(\sigma_{A,X}^B)^*=\sigma_{X,A^*}^{B^*}$). Now, $\Hom{}{X^\pm \otimes A \otimes X^\pm}{C}$ has two different orthonormal bases: \begin{align*} \cL & =\{ \sigma_{B,X}^C (\sigma_{X,A}^B \otimes \id) | B \subset X^\pm \otimes A \text{ and } C \subset B \otimes X^\pm \} \text{ and } \\ \cR & =\{ \sigma_{X,D}^C (\id \otimes \sigma_{A,X}^D ) | D \subset A \otimes X^\pm \text{ and } C \subset X^\pm \otimes D\}. \end{align*} Define $$ W(A,B,C,D)= \ip{\sigma_{B,X}^C (\sigma_{X,A}^B \otimes \id), \sigma_{X,D}^C (\id \otimes \sigma_{A,X}^D)} $$ Then, $W(A,B,C,D)$ is the coefficient of $\sigma_{X,D}^C (\id \otimes \sigma_{A,X}^D ) \in \cR$ when $ \sigma_{B,X}^C (\sigma_{X,A}^B \otimes \id)\in \cL$ is written as a linear combination of the basis $\cR$: $$\sigma_{B,X}^C (\sigma_{A,X}^B \otimes \id)= \sum_D W(A,B,C,D) \sigma_{X,D}^C (\id \otimes \sigma_{A,X}^D ) .$$ Unitarity follows from the fact that the bases $\cL_{A,C}$ are again orthonormal; the renormalization axiom follows from the good behavior of our bases $\sigma$ under duality and rotation. Flatness is unnecessary for this paper, so we say no more about it. \end{proof} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Triple Point Obstructions} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{The triple-single obstruction} Though to our knowledge the main result of this section is unpublished, we expect that this result or something like it was used by Haagerup to rule out the Asaeda-Haagerup vines beyond the first one (as we do in section \ref{sec:AH}). \begin{thm}[Triple-single obstruction]\label{thm:TripleSingle} Suppose we have a $4$-partite graph $\cG=(\sb{AA}\Gamma_{AB}\Gamma'_{BB}\Gamma'_{BA}\Gamma_{AA})$, and its component graphs $\Gamma$ and $\Gamma'$ have an odd triple point $\beta$/$\beta^*$ which is adjacent in one of the graphs to a degree-one vertex $\gamma_2$: $$ \begin{tikzpicture}[baseline,scale=.3] \draw[ultra thick] (0,0)--(3,0); \draw[ultra thick] (3,0)--(6,2); \draw[ultra thick] (3,0)--(6,-2); \filldraw[fill=white] (0,0) circle (.3cm); \filldraw (3,0) circle (.3cm); \filldraw[fill=white] (6,2) circle (.3cm); \filldraw[fill=white] (6,-2) circle (.3cm); \node at (0,0) [above] {$\alpha_1$}; \node at (3,0) [above] {$\beta$}; \node at (6,2) [right] {$\alpha_{2}$}; \node at (6,-2) [right] {$\alpha_{3}$}; \end{tikzpicture}\, ,\, \begin{tikzpicture}[baseline,scale=.3] \draw[ultra thick] (0,0)--(3,0); \draw[ultra thick] (3,0)--(6,2); \draw[ultra thick] (3,0)--(6,-2); \filldraw[fill=white] (0,0) circle (.3cm); \filldraw (3,0) circle (.3cm); \filldraw (6,2) circle (.3cm); \filldraw[fill=white] (6,-2) circle (.3cm); \node at (0,0) [above] {$\gamma_1$}; \node at (3,0) [above] {$\beta^*$}; \node at (6,2) [right] {$\gamma_{2}$}; \node at (6,-2) [right] {$\gamma_{3}$}; \end{tikzpicture}\, ,\, $$ Further suppose \begin{itemize} \item $\dim(\alpha_1)=\dim(\gamma_1)$; \item the only length-two paths (in $\mathcal{G}$) between $\alpha_1$ and $\gamma_{2}$ or $\gamma_{3}$ go through $\beta$ and $\beta^*$; \item the only length-two paths (in $\mathcal{G}$) between $\gamma_1$ and $\alpha_{2}$ or $\alpha_{3}$ go through $\beta$ and $\beta^*$. \end{itemize} If there is a biunitary connection $K$ on $\cG$, then \begin{equation}\label{eqn:TripleSingle} |\dim(\alpha_2)-\dim(\alpha_3)|\leq K(\beta, \alpha_1, \beta^*, \gamma_1) \dim(\beta). \end{equation} \end{thm} \begin{proof} The idea of this proof is to write down the three-by-three matrix $K(\beta, -, \beta^*, -)$; the conclusion will follow from unitarity. Let $a_i=\sqrt{\dim(\alpha_i)}$, $b=\sqrt{\dim(\beta)}$ and $c_i=\sqrt{\dim(\gamma_i)}$. By our hypotheses, we can find the norms of all entries of $K(\beta, -, \beta^*, -)$ except three. For example, $K( \alpha_2, \beta, \gamma_2, \beta^*)$ is a 1-by-1 unitary matrix, ie a complex unit; so by the renormalization axiom, $\abs{K(\beta, \alpha_2, \beta^*, \gamma_2)}=\dfrac{a_2 c_2}{b^2}$. This gives us that, up to phases, $$ K(\beta, -, \beta^*, -)=\frac{1}{b^2} \begin{pmatrix} ? & a_1 c_2 & a_1 c_3\\ a_2 c_1 &a_2c_2& ?\\ a_3 c_1 & a_3c_2& ? \end{pmatrix} $$ Taking the inner products of the first two columns and dividing by $\dfrac{c_1 c_2}{b^2}$, (recall $c_1 = a_1$), we have \begin{multline*} K(\beta, \alpha_1, \beta^*, \gamma_1) b^2 + e^{i \phi} a_2^2 + e^{i \psi} a_3^2 = \\ K(\beta, \alpha_1, \beta^*, \gamma_1) \dim(\beta) + e^{i \phi} \dim(\alpha_1) + e^{i \psi} \dim(\alpha_2) = 0 \end{multline*} for some phases $\phi$ and $\psi$. Then by the triangle inequality, we have $$\abs{\dim(\alpha_2)-\dim(\alpha_3)} \leq K(\beta, \alpha_1, \beta^*, \gamma_1) \dim(\beta).$$ \end{proof} Although the hypotheses of this theorem seem quite stringent, they are satisfied in some interesting cases -- for example, if $\beta$ is part of initial string. \begin{cor}\label{cor:TripleSingle} Suppose there is a biunitary connection on the $4$-partite graph $\cG=(\sb{AA}\Gamma_{AB}\Gamma'_{BB}\Gamma'_{BA}\Gamma_{AA})$. Suppose $\Gamma,\Gamma'$ are $(n-1)$-supertransitive (with $n$ even), there is a triple point $\beta$ at depth $n$, and one of the neighbors of $\beta$ or $\beta'$ is degree-one. Then \begin{equation}\label{eqn:InitialTripleSingle} \abs{\dim(\alpha_1)-\dim(\alpha_2)} \leq 1. \end{equation} \end{cor} \begin{proof} The hypotheses of Theorem \ref{thm:TripleSingle} are quickly verified. We find $K( \alpha_1, \beta, \gamma_1, \beta^*)=\dfrac{1}{[n-1]}$ by solving for the connection along the initial $A_n$ segment (this is a quick exercise which is sketched in \cite[p. 574-575]{MR1642584}). This gives us that $K(\beta, \alpha_1, \beta^*, \gamma_1)=\dfrac{1}{[n]}$. As $\dim(\beta)=[n]$, Theorem \ref{thm:TripleSingle} implies the desired inequality. \end{proof} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{The quadratic tangles obstruction} If an $n$-supertransitive principal graph has $n$-th annular multiplicity $1$, then it begins like $D_{n+3}$ (i.e. it starts with a `triple point'). We define the \emph{branch factor}, usually written $r$ (and $\check{r}$ for the branch factor of the dual principal graph), to be the ratio of the dimensions of the two vertices immediately past the branch point (where we take the larger divided by the smaller). If the next annular multiplicity is $0$, there are exactly three possibilities, the translates of: \begin{equation*} \cdots \bigraph{gbg1v1v1v1p1v1x0p0x1}\qquad\text{and}\qquad \cdots \bigraph{gbg1v1v1v1p1v1x0p1x0}. \end{equation*} Consider now a principal graph pair with annular multiplicities $*10$, and supertransitivity $m-1$. Haagerup proved in \cite{MR1317352}, using Ocneanu's triple point obstruction (see below), that the supertransitivity must be odd, and the principal and dual principal graphs must be different. For convenience, we'll always order the principal graph pair so the principal graph starts like the first graph above, and the dual principal graph starts like the second graph above. An improved version of the triple point obstruction was given by Jones in \cite{math/1007.1158} where he also gives the following formulas for $r$, $\check{r}$ and $\lambda$, the rotational eigenvalue of the unique weight $m$ lowest weight vector. \begin{align} r+\frac{1}{r} & = \frac{\lambda+\lambda^{-1}+2}{[m][m+2]}+2 \label{eq:QTequation}\\ \check{r} & = \frac{[m+2]}{[m]} \label{eq:QTrcheck} \end{align} The formula for $\check{r}$ follows from working out dimensions in the dual principal graph (see Example \ref{ex:AnnularMultiplicities*10}), but the formula for $r$ takes significantly more work. Since $\lambda$ must be an $m^\text{th}$ root of unity, we have the following inequalities which do not involve $\lambda$: \begin{equation}\label{eq:QTinequality} -4 \leq \left(r+\frac{1}{r}-2\right)[m][m+2]-4 \leq 0. \end{equation} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Ocneanu's triple point obstruction} For the sake of providing a thorough comparison of the available triple point obstructions, we briefly recall Ocneanu's obstruction, first described in Haagerup's paper \cite{MR1317352}. Note that the first paper of this series \cite{1007.1730} has already made use of this obstruction. There, we stated stronger results than those described by Haagerup (but which are proved by exactly the same technique), and we merely repeat these here. \todo{do so} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Applications} In this section we prove the main results of this paper by applying the triple point obstructions from the last section. For each of our applications we could use either the triple-single obstruction or the quadratic tangles obstructions. Neither obstruction is strictly stronger than the other, but typically the former will leave finitely many cases left over while using the latter approach you can apply \eqref{eq:QTequation} to eliminate the exceptions. In order to apply these triple point obstructions we need to first get a better handle of the dimensions of vertices. \subsection{Relative dimensions of vertices}\label{sec:relativedimensions} Suppose that $\Gamma$ an $n$-translate of an extension of $\Gamma_0$ and that this graph comes from a subfactor of index $(q+q^{-1})^2$. In order to apply the triple point obstructions from the last sections, we need to know the dimensions of the vertices as a function of $n$ and $q$. In general this is impossible, but if we're lucky and know a lot about the principal graphs, we may determine the dimensions of the vertices of each graph as functions of $n,q$ using the following three sets of equations. First, we set the dimension of the leftmost vertex of each graph to be $$[n+1]=\frac{q^{n+1}-q^{-n-1}}{q-q^{-1}}.