\documentclass{gtart} % to submit to AGT, uncomment %\agtart \newcommand{\pathtotrunk}{./} \input{text/article_preamble.tex} \input{text/top_matter.tex} \begin{document} \begin{abstract} This paper gives two applications of Jones's quadratic tangles techniques 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{index5-part1}. The two families we eliminate here each have principal graph pairs whose first branch point is a triple point, and which continue assymetrically past that. \end{abstract} \maketitle \todo{Change title: ``Subfactors of index less than 5, part 2: triple point obstructions"?} % remove table of contents for submitted version \tableofcontents \newcommand{\MagicNumbersOneZero}{\ensuremath{\left(\bigraph{bwd1v1v1v1p1v1x0p0x1duals1v1v1x2},\bigraph{bwd1v1v1v1p1v1x0p1x0duals1v1v1x2}\right)}} \newcommand{\crab}{\ensuremath{\left(\bigraph{bwd1v1v1v1p1v1x0p0x1v1x0p0x1p0x1v1x0x0p0x0x1duals1v1v1x2v2x1x3}, \bigraph{bwd1v1v1v1p1v1x0p1x0v0x1v1p1duals1v1v1x2v1}\right)}} \newcommand{\FSMp}{\bigraph{bwd1v1v1v1p1v1x0p0x1v1x0p1x0p0x1p0x1v0x1x0x0p0x0x1x0p0x0x0x1v1x0x0p1x0x0p0x1x0p0x0x1v1x0x0x0p0x0x1x0p0x0x0x1duals1v1v1x2v1x3x2x4v1x4x3x2}} \newcommand{\FSMd}{\bigraph{bwd1v1v1v1p1v1x0p1x0v1x0p0x1v0x1p1x0p0x1v1x0x0p0x1x0v0x1p1x0p0x1duals1v1v1x2v1x2v2x1}} \newcommand{\FSM}{\ensuremath{\left(\FSMp, \FSMd\right)}} \section{Introduction} Jones' index theorem for subfactors \cite{MR696688} states that the index of a subfactor lies in the range $\{ 4 \cos^2(\frac{\pi}{n}) | n =3, 4, \ldots \} \cup [4,\infty]$. All of these values are realized; however, ignoring subfactors with principal graph $A_\infty$, 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 $A_\infty$) 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 attempts to classify subfactors of index less than $5$. In the first paper \cite{index5-part1} we gave an initial classification result analogous to Haagerup's classification. {\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{index5-part1}}] 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 &= \left(\bigraph{bwd1v1v1v1p1v1x0p0x1v1x0p0x1p0x1v1x0x0p0x0x1duals1v1v1x2v2x1x3},\bigraph{bwd1v1v1v1p1v1x0p1x0v0x1v1p1duals1v1v1x2v1}\right), \\ \cF &=\left(\bigraph{bwd1v1v1v1p1v1x0p0x1v1x0p1x0p0x1p0x1v0x1x0x0p0x0x1x0p0x0x0x1v1x0x0p1x0x0p0x1x0p0x0x1v1x0x0x0p0x0x1x0p0x0x0x1duals1v1v1x2v1x3x2x4v1x4x3x2}, \bigraph{bwd1v1v1v1p1v1x0p1x0v1x0p0x1v0x1p1x0p0x1v1x0x0p0x1x0v0x1p1x0p0x1duals1v1v1x2v1x2v2x1}\right), \\ \cB &=\left(\bigraph{bwd1v1v1v1p1v1x0p0x1v1x0p1x0p0x1p0x1v0x1x0x0p0x0x1x0v1x0p0x1v1x0p0x1duals1v1v1x2v1x3x2x4v1x2}, \bigraph{bwd1v1v1v1p1v1x0p1x0v1x0p0x1v0x1p1x0v1x0p0x1v0x1p1x0duals1v1v1x2v1x2v2x1}\right), \\ \cQ &= \left(\bigraph{bwd1v1v1v1p1p1v0x0x1duals1v1v2x1x3}, \bigraph{bwd1v1v1v1p1p1v0x0x1duals1v1v2x1x3} \right), \\ \cQ' &= \left(\bigraph{bwd1v1v1v1p1p1v0x0x1duals1v1v1x2x3}, \bigraph{bwd1v1v1v1p1p1v0x0x1duals1v1v1x2x3} \right). \end{align*} \end{thm} (As in \cite{index5-part1}, the trivial bimodule always appears as the leftmost vertex of a principal graph, and dual pairs of bimodules are indicated by red tags.) In this paper, we use \emph{triple point obstructions}, i.e., conditions telling us which pairs of graphs containing trivalent vertices are allowable in principal graph pairs, to rule out families of possible principal graphs. In Section \ref{sec:connections}, we recall Ocneanu's triple point obstruction, originally appearing in Haagerup's classification \cite{MR1317352}. We then present a ``triple-single" obstruction, Theorem \ref{thm:TripleSingle}, and we use this obstruction to prove Theorem \ref{thm:AH}: strictly positive translates of the Asaeda-Haagerup principal graphs are not principal graphs of subfactors. This result was originally announced without proof in \cite{MR1317352}, and we presume that Haagerup's proof is similar to the one provided here. In Section \ref{sec:quadratic}, we recall the definition of \underline{annular multiplicities} of possible principal graphs along with Jones' quadratic tangles triple point obstruction \cite{math/1007.1158}. We deduce a powerful inequality for eliminating possible families of principal graphs with annular multiplicities $*10$. In particular, we prove: \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 subfactors, of any index, with principal graphs a translated extension of the pair $$\cF=\FSM.$$ \end{thm} \begin{rem} The techniques of this paper fail to rule out $$ \cB =\left(\bigraph{bwd1v1v1v1p1v1x0p0x1v1x0p1x0p0x1p0x1v0x1x0x0p0x0x1x0v1x0p0x1v1x0p0x1duals1v1v1x2v1x3x2x4v1x2}, \bigraph{bwd1v1v1v1p1v1x0p1x0v1x0p0x1v0x1p1x0v1x0p0x1v0x1p1x0duals1v1v1x2v1x2v2x1}\right) $$ because of the symmetry of its principal graph. See Example \ref{ex:badseeddims} for more details. This weed will be ruled out in a future paper \todo{add citation!}. \end{rem} \begin{rem} The approaches of this paper are also viable for ruling out a large subset of the vines described in the first paper \cite{index5-part1}. However, these approaches require a significant 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} 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{index5-part1} 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. \todo{Thank DARPA?} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Triple point obstructions from connections}\label{sec:connections} \todo{Mention connections here, our conventions==Evans and Kawahigashi's conventions} \subsection{Ocneanu's triple point obstruction} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{The triple-single obstruction} Though this is a previously unpublished result, it may have been known to Haagerup and used by him to rule out the Asaeda-Haagerup vines beyond the first one (as we do in subsection \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 path (in $\mathcal{G}$) between $\alpha_1$ and $\gamma_{2}$ or $\gamma_{3}$ goes through $\beta$ or $\beta^*$; \todo{this is just saying $\dim(\Hom{B-B}{X^*\otimes \alpha_1 \otimes X}{\gamma_i})=1$ for $i=2,3$} \item the only length-two path (in $\mathcal{G}$) between $\gamma_1$ and $\alpha_{2}$ or $\alpha_{3}$ goes through $\beta$, or $\beta^*$. \todo{this is just saying $\dim(\Hom{B-B}{X^*\otimes \alpha_i \otimes X}{\gamma_1})=1$ for $i=2,3$} \end{itemize} If there is a biunitary connection $K$ on the $4$-partite graph $\cG=(\sb{AA}\Gamma_{AB}\Gamma'_{BB}\Gamma'_{BA}\Gamma_{AA})$, 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 --\eep{is there a name for the swapping-entries-and-multiplying-by-some-dimensions relation on connections?}, $\abs{K(\beta, \alpha_2, \beta^*, \gamma_2)}=\dfrac{a_2 c_2}{b^2}$. 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\eep{or presumably we can just refer to Evans and Kawahigashi.}). 