$$ Second, if two vertices correspond to bimodules which are dual to each other, they must have the same dimension. Third, for each vertex $V$ which only connects to vertices which appear in the known segment of our graphs, we have an equation $$ \dim(V)=[2]\sum\limits_{\text{edges from $W$ to $V$}} \dim(W) $$ where $[2]=q+q^{-1}$. \begin{remark}\label{dimension} We do not assume that $\Gamma$ is finite depth nor that it is amenable. Thus the dimensions mentioned above need not be the Frobenius-Perron eigenvector for $\Gamma$, and the index of the subfactor with principal graph $\Gamma$ need not be the square of the Frobenius-Perron eigenvalue. Nonetheless, we note that the index of any subfactor with principal graph $\Gamma$ must be greater than or equal to $\|\Gamma_0\|^2$ (the square of the Frobenius-Perron eigenvalue for $\Gamma_0$). \end{remark} \begin{ex}\label{ex:AnnularMultiplicities*10} We label the vertices of the graphs $$ \MagicNumbersOneZero $$ by $V^i_{j,k}$ where $i$ is either $p$ or $d$ corresponding to the principal or dual principal graph, $j = 0,1,\ldots,5$ is the depth, and $k$ is the index of the vertex at that depth counting from the bottom to the top. We cannot solve explicitly for the dimensions in terms of $n,q$ for these graphs. Rather, there is a one parameter family of solutions. We set $\alpha=\dim(V^p_{4,2})$, and the dimensions of the vertices as functions of $n,q,\alpha$ are given by: {\scriptsize{ \begin{align*} \dim(V^p_{0,1})&=\dim(V^d_{0,1})=\frac{q^{-n} \left(q^{2 n+2}-1\right)}{q^2-1}& \mathllap{\dim(V^p_{1,1})=\dim(V^d_{1,1})=\frac{q^{-n-1} \left(q^{2 n+4}-1\right)}{q^2-1}}\displaybreak[1]\\\ \dim(V^p_{2,1})&=\dim(V^d_{2,1})=\frac{q^{-n-2} \left(q^{2 n+6}-1\right)}{q^2-1} & \mathllap{\dim(V^p_{3,1})=\dim(V^d_{3,1})=\frac{q^{-n-3} \left(q^{2 n+8}-1\right)}{q^2-1}}\displaybreak[1]\\\ \dim(V^p_{4,1})&=\frac{q^{-n-4} \left(\alpha q^{n+4}-\alpha q^{n+6}+q^{2 n+10}-1\right)}{q^2-1} & \mathllap{\dim(V^p_{4,2})=\alpha\hspace{.93in}}\displaybreak[1]\\\ \dim(V^d_{4,1})&=\frac{q^{-n-4} \left(q^{2 n+12}-1\right)}{q^4-1} & \mathllap{\dim(V^d_{4,2})=\frac{q^{-n-2} \left(q^{2 n+8}-1\right)}{q^4-1}}\displaybreak[1]\\\ \dim(V^p_{5,1})&=\dim(V^d_{5,1})=\frac{q^{-n-5} \left(\alpha q^{n+4}-\alpha q^{n+8}+q^{2 n+12}-1\right)}{q^2-1}\displaybreak[1]\\\ \dim(V^p_{5,2})&=\dim(V^d_{5,2})=-\frac{q^{-n-3} \left(\alpha q^{n+2}-\alpha q^{n+6}+q^{2 n+8}-1\right)}{q^2-1} \hspace{1.5in} \end{align*}}} \end{ex} \begin{ex}\label{ex:Crab} We can solve for the dimensions in $n,q$ for the graphs $$\crab ,$$ (which is an extension of the previous example) because we have the additional equation: $[2]V^d_{5,1} = V^d_{4,1}$. The dimensions of the vertices through depth $5$, as functions of $n,q$, are given by:\\ {\scriptsize{ \begin{align*} \dim(V^p_{0,1})&=\dim(V^d_{0,1})=\frac{q^{-n} \left(q^{2 n+2}-1\right)}{q^2-1} & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \mathllap{\dim(V^p_{1,1})=\dim(V^d_{1,1})=\frac{q^{-n-1}\left(q^{2 n+4}-1\right)}{q^2-1}}\displaybreak[1]\\ \dim(V^p_{2,1})&=\dim(V^d_{2,1})=\frac{q^{-n-2} \left(q^{2 n+6}-1\right)}{q^2-1} & \mathllap{\dim(V^p_{3,1})=\dim(V^d_{3,1})= \frac{q^{-n-3} \left(q^{2n+8}-1\right)}{q^2-1}}\displaybreak[1]\\\ \dim(V^p_{4,1})&= \frac{q^{-n-2} \left(q^{2n}(2q^{12}+2 q^{10}+q^{8})-q^4-2 q^2-2\right)}{\left(q^2-1\right) \left(q^2+1\right)^3} & \mathllap{\dim(V^p_{4,2})=\frac{q^{-n-4}\left(q^4+q^2+1\right) \left(q^{2 n+12}-1\right)}{\left(q^2-1\right) \left(q^2+1\right)^3}}\displaybreak[1]\\ \dim(V^d_{4,1})&=\frac{q^{-n-4} \left(q^{2n+12}-1\right)}{q^4-1} & \mathllap{\dim(V^d_{4,2})=\frac{q^{-n-2} \left(q^{2 n+8}-1\right)}{q^4-1}}\displaybreak[1]\\\ \dim(V^p_{5,1})&=\dim(V^d_{5,1})= \frac{q^{-n-3} \left(q^{2 n+12}-1\right)}{\left(q^2-1\right) \left(q^2+1\right)^2} \displaybreak[1]\\\ \dim(V^p_{5,2})&=\dim(V^d_{5,2})=\frac{q^{-n-5} \left(q^{2n}(q^{16}-q^{12}-q^{10})+q^6+q^4-1\right)}{\left(q^2-1\right) \left(q^2+1\right)^2}%\displaybreak[1]\\\ %\dim(V^p_{6,1})&= \dim(V^p_{6,2})=\frac{q^{-n-4} \left(q^{2n}(q^{16}-q^{12}-q^{10})+q^6+q^4-1\right)}{\left(q^2-1\right) \left(q^2+1\right)^3}\displaybreak[1]\\\ %\dim(V^p_{6,3})&=\frac{q^{-n-6} \left(q^4+q^2+1\right) \left(q^{2n}(q^{16}-q^{14}-q^{12}-q^{10})+q^6+q^4+q^2-1\right)}{\left(q^2-1\right)\left(q^2+1\right)^3}\displaybreak[1]\\\ %\dim(V^d_{6,1})&=\frac{q^{-n-6} \left(-q^{2 n+10}-q^{2 n+12}-q^{2 n+14}+q^{2n+16}+q^6+q^4+q^2-1\right)}{q^4-1}\displaybreak[1]\\\ %\dim(V^p_{7,1})&=\dim(V^d_{7,1})= \frac{q^{-n-5} \left(q^{2n}(q^{16}-q^{14}-q^{12}-q^{10})+q^6+q^4+q^2-1\right)}{\left(q^2-1\right)\left(q^2+1\right)^2}\displaybreak[1]\\\ %\dim(V^p_{7,2})&=\dim(V^d_{7,2})=\frac{q^{-n-7} \left(q^{2n}(q^{20}-q^{18}-q^{16}-2 q^{14}-q^{12}-q^{10})+q^{10}+q^8+2 q^6+q^4+q^2-1\right)}{\left(q^2-1\right)\left(q^2+1\right)^2}.\displaybreak[1]\\\ \end{align*}}} Thus, the branch factor for this principal graph as a function of $n$ and $q$ is $$ r(n,q)=\frac{\left(q^4+q^2+1\right) \left(q^{2 n+12}-1\right)}{q^2 \left(q^{2n}(2q^{12}+2 q^{10}+q^{8})-q^4-2 q^2-2\right)}. $$ \end{ex} \begin{ex} We can solve for the dimensions just past the branch points for the graphs $$\FSM ,$$ but we need to go all the way out to depth 7 to do so. Once we do this, we find {\scriptsize{ \begin{align*} \dim(V^p_{4,1}) = & \frac{q^{-4-n} \left(-1-q^2 \left(1+q^2\right) \left(2+q^2\right) \left(1+q^4\right)+q^{2 (5+n)} \left(1+3 q^2+3 q^4+3 q^6+2 q^8+q^{10}\right)\right)}{\left(1+q^2\right)^3 \left(-2+3 q^2-3 q^4+2 q^6\right)} \\ \dim(V^p_{4,2}) = & \frac{q^{-4-n} \left(-1-q^2 \left(3+2 q^2+2 q^4+2 q^6+q^8\right)+q^{2 (5+n)} \left(1+q^2 \left(1+q^2\right) \left(2+2 q^4+q^6\right)\right)\right)}{\left(1+q^2\right)^3 \left(-2+3 q^2-3 q^4+2 q^6\right)}. \end{align*}}} Thus, the branch factor for this principal graph as a function of $n$ and $q$ is $$ r(n,q)= \frac{q^{2n}\left(q^{20}+3 q^{18}+2 q^{16}+2 q^{14}+2 q^{12}+q^{10}\right)-q^{10}-2 q^8-2 q^6-2 q^4-3 q^2-1}{q^{2n}\left(q^{20}+2 q^{18}+3 q^{16}+3 q^{14}+3 q^{12}+q^{10}\right)-q^{10}-3 q^8-3 q^6-3 q^4-2 q^2-1}. $$ \end{ex} \begin{ex}\label{ex:badseeddims} For the graphs $$\cB =\left(\bigraph{bwd1v1v1v1p1v1x0p0x1v1x0p1x0p0x1p0x1v0x1x0x0p0x0x1x0v1x0p0x1v1x0p0x1duals1v1v1x2v1x3x2x4v1x2}, \bigraph{bwd1v1v1v1p1v1x0p1x0v1x0p0x1v0x1p1x0v1x0p0x1v0x1p1x0duals1v1v1x2v1x2v2x1}\right) $$ the branch factor $r(n,q)$ is equal to one. This is because in the principal graph, at depth six, we have a duality between two identical vertices on two identical branches, which implies that the dimensions are the same on both branches. Therefore, Inequalites \ref{eqn:InitialTripleSingle} and \ref{eq:QTinequality} (and indeed Equation \ref{eq:QTequation} with $\lambda = -1$) always holds for translations and extensions of these graphs, and none of these triple point obstructions can eliminate this weed. This weed is eliminated in Appendix \ref{sec:EliminateB}. \end{ex} \subsection{Eliminating the Asaeda-Haagerup vine}\label{sec:AH} We give a proof below of an unpublished result of Haagerup stated in \cite{MR1317352}. \begin{thm} There is no biunitary connection on the $4$-partite graph coming from any positive translate of the Asaeda-Haagerup principal graph pair $$\left(\bigraph{bwd1v1v1v1v1v1p1v1x0p0x1v1x0p0x1p0x1v1x0x0v1duals1v1v1v1x2v2x1x3v1}, \bigraph{bwd1v1v1v1v1v1p1v0x1p0x1v0x1v1duals1v1v1v1x2v1}\right).$$ \end{thm} \begin{proof} Suppose we translate the graphs by $j\geq 0$ so that the branch point is at depth $n=5+j$. Note the hypotheses of Corollary \ref{cor:TripleSingle} are verified at the branch point. Labeling the vertices/bimodules as in Example \ref{ex:AnnularMultiplicities*10}, we have $\dim(V_{5,1}^p)=[n]$, so $$\dim(V_{6+k,1}^p)=[k+1]\dim(V_{6,1}^p)-[k][n] \text{ for }1\leq k\leq 4.$$ As $\dim(V_{10,1}^p)=\dim(V_{9,1}^p)/[2]$, we have $\dim(V_{6,1}^p)=[5][n]/[6]$. By similar reasoning, we get the first and third equality below: $$ \frac{[3]\dim(V_{6,2}^p)-[2][n]}{2}=\dim(V_{8,2}^p)=\dim(V_{8,1}^p)=\frac{[3][5]}{[6]}[n]-[2][n], $$ and the second equality comes from duality. This means $$ \dim(V_{6,2}^p)=\left(2\frac{[5]}{[6]}-\frac{[2]}{[3]}\right)[n]=\frac{[5]+1}{[6]}[n]. $$ Now by Corollary \ref{cor:TripleSingle}, a biunitary connection can only exist if $$ |\dim(V_{6,2}^p)-\dim(V_{6,1}^p)|=\frac{[n]}{[6]}\leq 1, $$ which implies the result. \end{proof} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Eliminating $\cC$} \label{sec:odometer} \begin{prop}\label{prop:Crab1} Any subfactor with principal graphs a translated extension of the pair $$\cC=\crab$$ must have index at most $3+\sqrt{3}$. \end{prop} \begin{proof} Suppose a subfactor exists with principal graphs an extension of the pair translated by $n\in2\mathbb{Z}_{\geq 0}$, and let $(q+q^{-1})^2$ be the index. Plugging the branch factor $$ r(n,q)=\frac{\left(q^4+q^2+1\right) \left(q^{2 n+12}-1\right)}{q^2 \left(q^{2n}(2q^{12}+2 q^{10}+q^{8})-q^4-2 q^2-2\right)} $$ calculated in Example \ref{ex:Crab} into Inequality \eqref{eq:QTinequality} (with $m=n+4$), we get the following inequality: \begin{multline*} q^{-2 n-10} \left(q^{n+5}-1\right)^2 \left(q^{n+5}+1\right)^2 \times \\ \left(q^{n+10}-q^{n+8}-q^{n+6}-q^{n+4}-q^6-q^4-q^2+1\right) \times\qquad\qquad\qquad\\ \left(q^{n+10}-q^{n+8}-q^{n+6}-q^{n+4}+q^6+q^4+q^2-1\right) \times \\ \qquad\qquad\qquad(q-1)^{-2} (q+1)^{-2} \left(q^2-q+1\right)^{-1} \left(q^2+q+1\right)^{-1}\times \\ \left(q^{2 n+8}+2 q^{2 n+10}+2 q^{2 n+12}-q^4-2 q^2-2\right)^{-1} \leq 0. \end{multline*} All but the two longest factors in the numerator above (namely the factors on the second and third lines) are positive for all $q>1$. By Remark \ref{dimension}, after computing the graph norm, we see that any translated extension of the pair must satisfy $q>1.4533$, so $q^{10}-q^8 -q^6-q^4>0,$ and $$ q^n\left(q^{10}-q^{8}-q^{6}-q^{4}\right)+q^6+q^4+q^2-1\geq 0. $$ We conclude that Inequality \eqref{eq:QTinequality} is satisfied if and only if \begin{equation} q^n\left(q^{10}-q^{8}-q^{6}-q^{4}\right)-q^6-q^4-q^2+1 \leq 0.\label{eq:CrabInequality1} \end{equation} Note that the left hand side only increases as $n$ increases, so we examine the case $n=0$. The largest root of $$ q^{10}-q^{8}-2q^{6}-2q^{4}-q^2+1 $$ is the positive $q$ such that $(q+q^{-1})^2=3+\sqrt{3}$. Hence the index must be less than or equal to $3+\sqrt{3}$. \end{proof} \begin{rem} At this point, we could appeal to Haagerup's classification to index $3+\sqrt{3}$ to completely rule out all of these graphs. Since the published proof of his classification only covered the range up to index $3+\sqrt{2}$, for the sake of completeness we eliminate these graphs in \S \ref{sec:odometer}. \end{rem} \begin{prop} \label{prop:Crab2} Any subfactor with principal graphs a translated extension of the pair $$\crab$$ with index less than $3+\sqrt{3}$ is in fact a translate of one of the following graphs \begin{enumerate} \item \label{bwd1v1v1v1p1v1x0p0x1v1x0p0x1p0x1v1x0x0p0x0x1v1x0p1x0p0x1duals1v1v1x2v2x1x3v3x2x1} $ \left(\bigraph{bwd1v1v1v1p1v1x0p0x1v1x0p0x1p0x1v1x0x0p0x0x1v1x0p1x0p0x1duals1v1v1x2v2x1x3v3x2x1}, \bigraph{bwd1v1v1v1p1v1x0p1x0v0x1v1p1duals1v1v1x2v1}\right)$ \item \label{bwd1v1v1v1p1v1x0p1x0v0x1v1p1v0x1duals1v1v1x2v1v1} $\left(\bigraph{bwd1v1v1v1p1v1x0p0x1v1x0p0x1p0x1v1x0x0p0x0x1v1x1duals1v1v1x2v2x1x3v1}, \bigraph{bwd1v1v1v1p1v1x0p1x0v0x1v1p1v0x1duals1v1v1x2v1v1}\right)$ \item \label{bwd1v1v1v1p1v1x0p0x1v1x0p0x1p0x1v1x0x0p0x0x1v1x0p1x0p0x1p0x1v1x0x0x0v1duals1v1v1x2v2x1x3v4x2x3x1v1} $\left(\bigraph{bwd1v1v1v1p1v1x0p0x1v1x0p0x1p0x1v1x0x0p0x0x1v1x0p1x0p0x1p0x1v1x0x0x0v1duals1v1v1x2v2x1x3v4x2x3x1v1}, \bigraph{bwd1v1v1v1p1v1x0p1x0v0x1v1p1v0x1v1duals1v1v1x2v1v1}\right)$ \end{enumerate} \end{prop} \begin{proof} We run the odometer, as in \cite{1007.1730}, and find that it terminates after two steps. (Since the index bound is low, this computation can be easily verified by hand without using a computer. In particular, the index bound forces the dual graphs to have a particularly simple form.) The four weeds considered are shown in Figure \ref{fig:odometer}. Only the weed labelled \ref{bwd1v1v1v1p1v1x0p1x0v0x1v1p1v0x1duals1v1v1x2v1v1} satisfies the associativity test, giving case \ref{bwd1v1v1v1p1v1x0p1x0v0x1v1p1v0x1duals1v1v1x2v1v1} above. We next consider all the graphs obtained by extending one graph of a weed, staying below index $3+\sqrt{3}$ and satisfying the associativity test. The weeds at depth $+0$ and depth $+2$ each produce exactly one such graph, giving cases \ref{bwd1v1v1v1p1v1x0p0x1v1x0p0x1p0x1v1x0x0p0x0x1v1x0p1x0p0x1duals1v1v1x2v2x1x3v3x2x1} and \ref{bwd1v1v1v1p1v1x0p0x1v1x0p0x1p0x1v1x0x0p0x0x1v1x0p1x0p0x1p0x1v1x0x0x0v1duals1v1v1x2v2x1x3v4x2x3x1v1} above. \begin{figure}[!ht] \scalebox{0.8}{ \begin{tikzpicture} \tikzset{grow=right,level distance=170pt} \tikzset{every tree node/.style={draw,fill=white,rectangle,rounded corners,inner sep=2pt}} \Tree [.\node{$\!\!\begin{array}{c}\bigraph{bwd1v1v1v1p1v1x0p0x1v1x0p0x1p0x1v1x0x0p0x0x1duals1v1v1x2v2x1x3}\\\bigraph{bwd1v1v1v1p1v1x0p1x0v0x1v1p1duals1v1v1x2v1}\end{array}\!\!$\ref{bwd1v1v1v1p1v1x0p0x1v1x0p0x1p0x1v1x0x0p0x0x1v1x0p1x0p0x1duals1v1v1x2v2x1x3v3x2x1}}; [.\node{$\!\!\begin{array}{c}\bigraph{bwd1v1v1v1p1v1x0p1x0v0x1v1p1v0x1duals1v1v1x2v1v1}\\\bigraph{bwd1v1v1v1p1v1x0p0x1v1x0p0x1p0x1v1x0x0p0x0x1v1x1duals1v1v1x2v2x1x3v1}\end{array}\!\!$\ref{bwd1v1v1v1p1v1x0p1x0v0x1v1p1v0x1duals1v1v1x2v1v1}};] [.\node{$\!\!\begin{array}{c}\bigraph{bwd1v1v1v1p1v1x0p1x0v0x1v1p1v0x1duals1v1v1x2v1v1}\\\bigraph{bwd1v1v1v1p1v1x0p0x1v1x0p0x1p0x1v1x0x0p0x0x1v1x0p1x0p0x1p0x1duals1v1v1x2v2x1x3v4x2x3x1}\end{array}\!\!$}; [.\node{$\!\!\begin{array}{c}\bigraph{bwd1v1v1v1p1v1x0p1x0v0x1v1p1v0x1v1duals1v1v1x2v1v1}\\\bigraph{bwd1v1v1v1p1v1x0p0x1v1x0p0x1p0x1v1x0x0p0x0x1v1x0p1x0p0x1p0x1v1x0x0x0duals1v1v1x2v2x1x3v4x2x3x1}\end{array}\!\!$\ref{bwd1v1v1v1p1v1x0p0x1v1x0p0x1p0x1v1x0x0p0x0x1v1x0p1x0p0x1p0x1v1x0x0x0v1duals1v1v1x2v2x1x3v4x2x3x1v1}};]]] \end{tikzpicture} } \caption{Running the odometer for Proposition \ref{prop:Crab2}.} \label{fig:odometer} \end{figure} \end{proof} \begin{prop}\label{prop:Crab3} There are no subfactors with principal graphs a translation of the following pairs: \begin{enumerate} \item \label{bwd1v1v1v1p1v1x0p0x1v1x0p0x1p0x1v1x0x0p0x0x1v1x0p1x0p0x1duals1v1v1x2v2x1x3v3x2x1} $ \left(\bigraph{bwd1v1v1v1p1v1x0p0x1v1x0p0x1p0x1v1x0x0p0x0x1v1x0p1x0p0x1duals1v1v1x2v2x1x3v3x2x1}, \bigraph{bwd1v1v1v1p1v1x0p1x0v0x1v1p1duals1v1v1x2v1}\right)$ \item \label{bwd1v1v1v1p1v1x0p1x0v0x1v1p1v0x1duals1v1v1x2v1v1} $\left(\bigraph{bwd1v1v1v1p1v1x0p0x1v1x0p0x1p0x1v1x0x0p0x0x1v1x1duals1v1v1x2v2x1x3v1}, \bigraph{bwd1v1v1v1p1v1x0p1x0v0x1v1p1v0x1duals1v1v1x2v1v1}\right)$ \item \label{bwd1v1v1v1p1v1x0p0x1v1x0p0x1p0x1v1x0x0p0x0x1v1x0p1x0p0x1p0x1v1x0x0x0v1duals1v1v1x2v2x1x3v4x2x3x1v1} $\left(\bigraph{bwd1v1v1v1p1v1x0p0x1v1x0p0x1p0x1v1x0x0p0x0x1v1x0p1x0p0x1p0x1v1x0x0x0v1duals1v1v1x2v2x1x3v4x2x3x1v1}, \bigraph{bwd1v1v1v1p1v1x0p1x0v0x1v1p1v0x1v1duals1v1v1x2v1v1}\right)$ \end{enumerate} \end{prop} \begin{proof} Recall from above that for a subfactor with principal graphs a translation by $n$ of one of the above pairs and index $(q+q^{-1})^2$, we must have that $n,q$ satisfy Inequality \ref{eq:CrabInequality1} (which we recall for the reader's convenience): $$ q^n\left(q^{10}-q^{8}-q^{6}-q^{4}\right)-q^6-q^4-q^2+1 \leq 0. $$ For all three cases, $q>1.4817$ by Remark \ref{dimension}, so once again $$q^{10}-q^8 -q^6-q^4>0,$$ and the left hand side of Inequality \ref{eq:CrabInequality1} only increases as $n$ increases. Setting $n=2$, we have that the largest root of $$ q^{12}-q^{10}-q^{8}-2q^{6}-q^4-q^2+1 $$ is smaller than $1.45<1.4817$, so this expression is always positive. Thus there cannot be subfactors with principal graphs a translation by $n\geq 2$ of any of the above pairs. Finally, to check that these three possibilities cannot occur as principal graphs with translation $n=0$, we note that for each case, the dimension of the lower vertex at depth $4$ is not an algebraic integer. The appropriate information is contained in the table below: \[\begin{array}{c|c} \text{graph} & \text{minimal polynomial of dimension of vertex} \\ \hline 1 & 5 x^3-16 x^2-15 x+1 \\ \hline 2 & 3 x^5-19 x^4+25 x^3+18 x^2-25 x-13 \\ \hline 3 & 2 x^2-6 x-9 \end{array}\] \end{proof} \begin{proof}[Proof of Theorem \ref{thm:Crab}] The result is now an immediate consequence of Propositions \ref{prop:Crab1}, \ref{prop:Crab2}, and \ref{prop:Crab3}. \end{proof} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Eliminating $\cF$} \begin{prop}\label{FSMprop} Any subfactor with principal graphs a translated extension of the pair $$\cF=\FSM$$ must either \item[(1)] have principal graphs translated by $0$ and have rotational eigenvalue $\lambda$ and index $(q+q^{-1})^2$ where $\lambda$ and $q$ are either: \[\begin{array}{c|c|c} q & \text{minimal polynomial for $q$} & \lambda\\ \hline 2.0106... & x^{12}-3 x^{10}-3 x^8-4 x^6-3 x^4-3 x^2+1 & 1\\ \hline 1.8449... &x^{36}+x^{34}-2 x^{32}-17 x^{30}-46 x^{28}-91 x^{26}-144 x^{24}& \pm i\\ &-197 x^{22}-233 x^{20}-246 x^{18}-233 x^{16}-197 x^{14}\\ &-144 x^{12}-91 x^{10}-46 x^8-17 x^6-2 x^4+x^2+1 & \end{array}\] \item[(2)] have principal graphs translated by $2$ and have rotational eigenvalue $\lambda$ and index $(q+q^{-1})^2$ where $\lambda$ and $q$ are either: \[\begin{array}{c|c|c} q & \text{minimal polynomial for $q$} & \lambda\\ \hline 1.6341... & x^{16}-x^{14}-2 x^{12}-5 x^{10}-2 x^8-5 x^6-2 x^4-x^2+1 & 1\\ \hline 1.6069... & x^{52}-x^{48}-4 x^{46}-6 x^{44}-19 x^{42}-38 x^{40}-67 x^{38}& \exp(\pm \pi i/3)\\ &-98 x^{36}-139 x^{34}-178 x^{32}-218 x^{30}-238 x^{28}\\ &-246 x^{26}-238 x^{24}-218 x^{22}-178 x^{20}-139 x^{18}\\ &-98 x^{16}-67 x^{14}-38 x^{12}-19 x^{10}-6 x^8-4 x^6-x^4+1 & \end{array}\] \end{prop} \begin{rem} Note that both cases in (1) and the first case of (2) have index strictly greater than 5. \end{rem} \begin{proof} First note that the $q$ from any translated extension of this pair must be at least $1.5932$ by Remark \ref{dimension}. Proceeding as in Proposition \ref{prop:Crab1}, the branch factor as a function of $n$ and $q$ is given by $$ r(n,q)= \frac{q^{2n}\left(q^{20}+3 q^{18}+2 q^{16}+2 q^{14}+2 q^{12}+q^{10}\right)-q^{10}-2 q^8-2 q^6-2 q^4-3 q^2-1}{q^{2n}\left(q^{20}+2 q^{18}+3 q^{16}+3 q^{14}+3 q^{12}+q^{10}\right)-q^{10}-3 q^8-3 q^6-3 q^4-2 q^2-1}. $$ Plugging in $r(n,q)$ to Equation \eqref{eq:QTinequality}, we get the following inequality: {\scriptsize \begin{multline*} q^{-2 n-4} \left(q^{n+5}-1\right)^2 \left(q^{n+5}+1\right)^2 (q-1)^{-2} (q+1)^{-2} \times \\ \bigg(q^{2n}\left(q^{16}-q^{14}-q^{12}-q^{10}\right)+q^n\left(-2 q^{14}-3 q^{12}+3 q^{4}+2 q^{2}\right)+q^6+q^4+q^2-1\bigg) \times \\ \bigg(q^{2 n}\left(q^{16}-q^{14}-q^{12}-q^{10}\right)+q^n\left(2 q^{14}+3 q^{12}-3 q^{4}-2 q^{2}\right)+ q^6+q^4+q^2-1\bigg)\times \\ \qquad\quad\bigg(q^{2 n}\left(q^{20}+2 q^{18}+3 q^{16}+3 q^{14}+3 q^{12}+q^{10}\right)-q^{10}-3 q^8-3 q^6-3 q^4-2 q^2-1\bigg)^{-1} \times\\ \bigg(q^{2 n}\left(q^{20}+3 q^{18}+2 q^{16}+2 q^{14}+2 q^{12}+q^{10}\right)-q^{10}-2 q^8-2 q^6-2 q^4-3 q^2-1\bigg)^{-1}\leq 0. \end{multline*} } By similar analysis as above, this inequality is satisfied if and only if \begin{equation*} q^{2n}\left(q^{16}-q^{14}-q^{12}-q^{10}\right)+q^n\left(-2 q^{14}-3 q^{12}+3 q^{4}+2 q^{2}\right)+q^6+q^4+q^2-1\leq 0. \end{equation*} Let $p(n,q)$ denote the left hand side. If $n\geq 4$ and $q>1$, then \begin{align*} p(n,q)&\geq q^{2n}\left(q^{16}-q^{14}-q^{12}-q^{10}\right)+q^n\left(-2 q^{14}-3 q^{12}\right)\\ &\geq q^{2n}\left(-2q^{10}-3q^{8}+q^{16}-q^{14}-q^{12}-q^{10}\right)\\ &=q^{2n+8}\left(q^8-q^6-q^4-3q^2-3\right). \end{align*} The largest root of $$ q^8-q^6-q^4-3q^2-3 $$ is less than $1.5082<1.5932$, so there can be no subfactors with an $n$-translated extension of this pair of principal graphs for $n\geq 4$. Now suppose we have a subfactor with principal graphs an extension of this pair of principal graphs. Then $\lambda\in\{\pm 1,\pm i\}$ and $\lambda+\lambda^{-1}\in \{-2,0,2\}$. Solving Equation \eqref{eq:QTequation} for $q$ when $\lambda=-1$ shows that $q$ must be approximately $1.3123...$, with minimal polynomial $x^8-x^6-x^4-x^2+1$. This $q$ is smaller than $1.5932$ so we can ignore this case. Solving Equation \eqref{eq:QTequation} for $q$ when $\lambda\in\{1,\pm i\}$ gives the first table in the statement. Finally, suppose we have a subfactor with principal graphs a $2$-translated extension of this pair of principal graphs. Then $\lambda\in \{\pm 1,\exp(\pm2\pi i/3),\exp(\pm \pi i/3)\}$ and $\lambda+\lambda^{-1}\in \{-2,-1,1,2\}$. Solving Equation \eqref{eq:QTequation} for $q$ when $\lambda\in \{-1,\exp(\pm2\pi i/3)\}$ gives the cases \[\begin{array}{c|c|c} q & \text{minimal polynomial for $q$} & \lambda\\ \hline 1.3453...