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{Application: Eliminating the Asaeda-Haagerup family}\label{sec:AH} We give a proof below of an unpublished result of Haagerup stated in \cite{MR1317352}. \begin{thm}\label{thm:AH} There is no biunitary connection on the $4$-partite graph coming from any strictly positive translate of the Asaeda-Haagerup principal graph pair $$\left(\bigraph{bwd1v1v1v1v1v1p1v1x0p0x1v1x0p0x1p0x1v1x0x0v1duals1v1v1v1x2v2x1x3v1}, \bigraph{bwd1v1v1v1v1v1p1v0x1p0x1v0x1v1duals1v1v1v1x2v1}\right).$$ \end{thm} \begin{proof} Note the hypotheses of Corollary \ref{cor:TripleSingle} are verified. Labeling the 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} \section{Jones' quadratic tangles triple point obstruction}\label{sec:quadratic} %Most of the applications of these obstructions are to subfactors with annular multiplicities $*10$. (This is a fancy way of saying that the graph starts like $D_{2n}$, but we say it this way because it comes from a useful way of thinking about graphs.) We rapidly recall the language from \cite{MR1929335, math/1007.1158} to make sense of this statement and put it in context. We rapidly recall the language from \cite{MR1929335, math/1007.1158}. 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 $0$. 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 two possibilities, the translates of: \begin{equation*} \cdots \bigraph{bwd1v1v1v1p1v1x0p0x1duals1v1v1x2}\qquad\text{and}\qquad \cdots \bigraph{bwd1v1v1v1p1v1x0p1x0duals1v1v1x2}. \end{equation*} Consider now a principal graph pair with annular multiplicities $*10$, and supertransitivity $m-1$. \todo{refer to Ocneanu's triple point obstruction above. As we saw above...} Haagerup proved in \cite{MR1317352}, using Ocneanu's triple point obstruction, 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{Relative dimensions of vertices/bimodules}\label{sec:relativedimensions} Consider $\Gamma_0$ a weed with annular multiplicities $*10$. Now consider $\Gamma$ an $n$-translate of an extension of $\Gamma_0$. Suppose that this graph comes from a subfactor of index $(q+q^{-1})^2$. In order to apply the quadratic tangles inequality from the last subsection, we need to write $r$ 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, Inequality \ref{eq:QTinequality} (and indeed Equation \ref{eq:QTequation} with $\lambda = -1$) always holds for translations and extensions of these graphs, and the quadratic tangles obstruction cannot eliminate this weed. \end{ex} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Application: 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{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{proof}[Proof of Theorem \ref{thm:FSM}] 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 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{Application: 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{index5-part1}, and find that it terminates after two steps. 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] \begin{tikzpicture} [ level 1/.style={level distance=25mm, sibling distance=40mm,nodes={draw, fill=white, rectangle, rounded corners}}, level 2/.style={level distance=63mm, sibling distance=20mm}, level 3/.style={level distance=58mm, sibling distance=25mm}, level 4/.style={level distance=64mm}, grow=right]\node[draw, fill=white, rectangle, rounded corners] {$\begin{array}{c}\bigraph{bwd1v1v1v1p1v1x0p0x1v1x0p0x1p0x1v1x0x0p0x0x1duals1v1v1x2v2x1x3}\\\bigraph{bwd1v1v1v1p1v1x0p1x0v0x1v1p1duals1v1v1x2v1}\end{array}$\ref{bwd1v1v1v1p1v1x0p0x1v1x0p0x1p0x1v1x0x0p0x0x1v1x0p1x0p0x1duals1v1v1x2v2x1x3v3x2x1}} child {node {$\begin{array}{c}\bigraph{bwd1v1v1v1p1v1x0p0x1v1x0p0x1p0x1v1x0x0p0x0x1v1x1duals1v1v1x2v2x1x3v1}\\\bigraph{bwd1v1v1v1p1v1x0p1x0v0x1v1p1v0x1duals1v1v1x2v1v1}\end{array}$\ref{bwd1v1v1v1p1v1x0p1x0v0x1v1p1v0x1duals1v1v1x2v1v1}} }child {node {$\begin{array}{c}\bigraph{bwd1v1v1v1p1v1x0p0x1v1x0p0x1p0x1v1x0x0p0x0x1v1x0p1x0p0x1p0x1duals1v1v1x2v2x1x3v4x2x3x1}\\\bigraph{bwd1v1v1v1p1v1x0p1x0v0x1v1p1v0x1duals1v1v1x2v1v1}\end{array}$} child {node {$\begin{array}{c}\bigraph{bwd1v1v1v1p1v1x0p0x1v1x0p0x1p0x1v1x0x0p0x0x1v1x0p1x0p0x1p0x1v1x0x0x0duals1v1v1x2v2x1x3v4x2x3x1}\\\bigraph{bwd1v1v1v1p1v1x0p1x0v0x1v1p1v0x1v1duals1v1v1x2v1v1}\end{array}$\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} \section{Obsolete} Jones' index theorem for subfactors \cite{MR696688} states that the index of a subfactor lies in the range $\{ 4 \cos^2(\frac{\pi}{n}) | n =3, 4, \ldots \} \cup [4,\infty]$. All of these values are realized; however, ignoring subfactors with principal graph $A_\infty$, 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 $A_\infty$) 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 attempts to classify subfactors of index less than $5$. In the first paper \cite{index5-part1} we gave an initial classification result analogous to Haagerup's classification. 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{index5-part1}}] 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 &= \left(\bigraph{bwd1v1v1v1p1v1x0p0x1v1x0p0x1p0x1v1x0x0p0x0x1duals1v1v1x2v2x1x3},\bigraph{bwd1v1v1v1p1v1x0p1x0v0x1v1p1duals1v1v1x2v1}\right), \\ \cF &=\left(\bigraph{bwd1v1v1v1p1v1x0p0x1v1x0p1x0p0x1p0x1v0x1x0x0p0x0x1x0p0x0x0x1v1x0x0p1x0x0p0x1x0p0x0x1v1x0x0x0p0x0x1x0p0x0x0x1duals1v1v1x2v1x3x2x4v1x4x3x2}, \bigraph{bwd1v1v1v1p1v1x0p1x0v1x0p0x1v0x1p1x0p0x1v1x0x0p0x1x0v0x1p1x0p0x1duals1v1v1x2v1x2v2x1}\right), \\ \cB &=\left(\bigraph{bwd1v1v1v1p1v1x0p0x1v1x0p1x0p0x1p0x1v0x1x0x0p0x0x1x0v1x0p0x1v1x0p0x1duals1v1v1x2v1x3x2x4v1x2}, \bigraph{bwd1v1v1v1p1v1x0p1x0v1x0p0x1v0x1p1x0v1x0p0x1v0x1p1x0duals1v1v1x2v1x2v2x1}\right), \\ \cQ &= \left(\bigraph{bwd1v1v1v1p1p1v0x0x1duals1v1v2x1x3}, \bigraph{bwd1v1v1v1p1p1v0x0x1duals1v1v2x1x3} \right), \\ \cQ' &= \left(\bigraph{bwd1v1v1v1p1p1v0x0x1duals1v1v1x2x3}, \bigraph{bwd1v1v1v1p1p1v0x0x1duals1v1v1x2x3} \right). \end{align*} \end{thm} (As in \cite{index5-part1}, 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 result of this paper is to show that two of the above weeds do not appear as principal graphs of subfactors. These results are proved by applying a result of Jones' \cite{math/1007.1158} which uses ``quadratic tangles" planar algebra techniques. \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 subfactors, of any index, with principal graphs a translated extension of the pair $$\cF=\FSM.$$ \end{thm} \begin{rem} The techniques of this paper fail to rule out $$ \cB =\left(\bigraph{bwd1v1v1v1p1v1x0p0x1v1x0p1x0p0x1p0x1v0x1x0x0p0x0x1x0v1x0p0x1v1x0p0x1duals1v1v1x2v1x3x2x4v1x2}, \bigraph{bwd1v1v1v1p1v1x0p1x0v1x0p0x1v0x1p1x0v1x0p0x1v0x1p1x0duals1v1v1x2v1x2v2x1}\right) $$ because of the symmetry of its principal graph. See Example \ref{ex:badseeddims} for more details. \end{rem} \begin{rem} The approach of this paper is also viable for ruling out a large subset of the vines described in the first paper \cite{index5-part1}. 