& x^{16}-x^{14}-2 x^{10}-2 x^6-x^2+1 & -1\\ \hline 1.5203... & x^{52}-x^{48}-4 x^{46}-4 x^{44}-9 x^{42}-14 x^{40}-21 x^{38} & \exp(\pm2\pi i/3)\\ &-24 x^{36}-29 x^{34}-36 x^{32}-42 x^{30}-44 x^{28}-42 x^{26}\\ &-44 x^{24}-42 x^{22}-36 x^{20}-29 x^{18}-24 x^{16}-21 x^{14}\\ &-14 x^{12}-9 x^{10}-4 x^8-4 x^6-x^4+1 \end{array}\] which we ignore as $q$ is too small. Solving Equation \eqref{eq:QTequation} for $q$ when $\lambda\in \{1,\exp(\pm \pi i/3)\}$ gives the second table above. \end{proof} \begin{prop}\label{prop:ConnectionExistsFSM} In order for a connection to exist on any extension of an $(n-3)$-translate of $\cF$, it must have eigenvalue $d=q+q^{-1}$ where $q$ is the unique root greater than $1$ of \begin{equation}\label{eqn:ConsistencyConstraintFSM} ???????????? \end{equation} \end{prop} \begin{proof} The Ocneanu 4-partite graph between depth $n$ and $n+4$ is given by $$ \begin{tikzpicture}[baseline] \node at (0,2) {$\sb{A}{\sf{Mod}}_A$}; \filldraw (2,2) circle (1mm); \filldraw (3,2) circle (1mm); \filldraw (6,2) circle (1mm); \filldraw (7,2) circle (1mm); % \filldraw (8,2) circle (1mm); \filldraw (9,2) circle (1mm); \node at (0,1) {$\sb{A}{\sf{Mod}}_B$}; \filldraw (1,1) circle (1mm); \draw (1,1)--(2,2); \draw (1,1)--(2,0); \draw (1,1)--(3,2); \draw (1,1)--(3,0); % \filldraw (4,1) circle (1mm); \draw (4,1)--(2,2); % \draw (4,1)--(2,0); \draw (4,1)--(6,2); % \draw (4,1)--(8,2); \draw (4,1)--(6,0); \filldraw (5,1) circle (1mm); \draw (5,1)--(3,2); \draw (5,1)--(2,0); \draw (5,1)--(7,2); \draw (5,1)--(9,2); \draw (5,1)--(7,0); \filldraw (10,1) circle (1mm); \draw (10,1)--(8,2); \draw (10,1)--(7,0); \filldraw (11,1) circle (1mm); \draw (11,1)--(7,2); \draw (11,1)--(6,0); \filldraw (12,1) circle (1mm); \draw (12,1)--(9,2); \draw (12,1)--(7,0); \node at (0,0) {$\sb{B}{\sf{Mod}}_B$}; % \filldraw (2,0) circle (1mm); \filldraw (3,0) circle (1mm); \filldraw (6,0) circle (1mm); \filldraw (7,0) circle (1mm); \node at (0,-1) {$\sb{B}{\sf{Mod}}_A$}; \filldraw (1,-1) circle (1mm); \draw (1,-1)--(2,-2); \draw (1,-1)--(2,0); \draw (1,-1)--(3,-2); \draw (1,-1)--(3,0); \filldraw (4,-1) circle (1mm); \draw (4,-1)--(2,-2); \draw (4,-1)--(2,0); \draw (4,-1)--(6,-2); \draw (4,-1)--(7,-2); \draw (4,-1)--(6,0); % \filldraw (5,-1) circle (1mm); \draw (5,-1)--(3,-2); % \draw (5,-1)--(2,0); % \draw (5,-1)--(8,-2); \draw (5,-1)--(9,-2); \draw (5,-1)--(7,0); \filldraw (10,-1) circle (1mm); \draw (10,-1)--(7,-2); \draw (10,-1)--(7,0); \filldraw (11,-1) circle (1mm); \draw (11,-1)--(8,-2); \draw (11,-1)--(6,0); \filldraw (12,-1) circle (1mm); \draw (12,-1)--(9,-2); \draw (12,-1)--(7,0); \node at (0,-2) {$\sb{A}{\sf{Mod}}_A$}; \filldraw (2,-2) circle (1mm); \filldraw (3,-2) circle (1mm); \filldraw (6,-2) circle (1mm); \filldraw (7,-2) circle (1mm); % \filldraw (8,-2) circle (1mm); \filldraw (9,-2) circle (1mm); %blue \filldraw[blue] (8,2) circle (1mm); \filldraw[blue] (4,1) circle (1mm); \draw[blue] (4,1)--(2,0); \draw[blue] (4,1)--(8,2); \filldraw[blue] (2,0) circle (1mm); \filldraw[blue] (5,-1) circle (1mm); \draw[blue] (5,-1)--(2,0); \draw[blue] (5,-1)--(8,-2); \filldraw[blue] (8,-2) circle (1mm); \end{tikzpicture}$$ The loop $(V^p_{n+3,2},V^p_{n+2,1},V^d_{n+1,1},V^d_{n+2,2})$, in blue, appears in two different $1$-by-$1$ unitary matrices, so we can calculate the norm of the corresponding component of the connection in two different ways. In particular, we must have $$ \dim(V^p_{n+3,2})\dim(V^d_{n+1,1})=\dim(V^p_{n+2,1})\dim(V^d_{n+2,2}), $$ which, after expanding products of quantum numbers and canceling, gives ??????? \todo{finish} \end{proof} \begin{proof}[Proof of Theorem \ref{thm:FSM}] We will show the results of Propositions \ref{FSMprop} and \ref{prop:ConnectionExistsFSM} are mutually exclusive. \todo{finish} \end{proof} \subsection{Obsolete $\cF$} \begin{prop}\label{prop:FSM-odometer} There are no subfactors with index at most $4.96933$ with principal graphs an extension of $$\FSMF.$$ \end{prop} \begin{proof} We run the odometer on the $2$-translate of $\cF$, exactly as in the proof of Proposition \ref{prop:Crab2} above. This time, it is even simpler: there's a single further weed shown in Figure \ref{fig:FSM-odometer}, and no vines. \begin{figure}[!ht] \scalebox{0.92}{ \begin{tikzpicture} \tikzset{grow=right,level distance=220pt} \tikzset{every tree node/.style={draw,fill=white,rectangle,rounded corners,inner sep=2pt}} \Tree [.\node{$\!\!\begin{array}{c}\bigraph{bwd1v1v1v1v1v1p1v1x0p0x1v1x0p1x0p0x1p0x1v0x1x0x0p0x0x1x0p0x0x0x1v1x0x0p1x0x0p0x1x0p0x0x1v1x0x0x0p0x0x1x0p0x0x0x1duals1v1v1v1x2v1x3x2x4v1x4x3x2}\\\bigraph{bwd1v1v1v1v1v1p1v1x0p1x0v1x0p0x1v0x1p1x0p0x1v1x0x0p0x1x0v0x1p1x0p0x1duals1v1v1v1x2v1x2v2x1}\end{array}\!\!$}; [.\node{$\!\!\begin{array}{c}\bigraph{bwd1v1v1v1v1v1p1v1x0p1x0v1x0p0x1v0x1p1x0p0x1v1x0x0p0x1x0v0x1p1x0p0x1v1x0x0p0x1x0duals1v1v1v1x2v1x2v2x1v1x2}\\\bigraph{bwd1v1v1v1v1v1p1v1x0p0x1v1x0p1x0p0x1p0x1v0x1x0x0p0x0x1x0p0x0x0x1v1x0x0p1x0x0p0x1x0p0x0x1v1x0x0x0p0x0x1x0p0x0x0x1v1x0x0p0x1x0p0x1x0p0x0x1duals1v1v1v1x2v1x3x2x4v1x4x3x2v3x4x1x2}\end{array}\!\!$};]] \end{tikzpicture} } \caption{Running the odometer for Proposition \ref{prop:FSM-odometer}.} \label{fig:FSM-odometer} \end{figure} \end{proof} \begin{proof}[Proof of Theorem \ref{thm:FSM}] \todo{To rule out the amenable cases, we need to exhibit an object with dimension not an algebraic integer} For each the four allowed values of $q$ in Proposition \ref{FSMprop}, the index of the possible subfactor, $(q+q^{-1})^2$, is not cyclotomic. By \cite{MR2183279} this excludes the possibility of a finite depth subfactor. We can explicitly check this by calculating the discriminant of each index, then finding a prime $p$ which does not divide the discriminant, such that the minimal polynomial of the index does not have uniform degree irreducible factors mod $p$. We exhibit the appropriate data in the following table. \begin{center} \begin{tabular}{c|c|c|c} $n$ & $\lambda$ & $p$ & degrees of factors mod $p$ \\ \hline $0$ & $1$ & 5& 1, 2, 3\\ \hline $0$ & $\pm i$ &7 &1, 1, 2, 2, 4, 8 \\ \hline $2$ & 1 &3 &2, 2, 4 \\ \hline $2$ & $\exp(\pm \pi i/3)$ & 5&1, 2, 3, 3, 3, 6, 8 \\ \end{tabular} \end{center} \end{proof} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Eliminating $\cB$}\label{sec:EliminateB} Principal graphs coming from the weed $\cB$ are eliminated by showing they cannot have a connection, in the following way: we attempt to construct a connection, and obtain an equation on $q$. This equation is satisfied by the graph in which both arms at the end of $\cB$ are extended infinitely and simply (ie, without further branching). Some graph theoretic arguments, which we punt to an appendix, then show that no other potential principal graph extending $\cB$ has the needed graph norm. The infinite simple extension of $\cB$ cannot be a principal graph itself, because of a result of Popa's \cite{} prohibiting infinite legs. \begin{defn} Let $$(\Gamma_3, \Delta_3) = \BadSeed $$ %\left(\scalebox{1.5}{$\bigraph{bwd1v1v1v1p1v1x0p0x1v1x0p1x0p0x1p0x1v0x0x1x0p0x1x0x0duals1v1v1x2v1x3x2x4}$}, \scalebox{1.5}{$\bigraph{bwd1v1v1v1p1v1x0p1x0v1x0p0x1v1x0p0x1duals1v1v1x2v1x2}$} \right).$$ For $n> 3$, let $(\Gamma_n,\Delta_n)$ be the translation of $(\Gamma_3,\Delta_3)$ by $n-3$, so the branch point occurs at depth $n$. For $n\geq 3$, let $(\Gamma_{n,\infty},\Delta_{n,\infty})$ be the extension of $(\Gamma_n,\Delta_n)$ given by extending both arms infinitely and simply (we will not specify the dual data of the extension). \end{defn} (connective tissue) \begin{prop}\label{prop:ConnectionExists} In order for a connection to exist on any extension of an $(n-3)$-translate of $(\Gamma,\Delta)$, it must have eigenvalue $d=q+q^{-1}$ where $q$ is the unique root greater than $1$ of \begin{equation}\label{eqn:ConsistencyConstraint} q^{2 n+8}-2 q^{2 n+6}-q^{2 n+4}+q^4+1=0 \end{equation} \end{prop} \begin{proof} The proof is similar to Proposition \ref{prop:ConnectionExistsFSM}. The loop \todo{check!} $(V^p_{n+3,2},V^p_{n+2,1},V^d_{n+1,1},V^d_{n+2,2})$ %$(0062),(0151),(1141),(1052)$ appears in two different $1$-by-$1$ matrices, so we can calculate the norm of the corresponding component of the connection in two different ways. In particular, we must have $$ \dim(V^p_{n+3,2})\dim(V^d_{n+1,1})=\dim(V^p_{n+2,1})\dim(V^d_{n+2,2}), $$ %$$\dim(0062) \dim(1141) = \dim(0151) \dim(1052),$$ which, after expanding products of quantum numbers and canceling, gives $$ [2n+7] - [2n+5] - [2n+3] + [2n+1] -4 [n+1]^2=0. $$ Factoring the left hand side, we get $$ \frac{q^{-2 n-6} \left(q^{2 n+8}+q^{2 n+4}-q^4-2 q^2+1\right) \left(q^{2 n+8}-2q^{2 n+6}- q^{2 n+4}+q^4+1\right)}{(q-1)^2 (q+1)^2}=0. $$ Note that $q^{2 n+8}+q^{2 n+4}-q^4-2 q^2+1>0$ for all $q>1$, so the above equality is satisfied if and only if Equation \eqref{eqn:ConsistencyConstraint} holds. \end{proof} (connective tissue) \begin{cor}\label{cor:Bdies} No graph grown from $\cB$ is a principal graph. \end{cor} \begin{proof} Any graph grown from $\cB$ either \begin{enumerate} \item\label{item:Btwolegs} equals $\includegraphics[height=8mm]{Btwolegs},$ \item\label{item:Binfinite} equal $\Gamma_{n,\infty}=\includegraphics[height=8mm]{Binfinite},$ \item\label{item:Bmerger} contains $\includegraphics[height=8mm]{Bmerger},$ or \item\label{item:Bbadtriple} contains $ \includegraphics[height=8mm]{Bbadtriple}.$ \end{enumerate} The associativity lemma \ref{lem:assoc} tells us that (\ref{item:Btwolegs}) cannot be a principal graph, and any graph containing (\ref{item:Bbadtriple}) must actually contain \begin{itemize} \item[(4')] $\includegraphics[height=8mm]{Btriple}.$ \end{itemize} Recall that in order to be a principal graph, a graph grown from $\cB$ must have graph norm satisfying \ref{prop:ConnectionExists}. The graph (\ref{item:Binfinite}) has this norm (see Lemma \ref{lem:Binfinite}), and therefore graphs containing (\ref{item:Bmerger}) and (4') have norms which are too large -- this graph-theoretic fact follows from Lemmas \ref{lem:NormCycles} and \ref{lem:ExtraTriple} in the appendix. Finally, (\ref{item:Binfinite}) cannot be a principal graph by Theorem 4.5 of \cite{MR1334479}, or from Theorem 6.5 of \cite{0902.1294} and \cite{gpa}. \end{proof} Corollary \ref{cor:Bdies} establishes Theorem \ref{thm:BadSeed}. The proof of the corollary relied on two lemmas, one about the allowed extensions of graphs and the other about the norm of $\Gamma_{n,\infty}$. \begin{lem}[Associativity Lemma]\label{lem:assoc} Suppose the principal graphs $(\Gamma,\Delta)$ of a subfactor are an extension of \begin{align*} (\Gamma_{\text{odd}},\Delta_{\text{odd}})&=\left(\, \begin{tikzpicture}[baseline=.8cm, scale=.6] \filldraw[black] (1,.5) rectangle (3,2.5); \node at (2,1.5) { \textcolor{white}{$\bf \Gamma_{\text{odd}}'$} }; \draw (3,1)--(9,1); \filldraw (5,1) circle (1mm); \filldraw (7,1) circle (1mm); \filldraw (9,1) circle (1mm); \draw (3,2)--(9,2); \filldraw (5,2) circle (1mm); \filldraw (7,2) circle (1mm); \filldraw (9,2) circle (1mm); \draw[thick,red] (7,1)--(7,1.2); \draw[thick,red] (7,2)--(7,2.2); \end{tikzpicture} \,,\, \begin{tikzpicture}[baseline=.8cm, scale=.6] \filldraw[black] (1,.5) rectangle (3,2.5); \node at (2,1.5) { \textcolor{white}{$\bf \Delta_{\text{odd}}'$} }; \draw (3,1)--(7,1)--(9,2); \filldraw (5,1) circle (1mm); \filldraw (7,1) circle (1mm); \filldraw (9,1) circle (1mm); \draw (3,2)--(7,2)--(9,1); \filldraw (5,2) circle (1mm); \filldraw (7,2) circle (1mm); \filldraw (9,2) circle (1mm); \draw[thick,red] (7,1)--(7.3,1.5)--(7,2); \end{tikzpicture} \right)\text{ or}\\ (\Gamma_{\text{even}},\Delta_{\text{even}})&=\left(\, \begin{tikzpicture}[baseline=.8cm, scale=.6] \filldraw[black] (1,.5) rectangle (3,2.5); \node at (2,1.5) { \textcolor{white}{$\bf \Gamma_{\text{even}}'$} }; \draw (3,1)--(9,1); \filldraw (5,1) circle (1mm); \filldraw (7,1) circle (1mm); \filldraw (9,1) circle (1mm); \draw (3,2)--(9,2); \filldraw (5,2) circle (1mm); \filldraw (7,2) circle (1mm); \filldraw (9,2) circle (1mm); \draw[thick,red] (5,1)--(5,1.2); \draw[thick,red] (5,2)--(5,2.2); \draw[thick,red] (9,1)--(9.3,1.5)--(9,2); \end{tikzpicture} \,,\, \begin{tikzpicture}[baseline=.8cm, scale=.6] \filldraw[black] (1,.5) rectangle (3,2.5); \node at (2,1.5) { \textcolor{white}{$\bf \Delta_{\text{even}}'$} }; \draw (3,1)--(5,1)--(7,2)--(9,2); \filldraw (5,1) circle (1mm); \filldraw (7,1) circle (1mm); \filldraw (9,1) circle (1mm); \draw (3,2)--(5,2)--(7,1)--(9,1); \filldraw (5,2) circle (1mm); \filldraw (7,2) circle (1mm); \filldraw (9,2) circle (1mm); \draw[thick,red] (9,1)--(9,1.2); \draw[thick,red] (9,2)--(9,2.2); \draw[thick,red] (5,1)--(5.3,1.5)--(5,2); \end{tikzpicture} \right) \end{align*} where we only see the vertices at depths $k-2$ through $k$. Then each vertex at depth $k$ of $(\Gamma,\Delta)$ must attach to a vertex at depth $k+1$. \todo{and, if the legs of one extend boringly, the others are boring too. And include dual data if needed.} \end{lem} \begin{proof} We only give the proof for $(\Gamma_{\text{odd}},\Delta_{\text{odd}})$ as the proof for $(\Gamma_{\text{even}},\Delta_{\text{even}})$ is similar. The pair $(\Gamma_{\text{odd}},\Delta_{\text{odd}})$ fails the associativity test. Its Ocneanu 4-partite graph, starting at depth $n+4$, is given by $$ \begin{tikzpicture}[baseline] \node at (-1,2) {$\sb{A}{\sf{Mod}}_A$}; \filldraw (3,2) circle (1mm); \filldraw (4,2) circle (1mm); \node at (-1,1) {$\sb{A}{\sf{Mod}}_B$}; \filldraw (0,1) circle (1mm); \draw (0,1)--(3,2); \draw (0,1)--(4,0); \filldraw (1,1) circle (1mm); \draw (1,1)--(4,2); \draw (1,1)--(3,0); \filldraw (6,1) circle (1mm); \node at (6,1.5) {$\beta_1$}; \draw (6,1)--(3,2); \draw (6,1)--(3,0); \filldraw (7,1) circle (1mm); \node at (7,1.5) {$\beta_2$}; \draw (7,1)--(4,2); \draw (7,1)--(4,0); \node at (-1,0) {$\sb{B}{\sf{Mod}}_B$}; \filldraw (3,0) circle (1mm); \filldraw (4,0) circle (1mm); \node at (-1,-1) {$\sb{B}{\sf{Mod}}_A$}; \filldraw (0,-1) circle (1mm); \draw (0,-1)--(3,-2); \draw (0,-1)--(3,0); \filldraw (1,-1) circle (1mm); \draw (1,-1)--(4,-2); \draw (1,-1)--(4,0); \filldraw (6,-1) circle (1mm); \node at (6,-.5) {$\beta_1^*$}; \draw (6,-1)--(3,-2); \draw (6,-1)--(4,0); \filldraw (7,-1) circle (1mm); \node at (7,-.5) {$\beta_2^*$}; \draw (7,-1)--(4,-2); \draw (7,-1)--(3,0); \node at (-1,-2) {$\sb{A}{\sf{Mod}}_A$}; \filldraw (3,-2) circle (1mm); \filldraw (4,-2) circle (1mm); \end{tikzpicture} $$ Notice there are two problems with associativity: \begin{enumerate} \item[(1)] There is a path from $\beta_1$ to $\beta_2^*$ through a $B-B$ bimodule, but not through an $A-A$ bimodule. The same is true for $\beta_2$ and $\beta_1^*$. \item[(2)] There is a path from $\beta_1$ to $\beta_1^*$ through an $A-A$ bimodule, but not through a $B-B$ bimodule. The same is true for $\beta_2$ and $\beta_2^*$. \end{enumerate} Hence, to obtain an extension of $(\Gamma_{\text{odd}},\Delta_{\text{odd}})$ which fixes these problems, we must attach at least one vertex at depth $k+1$ to each vertex at depth $k$. \end{proof} \begin{rem} This lemma shows that you can't extend one arm of $\cB$ without extending the other. In fact, if we just extend \todo{pictures for dual data} \end{rem} We also needed the following lemma about the norm of $\Gamma_{n,\infty}$: \begin{lem}\label{lem:Binfinite} The graph $\Gamma_{n,\infty}$ has norm $d=q+q^{-1}$ where $q$ is the unique root greater than $1$ of Equation \eqref{eqn:ConsistencyConstraint}. The graph $\Gamma_{n,\infty}$ has a totally positive $\ell^2$-eigenvector $\bf{v}$, where $v_{i,j}$ is the value of the eigenvector at the $j$-th vertex at depth $i$ (the branch point is at depth $n$): %{\scriptsize{ \begin{align*} v_{k,1}&=[k+1] \text{ for all $k\leq n$ } \hspace{.5in}& v_{n+1,1}&=v_{n+1,2}=\frac{[n+2]}{2}\displaybreak[1]\\\ v_{n+2,1}&=v_{n+2,2}=\frac{[2][n+2]}{2q^2} &v_{n+3,1}&=v_{n+3,4}=\frac{[n+2]}{2q^2}\displaybreak[1]\\\ v_{n+3,2}&=v_{n+3,3}=\frac{[2][n+2]}{2q^3} & v_{n+k,1}&=v_{n+k,2}= \frac{[2][n+2]}{2q^{k}} \text{ for all $k\geq 4$.} \end{align*}%}} with eigenvalue $d$ as above. %$\Gamma_{n,\infty}$ has the following Frobenius-Perron $\ell^2$-eigenvector $v$, where $v_{i,j}$ is the value of the eigenvector at the $j$-th vertex at depth $i$ (the branch point is at depth $n$): %%{\scriptsize{ %\begin{align*} %v_{k,1}&=[k+1] \text{ for all $k\leq n$ } \hspace{.5in}& v_{n+1,1}&=v_{n+1,2}=\frac{[n+2]}{2}\displaybreak[1]\\\ %v_{n+2,1}&=v_{n+2,2}=\frac{[2][n+2]}{2q^2} &v_{n+3,1}&=v_{n+3,4}=\frac{[n+2]}{2q^2}\displaybreak[1]\\\ %v_{n+3,2}&=v_{n+3,3}=\frac{[2][n+2]}{2q^3} & v_{n+k,1}&=v_{n+k,2}= \frac{[2][n+2]}{2q^{k}} \text{ for all $k\geq 4$.} %\end{align*}%}} %with eigenvalue $d=q+q^{-1}$ where $q$ is the unique root greater than $1$ of %Equation \eqref{eqn:ConsistencyConstraint}. Note that $d=\|\Gamma_{n,\infty}\|$ by Theorem \ref{thm:InfiniteEigenvector}. % %$\Delta_{n,\infty}$ has the following Frobenius-Perron $\ell^2$-eigenvector $w$, where $w_{i,j}$ is the value of the eigenvector at the $j$-th vertex at depth $i$: %\begin{align*} %w_{k,1}&=[k+1] \text{ for all $k\leq n$ } \hspace{.5in}& w_{n+1,1}&=\frac{[n]}{2}\displaybreak[1]\\\ %w_{n+1,2}&=\frac{[n+2]}{[2]} &v_{n+k,1}&=w_{n+k,2}=\frac{[n+2]}{q[2]}\text{ for all $k\geq 2$} %\end{align*} %with the same eigenvalue as $\Gamma_{n,\infty}$. Note $\|\Delta_{n,\infty}\|=\|\Gamma_{n,\infty}\|$. \end{lem} \begin{proof} By Theorem \ref{thm:InfiniteEigenvector}, $d=\|\Gamma_{n,\infty}\|$. So showing that $\bf{v}$ is in fact an eigenvector for $\Gamma_{n,\infty}$, with eigenvalue $d$, also proves the first part of this claim. \todo{I think the following can be simplified a lot-E} By the Frobenius-Perron eigenvalue equations, we know $v_{k,1}$ for $k\leq n$ and $v_{n+1,j}$ for $j=1,2$. Let $x=v_{n+2,2}=v_{n+2,3}$. Then clearly $v_{n+3,1}=v_{n+3,4}=x/[2]$, $v_{n+3,2}=v_{n+3,3}=xq^{-1}$, and $v_{n+k,1}=v_{n+k,2}=xq^{2-k}$ for all $k\geq 4$ by Lemma \ref{lem:geometric}. Hence it suffices to calculate $x$. We may do so via 2 different eigenvalue equations: \begin{align*} v_{n+2,2}+v_{n,1}=[2]v_{n+1,1} &\Longrightarrow x+[n+1] = [2]\frac{[n+2]}{2}\text{ and}\\ v_{n+1,1}+v_{n+3,1}+v_{n+3,2}=[2]v_{n+2,2} &\Longrightarrow \frac{[n+2]}{2}+\frac{x}{[2]}+q^{-1}x = [2]x. \end{align*} After a little algebra, eliminating $x$ in the above equations yields $$ \frac{q^2[n+1]}{q^2-1}-\frac{[2][n+2]}{2}=\frac{q^{-n-2} \left(q^{2 n+8}-2 q^{2 n+6}-q^{2 n+4}+q^4+1\right)}{2 (q-1)^2 (q+1)^2}=0, $$ which is satisfied if and only if Equation \eqref{eqn:ConsistencyConstraint} holds. %The proof for $\Delta_{n,\infty}$ is similar. \end{proof} %\begin{cor}\label{cor:NoInfiniteArms} %The graphs $(\Gamma_{n,\infty},\Delta_{n,\infty})$ are not the principal graphs of any subfactor. %\end{cor} %\begin{proof} %By (3) on page 184 of \cite{MR1278111}, if an infinite graph has an $\ell^2$ Frobenius-Perron eigenvector, then it is not the principal graph of any subfactor. %\end{proof} % %\begin{cor}\label{cor:NotGraph} %The graphs $(\Gamma_n,\Delta_n)$ are not the principal graphs of any subfactor as $\|\Gamma_n\|=\Delta_n\|$ is too small. % %The graphs \todo{add top leg to first graph, $\Delta$ graphs...} %$$\includegraphics[height=8mm]{Btriple}, \includegraphics[height=8mm]{Bmerger},$$ %are not the principal graphs of any subfactor as their norms are too large. %\end{cor} %\begin{proof} %That $\|\Gamma_n\|=\Delta_n\|$ is too small follows immediately from Proposition \ref{prop:ConnectionExists}, Lemma \ref{lem:Binfinity}, and Theorem \ref{thm:NormsConverge}. % %That the other norms are too large follows from Proposition \ref{prop:ConnectionExists} and Lemmas \ref{lem:Binfinity}, \ref{lem:NormCycles}, and \ref{lem:ExtraTriple}. %\end{proof} % %\begin{proof}[Proof of Theorem \ref{thm:BadSeed}] %Fix $n\geq 3$. Suppose we have a subfactor $A\subset B$ whose principal graphs $(\Gamma,\Delta)$ are an extension of $(\Gamma_n,\Delta_n)$. We know that $\Gamma\neq\Gamma_n$ for two reasons. First, $\|\Gamma_n\|$ is too small by Corollary \ref{cor:NotGraph}. Second, $(\Gamma_n,\Delta_n)$ fails the associativity test. Its Ocneanu 4-partite graph, starting at depth $n+4$, is given by % %$$ %\begin{tikzpicture}[baseline] % \node at (-1,2) {$\sb{A}{\sf{Mod}}_A$}; % \filldraw (3,2) circle (1mm); % \filldraw (4,2) circle (1mm); % \node at (-1,1) {$\sb{A}{\sf{Mod}}_B$}; % \filldraw (0,1) circle (1mm); % \draw (0,1)--(3,2); % \draw (0,1)--(4,0); % \filldraw (1,1) circle (1mm); % \draw (1,1)--(4,2); % \draw (1,1)--(3,0); % \filldraw (6,1) circle (1mm); \node at (6,1.5) {$\beta_1$}; % \draw (6,1)--(3,2); % \draw (6,1)--(3,0); % \filldraw (7,1) circle (1mm); \node at (7,1.5) {$\beta_2$}; % \draw (7,1)--(4,2); % \draw (7,1)--(4,0); % \node at (-1,0) {$\sb{B}{\sf{Mod}}_B$}; % \filldraw (3,0) circle (1mm); % \filldraw (4,0) circle (1mm); % \node at (-1,-1) {$\sb{B}{\sf{Mod}}_A$}; % \filldraw (0,-1) circle (1mm); % \draw (0,-1)--(3,-2); % \draw (0,-1)--(3,0); % \filldraw (1,-1) circle (1mm); % \draw (1,-1)--(4,-2); % \draw (1,-1)--(4,0); % \filldraw (6,-1) circle (1mm); \node at (6,-.5) {$\beta_1^*$}; % \draw (6,-1)--(3,-2); % \draw (6,-1)--(4,0); % \filldraw (7,-1) circle (1mm); \node at (7,-.5) {$\beta_2^*$}; % \draw (7,-1)--(4,-2); % \draw (7,-1)--(3,0); % \node at (-1,-2) {$\sb{A}{\sf{Mod}}_A$}; % \filldraw (3,-2) circle (1mm); % \filldraw (4,-2) circle (1mm); %\end{tikzpicture} %$$ % %Notice there are two problems with associativity: %\begin{enumerate} %\item[(1)] %There is a path from $\beta_1$ to $\beta_2^*$ through a $B-B$ bimodule, but not through an $A-A$ bimodule. The same is true for $\beta_2$ and $\beta_1^*$. %\item[(2)] %There is a path from $\beta_1$ to $\beta_1^*$ through an $A-A$ bimodule, but not through a $B-B$ bimodule. The same is true for $\beta_2$ and $\beta_2^*$. %\end{enumerate} % %Hence, to obtain an extension of $(\Gamma_n,\Delta_n)$ which fixes these problems, we must attach at least one vertex at depth $n+7$ to each vertex of $\Gamma_n$ and $\Delta_n$ at depth $n+6$. There are three cases for each graph. We give the three cases for extending $\Gamma_n$, and the three cases for extending $\Delta_n$ are similar. %\begin{enumerate} %\item[(Case 1)] We could attach one distinct vertex at depth $n+7$ to each vertex at depth $n+6$: \todo{image} %\item[(Case 2)] The extension of $\Gamma_n$ contains the subgraph \todo{add vertex to upper arm, no dots} %$$\includegraphics[height=8mm]{Btriple}.$$ %\item[(Case 3)] The extension of $\Gamma_n$ contains the subgraph \todo{no dots} %$$\includegraphics[height=8mm]{Bmerger}.$$ %\end{enumerate} %Note that by Corollary \ref{cor:NotGraph}, the second and third cases cannot occur for either $\Gamma_n$ or $\Delta_n$ as the norms of the graphs are too big for connections to exist. % %Hence, we must be in the first case, which by path counting considerations, yields the following Ocneanu 4-partite graph: % %$$ %\begin{tikzpicture}[baseline] % \node at (-1,2) {$\sb{A}{\sf{Mod}}_A$}; % \filldraw (3,2) circle (1mm); % \filldraw (4,2) circle (1mm); % \filldraw (9,2) circle (1mm); \node at (9,2.5) {$\alpha_1$}; % \filldraw (10,2) circle (1mm); \node at (10,2.5) {$\alpha_2$}; % \node at (-1,1) {$\sb{A}{\sf{Mod}}_B$}; % \filldraw (0,1) circle (1mm); % \draw (0,1)--(3,2); % \draw (0,1)--(4,0); % \filldraw (1,1) circle (1mm); % \draw (1,1)--(4,2); % \draw (1,1)--(3,0); % \filldraw (6,1) circle (1mm); \node at (6,1.5) {$\beta_1$}; % \draw (6,1)--(3,2); % \draw (6,1)--(3,0); % \draw (6,1)--(9,2); % \draw (6,1)--(9,0); % \filldraw (7,1) circle (1mm); \node at (7,1.5) {$\beta_2$}; % \draw (7,1)--(4,2); % \draw (7,1)--(4,0); % \draw (7,1)--(10,2); % \draw (7,1)--(10,0); % \node at (-1,0) {$\sb{B}{\sf{Mod}}_B$}; % \filldraw (3,0) circle (1mm); % \filldraw (4,0) circle (1mm); % \filldraw (9,0) circle (1mm); \node at (9,.5) {$\gamma_1$}; % \filldraw (10,0) circle (1mm); \node at (10,.5) {$\gamma_2$}; % \node at (-1,-1) {$\sb{B}{\sf{Mod}}_A$}; % \filldraw (0,-1) circle (1mm); % \draw (0,-1)--(3,-2); % \draw (0,-1)--(3,0); % \filldraw (1,-1) circle (1mm); % \draw (1,-1)--(4,-2); % \draw (1,-1)--(4,0); % \filldraw (6,-1) circle (1mm); \node at (6,-.5) {$\beta_1^*$}; % \draw (6,-1)--(3,-2); % \draw (6,-1)--(4,0); % \draw (6,-1)--(10,-2); % \draw (6,-1)--(9,0); % \filldraw (7,-1) circle (1mm); \node at (7,-.5) {$\beta_2^*$}; % \draw (7,-1)--(4,-2); % \draw (7,-1)--(3,0); % \draw (7,-1)--(9,-2); % \draw (7,-1)--(10,0); % \node at (-1,-2) {$\sb{A}{\sf{Mod}}_A$}; % \filldraw (3,-2) circle (1mm); % \filldraw (4,-2) circle (1mm); % \filldraw (9,-2) circle (1mm); \node at (9,-1.5) {$\alpha_1$}; % \filldraw (10,-2) circle (1mm); \node at (10,-1.5) {$\alpha_2$}; %\end{tikzpicture} %$$ %where we could have chosen to cross the paths from the $\beta_i$ to the $\alpha_j$ instead of crossing the paths from the $\beta_i^*$ to the $\alpha_j$ (this means $\alpha_2=\alpha_1^*$ and $\gamma_i^*=\gamma_i$ for $i=1,2$). % %However, in fixing the two aforementioned associativity problems, we have created two more: %\begin{enumerate} %\item[(1)] %There is a path from $\alpha_1$ to $\gamma_1$ through an $A-B$ bimodule, but not through a $B-A$ bimodule. The same is true for $\alpha_2$ and $\gamma_2$. %\item[(2)] %There is a path from $\alpha_1$ to $\gamma_2$ through a $B-A$ bimodule, but not through an $A-B$ bimodule. The same is true for $\alpha_2$ and $\gamma_1$. %\end{enumerate} % %Repeating a similar analysis as above, we see that our only choice for extending our graphs is to attach one distinct vertex at depth $n+8$ to each vertex at depth $n+7$. But this forces us to extend once again, and by an induction argument, we have that $(\Gamma,\Delta)=(\Gamma_{n,\infty},\Delta_{n,\infty})$, where the duality is given periodically by % %\todo{image} % %But $(\Gamma_{n,\infty},\Delta_{n,\infty})$ are not the principal graphs of a subfactor by Corollary \ref{cor:NoInfiniteArms}, a contradiction. %\end{proof} % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\subsection{Emily's Eliminating $\cB$}\label{sec:EliminateB} % %Principal graphs coming from the weed $\cB$ are eliminated by showing they cannot have a connection. We first need a simple result restricting slightly the possible principal graphs. Then we deduce, from attempting to construct a connection, an equation which $q$ must satisfy. This equation is satisfied by the graph in which both arms at the end of $\cB$ are extended infinitely and simply (ie, without further branching). Some graph theoretic arguments, which we punt to an appendix, then show that no other potential principal graph extending $\cB$ has the needed graph norm. % %\begin{defn} %Let %$$(\Gamma, \Delta) = \BadSeed %$$ %%\left(\scalebox{1.5}{$\bigraph{bwd1v1v1v1p1v1x0p0x1v1x0p1x0p0x1p0x1v0x0x1x0p0x1x0x0duals1v1v1x2v1x3x2x4}$}, \scalebox{1.5}{$\bigraph{bwd1v1v1v1p1v1x0p1x0v1x0p0x1v1x0p0x1duals1v1v1x2v1x2}$} \right).$$ % %For $n\geq 3$, let $(\Gamma_n,\Delta_n)$ be the translation of $(\Gamma,\Delta)$ by $n-3$, so the branch point occurs at depth $n$. For $n\geq 3$, let $(\Gamma_{n,\infty},\Delta_{n,\infty})$ be the extension of $(\Gamma_n,\Delta_n)$ given by extending both arms infinitely and simply (we will not specify the dual data of the extension). %\end{defn} % % %\begin{lem}\label{lem:Bdual} %In a principal graph grown from $\cB$: If both legs extend simply, the dual data is consist of repetitions of (image) . %If there is no branch or merger at depth $n$, then there are two vertices at depth $n+1$ which simply extend the previous legs. %\end{lem} % %\begin{proof} %associativity on 4-partite graph computation. %\end{proof} % %\begin{thm}\label{thm:Bgrowth} %$$\includegraphics[height=8mm]{Btwolegs},$$ %A subfactor whose principal graph is a translated extension of $\Gamma$ either has principal graph %$$\Gamma_{n, \infty} = \includegraphics[height=8mm]{Binfinite},$$ %or contains %$$\Gamma_{\text{merge}} = \includegraphics[height=8mm]{Bmerger}$$ %or %$$\Gamma_{\text{equal legs, branch}} = \includegraphics[height=8mm]{Btriple}.$$ %\end{thm} % %\begin{proof} %Any graph which is a translated extensions of $\cB$ either is %$$\includegraphics[height=8mm]{Btwolegs},$$ %which cannot be a principal graph because of Lemma \ref{lem:Bdual}, or is %$\Gamma_{n, \infty}$, or contains $\Gamma_{\text{merge}}$ or %$$\Gamma_{\text{general branch}} = \includegraphics[height=8mm]{Bbadtriple}.$$ % So we need to argue that $\Gamma_{\text{general branch}} $ cannot be contained in a principal graph unless it (the principal graph) also contains $\Gamma_{\text{equal legs, branch}}$. This is again Lemma \ref{lem:Bdual} %\end{proof} % %We will now use connections to show that none of these except $\Gamma_{n, \infty}$ can be principal graphs; $\Gamma_{n, \infty}$ is ruled out by Popa / eep thesis / objects have dimension less than 1. % %\begin{prop}\label{prop:ConnectionExists} %In order for a connection to exist on any extension of an $(n-3)$-translate of $(\Gamma,\Delta)$, %it must have eigenvalue $d=q+q^{-1}$ where $q$ is the unique root greater than $1$ of %\begin{equation}\label{eqn:ConsistencyConstraint} %q^{2 n+8}-2 q^{2 n+6}-q^{2 n+4}+q^4+1=0 %\end{equation} %\end{prop} %\begin{proof} %The loop $(0062),(0151),(1141),(1052)$ appears in two different $1$-by-$1$ matrices, so we can calculate the norm of the corresponding component of the connection in two different ways. In particular, we must have %$$\dim(0062) \dim(1141) = \dim(0151) \dim(1052),$$ %which, after expanding products of quantum numbers and canceling, gives %$$ %[2n+7] - [2n+5] - [2n+3] + [2n+1] -4 [n+1]^2=0. %$$ %Factoring the left hand side, we get %$$ %\frac{q^{-2 n-6} \left(q^{2 n+8}+q^{2 n+4}-q^4-2 q^2+1\right) \left(q^{2 n+8}-2q^{2 n+6}- q^{2 n+4}+q^4+1\right)}{(q-1)^2 (q+1)^2}=0. %$$ %Note that $q^{2 n+8}+q^{2 n+4}-q^4-2 q^2+1>0$ for all $q>1$, so the above equality is satisfied if and only if Equation \eqref{eqn:ConsistencyConstraint} holds. %\end{proof} % %\begin{lem}\label{lem:Binfinity} %$\Gamma_{n,\infty}$ has the following Frobenius-Perron $\ell^2$-eigenvector $v$, where $v_{i,j}$ is the value of the eigenvector at the $j$-th vertex at depth $i$: (the branch point is at depth $n$): %%{\scriptsize{ %\begin{align*} %v_{k,1}&=[k+1] \text{ for all $k\leq n$ } \hspace{.5in}& v_{n+1,1}&=v_{n+1,2}=\frac{[n+2]}{2}\displaybreak[1]\\\ %v_{n+2,1}&=v_{n+2,2}=\frac{[2][n+2]}{2q^2} &v_{n+3,1}&=v_{n+3,4}=\frac{[n+2]}{2q^2}\displaybreak[1]\\\ %v_{n+3,2}&=v_{n+3,3}=\frac{[2][n+2]}{2q^3} & v_{n+k,1}&=v_{n+k,2}= \frac{[2][n+2]}{2q^{k}} \text{ for all $k\geq 4$.