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. Section \ref{sec:annularandquadratic} recalls some key results from \cite{MR1929335, math/1007.1158} about the structure of annular Temperley-Lieb modules and about two equations involving the `chirality' (a certain rotational eigenvalue) and the `branch factors' (a certain ratio of two dimensions) of principal graphs. The main observation in this paper is that since the chirality of a principal graph is always a root of unity, these equations produce an inequality which can be used to eliminate certain weeds. Section \ref{sec:relativedimensions} describes what we can deduce about the dimensions of bimodules from incomplete graph information for potential principal graphs and calculates these dimensions for graphs coming from the weeds $\cC$ and $\cF$. In Section \ref{sec:indexinequalities}, we argue that the inequality of Section \ref{sec:annularandquadratic} cannot be satisfied for graphs coming from the weeds $\cC$ and $\cF$, except for a few exceptions. In Section \ref{sec:odometer} we eliminate the exceptions, which are all graphs with index below $3+\sqrt{3}$, by running the odometer described in \cite{index5-part1}. Bundled with the arXiv sources of this article are two Mathematica notebooks, \code{Crab.nb} and \code{FSM.nb}, which contain all relevant calculations for what follows. These make use of a package called FusionAtlas; see \cite{index5-part1} 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 at the University of New Hampshire, and Noah Snyder was supported in part by RTG grant DMS-0354321 and in part by an NSF Postdoctoral Fellowship at Columbia University. \subsection{Background on quadratic tangles and annular multiplicities $*10$}%\label{sec:annularandquadratic} The main technique of this paper is to apply the formula given by Jones in \cite{math/1007.1158} for the rotational eigenvalue of the annular low-weight vector with weight $n+1$ for an $n$-supertransitive subfactor with 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 $n$-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 (i.e. the principal graph begins with $n$ edges before the first branch point or multiple edge), 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 $0$. 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 two possibilities, the translates of: \begin{equation*} \cdots \bigraph{bwd1v1v1v1p1v1x0p0x1duals1v1v1x2}\qquad\text{and}\qquad \cdots \bigraph{bwd1v1v1v1p1v1x0p1x0duals1v1v1x2}. \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, 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{Relative dimensions of vertices/bimodules}\label{sec:relativedimensions} Consider $\Gamma_0$ a weed with annular multiplicities $*10$. Now consider $\Gamma$ an $n$-translate of an extension of $\Gamma_0$. Suppose that this graph comes from a subfactor of index $(q+q^{-1})^2$. In order to apply the quadratic tangles inequality from the last section, we need to write $r$ 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). \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, Inequality \ref{eq:QTinequality} (and indeed Equation \ref{eq:QTequation} with $\lambda = -1$) always holds for translations and extensions of these graphs, and the techniques of this paper cannot eliminate this weed. \end{ex} % ---------------------------------------------------------------- \newcommand{\urlprefix}{} \bibliographystyle{gtart} %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} % ----------------------------------------------------------------