} %\end{align*}%}} %with eigenvalue $d=q+q^{-1}$ where $q$ is the unique root greater than $1$ of %Equation \eqref{eqn:ConsistencyConstraint}. %\end{lem} %\begin{proof} %By the Frobenius-Perron eigenvalue equations, we know $v_{k,1}$ for $k\leq n$ and $v_{n+1,j}$ for $j=1,2$. Let $x=v_{n+2,2}=v_{n+2,3}$. Then clearly $v_{n+3,1}=v_{n+3,4}=x/[2]$, $v_{n+3,2}=v_{n+3,3}=xq^{-1}$, and $v_{n+k,1}=v_{n+k,2}=xq^{2-k}$ for all $k\geq 4$ by Lemma \ref{lem:geometric}. Hence it suffices to calculate $x$. We may do so via 2 different eigenvalue equations: %\begin{align*} %v_{n+2,2}+v_{n,1}=[2]v_{n+1,1} &\Longrightarrow x+[n+1] = [2]\frac{[n+2]}{2}\text{ and}\\ %v_{n+1,1}+v_{n+3,1}+v_{n+3,2}=[2]v_{n+2,2} &\Longrightarrow \frac{[n+2]}{2}+\frac{x}{[2]}+q^{-1}x = [2]x. %\end{align*} %After a little algebra, eliminating $x$ in the above equations yields %$$ %\frac{q^2[n+1]}{q^2-1}-\frac{[2][n+2]}{2}=\frac{q^{-n-2} \left(q^{2 n+8}-2 q^{2 n+6}-q^{2 n+4}+q^4+1\right)}{2 (q-1)^2 (q+1)^2}=0, %$$ %which is satisfied if and only if Equation \eqref{eqn:ConsistencyConstraint} holds. %\end{proof} % %\todo{see (3) on 184 of Popa's classification of amenable subfactors of type II} % % %\begin{cor}\label{cor:NormskillB} %$q$ for the graph $\Gamma_{n, \text{ simple legs}}$ is too small for $\Gamma_{n, \text{ simple legs}}$ to be a principal graph. $q$ for the graphs $\Gamma_{\text{merge}}$ and $\Gamma_{\text{equal legs, branch}}$ are too large for them to be principal graphs, so any graph containing one of these cannot be a principal graph either. %\end{cor} % %\begin{proof} %$\Gamma_{n, \text{ simple legs}}$ is properly contained in $\Gamma_{n,\infty}$, and thus has smaller associated $q$. % %$\Gamma_{\text{merge}}$ and $\Gamma_{\text{equal legs, branch}}$ have larger associated $q$ than $\Gamma_{n,\infty}$. This follows from general facts about norms of infinite graphs, \ref{lem:NormCycles} and \ref{lem:ExtraTriple}, proven in the Appendix. %\end{proof} % %\begin{prop}\label{prop:NoInfiniteArms} %The infinite graph $\Gamma_{n,\infty}$ can't be the principal graph of an infinite depth subfactor. %\end{prop} %\begin{proof} %This follows from Theorem 4.5 of \cite{MR1334479} or from Theorem 6.5 of \cite{0902.1294} and \todo{fix citation for MW}%\cite{gpa}. %\end{proof} % %Thus, \ref{thm:Bgrowth} and \ref{cor:NormskillB} prove Theorem \ref{thm:BadSeed}, that no translated extension of $\cB$ can be a principal graph. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \appendix \section{Facts about norms for some infinite graphs}\label{app:infinitenorms}%terrible section title %\todo{replace with graphs for which $\Gamma$ does not have a crossing. Make graphs for $\cB$ in intro equal to this one. Then remove this theorem as it is in the intro.} %\begin{thm}\label{thm:EliminateBadSeed} %There are no subfactors with principal graph pair a translated extension of %$$(\Gamma, \Delta) = \left(\scalebox{1.5}{$\bigraph{bwd1v1v1v1p1v1x0p0x1v1x0p1x0p0x1p0x1v0x0x1x0p0x1x0x0duals1v1v1x2v1x3x2x4}$}, \scalebox{1.5}{$\bigraph{bwd1v1v1v1p1v1x0p1x0v1x0p0x1v1x0p0x1duals1v1v1x2v1x2}$} \right).$$\dave{I switched $\Delta$ and $\Gamma$.} %\end{thm} % %\begin{defn} %For $n\geq 3$, let $(\Gamma_n,\Delta_n)$ be the extension of $(\Gamma,\Delta)$ by $n-3$, so the branch point occurs at depth $n$. For $n\geq 3$, let $(\Gamma_{n,\infty},\Delta_{n,\infty})$ be the extension of $(\Gamma_n,\Delta_n)$ given by extending both arms infinitely (we will not specify the dual data of the extension). %\end{defn} % %\subsection{Generalities} $G$ will always denote a locally finite graph, $A(G)$ its adjacency matrix, and $r(G)$ its spectral radius/graph norm. \begin{thm}\label{thm:NormsConverge}[Mohar and Woess, 4.13] Suppose subgraphs $G_n$ converge to $G$. Then $r(G_n) \nearrow r(G)$.\dave{i think we should replace $r(G)$ by $\|G\|$ everywhere} \end{thm} \begin{thm}\label{thm:InfiniteEigenvector} If an infinite graph $G$ has a Frobenius-Perron eigenvector $\bf{v}\in\ell^2$ (an $\ell^2$-eigenvector with strictly positive entries) corresponding to eigenvalue $d$, then $r(G)=\|G\|= d$. \end{thm} \begin{proof} This follows from Theorems 4.4 and 6.2 of Mohar and Woess. \end{proof} \todo{We might want to quote a remark on page 183 of Popa's classification of amenable subfactors of type II.} \begin{lem}\label{lem:geometric} Suppose $G$ is of the form $$\begin{tikzpicture} \filldraw[black] (1,.5) rectangle (3,1.5); \draw (3,1)--(7,1); \filldraw (4,1) circle (1mm); \filldraw (5,1) circle (1mm); \filldraw (6,1) circle (1mm); \node at (7,1) [right] {$\cdots$}; \end{tikzpicture}$$ If $\bf{v}$ is a Frobenius-Perron $\ell^2$-eigenvector for $G$ with eigenvalue $d=(q+q^{-1})>2$, with entries $$\begin{tikzpicture} \filldraw[black] (1,.5) rectangle (3,1.5); \draw (3,1)--(7,1); \filldraw (4,1) circle (1mm) node [above] {$a_0$}; \filldraw (5,1) circle (1mm) node [above] {$a_1$}; \filldraw (6,1) circle (1mm) node [above] {$a_2$}; \node at (7,1) [right] {$\cdots$}; \end{tikzpicture}$$ then $a_n=q^{-n} a_0$. \end{lem} \begin{proof} From $a_0$ and $a_1$ and the relation $[2] a_k = a_{k-1}+a_{k+1}$, one shows inductively that $a_n = [n] a_{1} - [n-1] a_{0}$. For $\bf v$ to be an $\ell^2$-eigenvector, we need $a_n \rightarrow 0$. There's some $\epsilon$ such that $a_0=(q+\epsilon) a_1$; Expanding out $a_n$ using this relation, we find $$a_n = [n] a_1 - [n-1] a_0= a_2 \dfrac{-\epsilon q^{n-2} + q^{-n+3} + \epsilon q^{-n+2} - q^{-n+1} }{q-q^{-1}}.$$ Since $q>1$ and $a_n \rightarrow 0$, we must have $\epsilon =0$. So we know $a_0=q a_1$. Now $a_n = [n] a_{1} - [n-1] a_{0} = (q^{-1} [n] -[n-1])a_0=q^{-n} a_0 $. \end{proof} \begin{lem}\label{lem:NormCycles} If $$\begin{tikzpicture}[scale=.6] \filldraw[black] (1,.5) rectangle (3,2.5); \draw (3,1)--(7,1); \filldraw (4,1) circle (1mm) node [below] {$a$}; \filldraw (5,1) circle (1mm); \filldraw (6,1) circle (1mm); \node at (7,1) [right] {$\cdots$}; \draw (3,2)--(7,2); \filldraw (4,2) circle (1mm) node [above] {$a$}; \filldraw (5,2) circle (1mm); \filldraw (6,2) circle (1mm); \node at (7,2) [right] {$\cdots$}; \end{tikzpicture}$$ has an $\ell^2$-eigenvector with eigenvalue $d>2$, then $$r \left( \begin{tikzpicture}[baseline=.8cm, scale=.6] \filldraw[black] (1,.5) rectangle (3,2.5); \draw (3,1)--(4,1)--(5,1.5)--(4,2)--(3,2); \filldraw (4,1) circle (1mm); \filldraw (5,1.5) circle (1mm); \filldraw (4,2) circle (1mm); \end{tikzpicture} \right) > r \left( \begin{tikzpicture}[baseline=.8cm, scale=.6] \filldraw[black] (1,.5) rectangle (3,2.5); \draw (3,1)--(7,1); \filldraw (4,1) circle (1mm) node [below] {$a$}; \filldraw (5,1) circle (1mm); \filldraw (6,1) circle (1mm); \node at (7,1) [right] {$\cdots$}; \draw (3,2)--(7,2); \filldraw (4,2) circle (1mm) node [above] {$a$}; \filldraw (5,2) circle (1mm); \filldraw (6,2) circle (1mm); \node at (7,2) [right] {$\cdots$}; \end{tikzpicture} \right) $$ \end{lem} \begin{proof} By Lemma \ref{lem:geometric}, the $\ell^2$-eigenvector is of the form $$\begin{tikzpicture}[scale=.6] \filldraw[black] (1,.5) rectangle (3,2.5); \node at (2,1.5) { \textcolor{white}{$\bf v$} }; \draw (3,1)--(7,1); \filldraw (4,1) circle (1mm) node [below] {$a$}; \filldraw (5,1) circle (1mm) node [below] {$\dfrac{a}{q}$}; \filldraw (6,1) circle (1mm) node [below] {$\dfrac{a}{q^2}$}; \node at (7,1) [right] {$\cdots$}; \draw (3,2)--(7,2); \filldraw (4,2) circle (1mm) node [above] {$a$}; \filldraw (5,2) circle (1mm) node [above] {$\dfrac{a}{q}$}; \filldraw (6,2) circle (1mm) node [above] {$\dfrac{a}{q^2}$}; \node at (7,2) [right] {$\cdots$}; \end{tikzpicture}$$ with eigenvalue $d=(q+q^{-1})$ for $q>1$. Now, consider this vector ``restricted" to the finite graph: $$\begin{tikzpicture}[baseline=.8cm, scale=.6] \filldraw[black] (1,.5) rectangle (3,2.5); \node at (2,1.5) { \textcolor{white}{$\bf v$} }; \draw (3,1)--(4,1)--(5,1.5)--(4,2)--(3,2); \filldraw (4,1) circle (1mm) node [below] {$a$}; \filldraw (5,1.5) circle (1mm) node [right] {$\dfrac{a}{q}$}; \filldraw (4,2) circle (1mm) node [above] {$a$}; \end{tikzpicture} $$ By Frobenius-Perron eigentheory, we know \begin{align*} r \left( \begin{tikzpicture}[scale=.6, baseline=.8cm] \filldraw[black] (1,.5) rectangle (3,2.5); \draw (3,1)--(4,1)--(5,1.5)--(4,2)--(3,2); \filldraw (4,1) circle (1mm); \filldraw (5,1.5) circle (1mm); \filldraw (4,2) circle (1mm); \end{tikzpicture} \right) & \geq \frac{\norm{A \left( \begin{tikzpicture}[baseline=.8cm, scale=.6] \filldraw[black] (1,.5) rectangle (3,2.5); \node at (2,1.5) { \textcolor{white}{$\bf v$} }; \draw (3,1)--(4,1)--(5,1.5)--(4,2)--(3,2); \filldraw (4,1) circle (1mm) node [below] {$a$}; \filldraw (5,1.5) circle (1mm) node [right] {$\dfrac{a}{q}$}; \filldraw (4,2) circle (1mm) node [above] {$a$}; \end{tikzpicture} \right) }} {\norm{ \begin{tikzpicture}[baseline=.8cm, scale=.6] \filldraw[black] (1,.5) rectangle (3,2.5); \node at (2,1.5) { \textcolor{white}{$\bf v$} }; \draw (3,1)--(4,1)--(5,1.5)--(4,2)--(3,2); \filldraw (4,1) circle (1mm) node [below] {$a$}; \filldraw (5,1.5) circle (1mm) node [right] {$\dfrac{a}{q}$}; \filldraw (4,2) circle (1mm) node [above] {$a$}; \end{tikzpicture} }} = \frac{\norm{ \begin{tikzpicture}[baseline=.8cm, scale=.6] \filldraw[black] (1,.5) rectangle (3,2.5); \node at (2,1.5) { \textcolor{white}{$d {\bf v}$} }; \draw (3,1)--(4,1)--(5,1.5)--(4,2)--(3,2); \filldraw (4,1) circle (1mm) node [below] {$d a$}; \filldraw (5,1.5) circle (1mm) node [right] {$2a$}; \filldraw (4,2) circle (1mm) node [above] {$d a$}; \end{tikzpicture} }} {\norm{ \begin{tikzpicture}[baseline=.8cm, scale=.6] \filldraw[black] (1,.5) rectangle (3,2.5); \node at (2,1.5) { \textcolor{white}{$\bf v$} }; \draw (3,1)--(4,1)--(5,1.5)--(4,2)--(3,2); \filldraw (4,1) circle (1mm) node [below] {$a$}; \filldraw (5,1.5) circle (1mm) node [right] {$\dfrac{a}{q}$}; \filldraw (4,2) circle (1mm) node [above] {$a$}; \end{tikzpicture} }}\\ &=\frac{\sqrt{d^2 \norm{ \begin{tikzpicture}[baseline=.8cm, scale=.6] \filldraw[black] (1,.5) rectangle (3,2.5); \node at (2,1.5) { \textcolor{white}{$\bf v$} }; \end{tikzpicture} }^2 +2 d^2 a^2 + (2a)^2}}% {\sqrt{\norm{ \begin{tikzpicture}[baseline=.8cm, scale=.6] \filldraw[black] (1,.5) rectangle (3,2.5); \node at (2,1.5) { \textcolor{white}{$\bf v$} }; \end{tikzpicture} }^2 +2 a^2 + (\dfrac{a}{q})^2}} >d \end{align*} because $2>d q^{-1}$ (since $2q>d=q+q^{-1}$). By Theorem \ref{thm:InfiniteEigenvector}, we are finished. \end{proof} \begin{lem}\label{lem:ExtraTriple} If $$\begin{tikzpicture}[scale=.6] \filldraw[black] (1,.5) rectangle (3,2.5); \draw (3,1)--(7,1); \filldraw (4,1) circle (1mm) node [below] {$a$}; \filldraw (5,1) circle (1mm); \filldraw (6,1) circle (1mm); \node at (7,1) [right] {$\cdots$}; \draw (3,2)--(7,2); \filldraw (4,2) circle (1mm) node [above] {$a$}; \filldraw (5,2) circle (1mm); \filldraw (6,2) circle (1mm); \node at (7,2) [right] {$\cdots$}; \end{tikzpicture}$$ has a Frobenius-Perron $\ell^2$-eigenvector with eigenvalue $d=q+q^{-1}$ where $2q^2-3-3q^{-2}>0$ (which is true for $q>1.48$), then $$\norm{ \begin{tikzpicture}[baseline=.8cm, scale=.6] \filldraw[black] (1,.5) rectangle (3,2.5); \draw (3,1)--(4,1)--(5,.5); \draw (3,2)--(4,2)--(5,1.5); \draw (4,2)--(5,2.5); \filldraw (4,1) circle (1mm); \filldraw (5,.5) circle (1mm); \filldraw (4,2) circle (1mm); \filldraw (5,1.5) circle (1mm); \filldraw (5,2.5) circle (1mm); \end{tikzpicture} } > \norm{ \begin{tikzpicture}[baseline=.8cm, scale=.6] \filldraw[black] (1,.5) rectangle (3,2.5); \draw (3,1)--(7,1); \filldraw (4,1) circle (1mm) node [below] {$a$}; \filldraw (5,1) circle (1mm); \filldraw (6,1) circle (1mm); \node at (7,1) [right] {$\cdots$}; \draw (3,2)--(7,2); \filldraw (4,2) circle (1mm) node [above] {$a$}; \filldraw (5,2) circle (1mm); \filldraw (6,2) circle (1mm); \node at (7,2) [right] {$\cdots$}; \end{tikzpicture} }.$$ \end{lem} \begin{proof} This is similar to the proof of Lemma \ref{lem:NormCycles}. We consider the Frobenius-Perron vector ``restricted" to the finite graph: $$ \widetilde{\bf{v}}= \begin{tikzpicture}[baseline=.8cm, scale=.6] \filldraw[black] (1,.5) rectangle (3,2.5); \draw (3,1)--(4,1)--(5,.5); \draw (3,2)--(4,2)--(5,1.5); \draw (4,2)--(5,2.5); \node at (2,1.5) { \textcolor{white}{$\bf v$} }; \filldraw (4,1) circle (1mm) node [below] {$a$}; \filldraw (5,.5) circle (1mm) node [right] {$aq^{-1}$}; \filldraw (4,2) circle (1mm) node [above] {$a$}; \filldraw (5,1.5) circle (1mm) node [right] {$aq^{-1}$}; \filldraw (5,2.5) circle (1mm) node [right] {$aq^{-1}$}; \end{tikzpicture} $$ Then we have \begin{align*} \norm{ \begin{tikzpicture}[baseline=.8cm, scale=.6] \filldraw[black] (1,.5) rectangle (3,2.5); \draw (3,1)--(4,1)--(5,.5); \draw (3,2)--(4,2)--(5,1.5); \draw (4,2)--(5,2.5); \filldraw (4,1) circle (1mm); \filldraw (5,.5) circle (1mm); \filldraw (4,2) circle (1mm); \filldraw (5,1.5) circle (1mm); \filldraw (5,2.5) circle (1mm); \end{tikzpicture} }^2 & \geq \frac{\norm{A \left(\widetilde{\bf{v}}\right)}^2} {\norm{\widetilde{\bf{v}}}^2} = \frac{\norm{ \begin{tikzpicture}[baseline=1.2cm, scale=.6] \filldraw[black] (1,.5) rectangle (3,2.5); \draw (3,1)--(4,1)--(5,.5); \draw (3,2)--(4,2)--(5,1.5); \draw (4,2)--(5,2.5); \node at (2,1.5) { \textcolor{white}{$\bf dv$} }; \filldraw (4,1) circle (1mm) node [below] {$da$}; \filldraw (5,.5) circle (1mm) node [right] {$a$}; \filldraw (4,2) circle (1mm) node [above] {$b$}; \node at (4,3.4) {$\underbrace{da+aq^{-1}}$}; \filldraw (5,1.5) circle (1mm) node [right] {$a$}; \filldraw (5,2.5) circle (1mm) node [right] {$a$}; \end{tikzpicture} }^2} {\norm{\widetilde{\bf{v}}}^2}\\ &= \frac{ d^2 \norm{ \begin{tikzpicture}[baseline=.8cm, scale=.6] \filldraw[black] (1,.5) rectangle (3,2.5); \node at (2,1.5) { \textcolor{white}{$\bf v$} }; \end{tikzpicture}}^2 + 2d^2 a^2 + 2\dfrac{da^2}{q} + \dfrac{a^2}{q^2} +3a^2 }% {\norm{ \begin{tikzpicture}[baseline=.8cm, scale=.6] \filldraw[black] (1,.5) rectangle (3,2.5); \node at (2,1.5) { \textcolor{white}{$\bf v$} }; \end{tikzpicture} }^2 +2 a^2 + 3\dfrac{a^2}{q^2} } > d^2 \end{align*} because the inequality $$ d^2 a^2 + 2\dfrac{da^2}{q} + \dfrac{a^2}{q^2} +3a^2 > d^2 \left(2 a^2 + 3\dfrac{a^2}{q^2}\right) $$ is equivalent to $2q^2-3-3q^{-2}>0$. By Theorem \ref{thm:InfiniteEigenvector}, we are finished. \end{proof} %\todo{If we extend one arm and attach a triple point, we need to extend the other arm, or the resulting inequality won't work. we need more $q$'s on the denominator of the first term of the right hand side of the inequality. How can we rule out extending one arm and adding a triple point and leaving the other arm alone? I suppose this messes up the dual data...?} % % % %\subsection{Particulars} % %\begin{lem} %On any extension of an $n-3$ translate, the dimensions of the vertices are given by the following rational functions in $q$ and $q^n$. %\todo{}\dave{is this lemma necessary?} %\end{lem} % % %\begin{lem}\label{lem:Binfinity} %$\Gamma_{n,\infty}$ has the following Frobenius-Perron $\ell^2$-eigenvector $v$, where $v_{i,j}$ is the value of the eigenvector at the $j$-th vertex at depth $i$: (the branch point is at depth $n$): %%{\scriptsize{ %\begin{align*} %v_{k,1}&=[k+1] \text{ for all $k\leq n$ } \hspace{.5in}& v_{n+1,1}&=v_{n+1,2}=\frac{[n+2]}{2}\displaybreak[1]\\\ %v_{n+2,1}&=v_{n+2,2}=\frac{[2][n+2]}{2q^2} &v_{n+3,1}&=v_{n+3,4}=\frac{[n+2]}{2q^2}\displaybreak[1]\\\ %v_{n+3,2}&=v_{n+3,3}=\frac{[2][n+2]}{2q^3} & v_{n+k,1}&=v_{n+k,2}= \frac{[2][n+2]}{2q^{k}} \text{ for all $k\geq 4$.} %\end{align*}%}} %with eigenvalue $d=q+q^{-1}$ where $q>1$ such that %\begin{equation}\label{eqn:ConsistencyConstraint} %q^{2 n+8}-2 q^{2 n+6}-q^{2 n+4}+q^4+1=0 %\end{equation} %(such a $q$ is clearly unique). %\end{lem} %\begin{proof} %By the Frobenius-Perron eigenvalue equations, we know $v_{k,1}$ for $k\leq n$ and $v_{n+1,j}$ for $j=1,2$. Let $x=v_{n+2,2}=v_{n+2,3}$. Then clearly $v_{n+3,1}=v_{n+3,4}=x/[2]$, $v_{n+3,2}=v_{n+3,3}=xq^{-1}$, and $v_{n+k,1}=v_{n+k,2}=xq^{2-k}$ for all $k\geq 4$ by Lemma \ref{lem:geometric}. Hence it suffices to calculate $x$. We may do so via 2 different eigenvalue equations: %\begin{align*} %v_{n+2,2}+v_{n,1}=[2]v_{n+1,1} &\Longrightarrow x+[n+1] = [2]\frac{[n+2]}{2}\text{ and}\\ %v_{n+1,1}+v_{n+3,1}+v_{n+3,2}=[2]v_{n+2,2} &\Longrightarrow \frac{[n+2]}{2}+\frac{x}{[2]}+q^{-1}x = [2]x. %\end{align*} %After a little algebra, eliminating $x$ in the above equations yields %$$ %\frac{q^2[n+1]}{q^2-1}-\frac{[2][n+2]}{2}=\frac{q^{-n-2} \left(q^{2 n+8}-2 q^{2 n+6}-q^{2 n+4}+q^4+1\right)}{2 (q-1)^2 (q+1)^2}=0, %$$ %which is satisfied if and only if Equation \eqref{eqn:ConsistencyConstraint} holds. %\end{proof} % %\todo{see (3) on 184 of Popa's classification of amenable subfactors of type II} % % %\begin{prop}\label{prop:ConnectionExists} %In order for a connection to exist on any extension of an $(n-3)$-translate of $(\Gamma,\Delta)$, Equation \eqref{eqn:ConsistencyConstraint} must hold. %\end{prop} %\begin{proof} %The loop $(0062),(0151),(1141),(1052)$ appears in two different $1$-by-$1$ matrices, so we can calculate the norm of the corresponding component of the connection in two different ways. In particular, we must have %$$\dim(0062) \dim(1141) = \dim(0151) \dim(1052),$$ %which, after expanding products of quantum numbers and canceling, gives %$$ %[2n+7] - [2n+5] - [2n+3] + [2n+1] -4 [n+1]^2=0. %$$ %Factoring the left hand side, we get %$$ %\frac{q^{-2 n-6} \left(q^{2 n+8}+q^{2 n+4}-q^4-2 q^2+1\right) \left(q^{2 n+8}-2q^{2 n+6}- q^{2 n+4}+q^4+1\right)}{(q-1)^2 (q+1)^2}=0. %$$ %Note that $q^{2 n+8}+q^{2 n+4}-q^4-2 q^2+1>0$ for all $q>1$, so the above equality is satisfied if and only if Equation \eqref{eqn:ConsistencyConstraint} holds. %\end{proof} % %\begin{fact}\label{fact:Extension} %Any graph which is an extension of $\Gamma_n$ either %\begin{itemize} %\item extends the two arms with boring chains of length $j,k$, with $j\leq k\leq \infty$. Call these $\Gamma_{n,(j,k)}$ (and $\Gamma_{n,(\infty,\infty)}=\Gamma_{n,\infty}$). %\item contains the subgraph $\Gamma'_{n,k}$ (translate by $n$, leave the first arm alone, extend the second arm by $k$, and attach two vertices to the final vertex on the second arm). %\item contains the subgraph $\Gamma''_{n,k}$ (translate by $n$, extend the two arms with boring chains of length $k$, and have the arms connect at the final depth) %\end{itemize} %\end{fact} % % %\begin{prop}\label{prop:TooSmall} %For the graphs $\Gamma_{n,(j,k)}$ with $j<\infty$, $\|\Gamma_{n,(j,k)}\|=q+q^{-1}$ is too small for a connection to exist.\dave{we should probably have $\Delta$ and $\Gamma$ here, as connections exist on 4-partite graphs...} %\end{prop} %\begin{proof} %This follows from Theorem \ref{thm:NormsConverge} and Proposition \ref{prop:ConnectionExists}. %\end{proof} % %\begin{prop}\label{prop:TooLarge} %For the graphs $\Gamma'_{n,k}$, $\|\Gamma'_{n,k}\|=q+q^{-1}$ is too large for a connection to exist. %\end{prop} %\begin{proof} %In \ref{lem:Binfinity} we showed that $\Gamma_{n,\infty}$ has an $\ell^2$-eigenvector. As $2q^2-3-3q^{-2}>0$ when $q\geq \|\Gamma\|>1.53$, the hypotheses of Lemma \ref{lem:ExtraTriple} hold, $\Gamma'_{n,k}$ has a larger norm than $\Gamma_{n,\infty}$, and Equation \eqref{eqn:ConsistencyConstraint} cannot hold. The result follows from Proposition \ref{prop:ConnectionExists}. %\end{proof} % %\begin{prop}\label{prop:Bcycles} %For the graphs $\Gamma''_{n,k}$, $\|\Gamma''_{n,k}\|=q+q^{-1}$ is too large for a connection to exist. %\end{prop} %\begin{proof} %In \ref{lem:Binfinity} we showed that $\Gamma_{n,\infty}$ has an $\ell^2$-eigenvector. Thus, the hypotheses of Lemma \ref{lem:NormCycles} hold, $\Gamma''_{n,k}$ has a larger norm than $\Gamma_{n,\infty}$, and Equation \eqref{eqn:ConsistencyConstraint} cannot hold. The result follows from Proposition \ref{prop:ConnectionExists}. %\end{proof} % %\begin{prop}\label{prop:NoInfiniteArms} %The infinite graph $\Gamma_{n,\infty}$ can't be the principal graph of an infinite depth subfactor. %\end{prop} %\begin{proof} %This follows from Theorem 4.5 of \cite{MR1334479} or from Theorem 6.5 of \cite{0902.1294} and \todo{fix citation for MW}%\cite{gpa}. %\end{proof} % %\begin{proof}[Proof of Theorem \ref{thm:EliminateBadSeed}] %Follows immediately from Fact \ref{fact:Extension} and Propositions \ref{prop:TooSmall}, \ref{prop:TooLarge}, \ref{prop:Bcycles}, and \ref{prop:NoInfiniteArms}. %\end{proof} % ---------------------------------------------------------------- \newcommand{\urlprefix}{} \bibliographystyle{alpha} %Included for winedt: %input "bibliography/bibliography.bib" \bibliography{../../bibliography/bibliography} % ---------------------------------------------------------------- This paper is available online at \arxiv{1007.2240}, and at \url{http://tqft.net/index5-part2}. % A GTART necessity: % \Addresses % ---------------------------------------------------------------- \end{document} % ----------------------------------------------------------------