%!TEX root = ../article.tex

The goal of this paper is to construct knot and link invariants from the $\mathcal{D}_{2n}$ subfactor planar algebras, and to use these invariants to prove new identities between quantum group knot polynomials. These identities relate certain specializations of colored Jones polynomials to specializations of other knot polynomials.  In particular we prove that for any knot $K$ (but
not for a link!),
\begin{align}
\restrict{\J{\SL{2}}{(2)}(K)}{q=\exp(\frac{2\pi i}{12})} & = 2 \J{\mathcal{D}_4}{P}(K)  \notag \\
& = 2, \tag{Theorem \ref{thm:identities-2}}\\
\restrict{\J{\SL{2}}{(4)}(K)}{q=\exp(\frac{2\pi i}{20})} & = 2 \J{\mathcal{D}_6}{P}(K) \notag \\ & = 2 \restrict{\J{\SL{2}}{(1)}(K)}{q=\exp(- \frac{2\pi i}{10})}, \tag{Theorem \ref{thm:identities-3}}\\
\restrict{\J{\SL{2}}{(6)}(K)}{q=\exp(\frac{2\pi i}{28})} & = 2 \J{\mathcal{D}_8}{P}(K) \notag \\
                & = 2 \HOMFLY(K)(\exp(2\pi i\frac{5}{7}), \exp(- \frac{2\pi i}{14}) - \exp(\frac{2\pi i}{14})), \tag{Theorem \ref{thm:identities-4}} \displaybreak[1] \\
\restrict{\J{\SL{2}}{(8)}(K)}{q=\exp(\frac{2\pi i}{36})} & = 2 \J{\mathcal{D}_{10}}{P}(K) \notag \\
        & = 2  \Kauffman(K) (\exp(2 \pi i \frac{31}{36}), \exp(2 \pi i \frac{25}{36}) + \exp(2 \pi i \frac{11}{36}))\notag  \\
                & = 2 \restrict{\Kauffman(K) (-iq^7,i(q-q^{-1}))}{q=-\exp (\frac{-2 \pi i}{18})}  \tag{Theorem \ref{thm:identities-5}}  \\
\intertext{and}                
\restrict{\J{\SL{2}}{(12)}(K)}{q=\exp(\frac{2\pi i}{52})} & = 2 \J{\mathcal{D}_{14}}{P}(K) \notag \\
& = 2\restrict{\J{G_2}{V_{(10)}}(K)}{q=\exp{2\pi i \frac{23}{26}}}, \tag{Theorem \ref{thm:G2-links}}
\end{align}
where $\J{\SL{2}}{(k)}$ denotes the $k^{\text{th}}$ colored Jones polynomial, $\J{G_2}{V_{(1,0)}}$ denotes the knot invariant associated to the $7$-dimensional representation of $G_2$ and $\J{\mathcal{D}_{2n}}{P}$ is the $\mathcal{D}_{2n}$ link invariant for which we give a skein-theoretic construction.\footnote{Beware, the $\mathcal{D}_{2n}$ planar algebra is  not related to the lie algebra $\mathfrak{so}_{4n}$ with Dynkin diagram $D_{2n}$ but is instead a quantum subgroup of $U_q(\mathfrak{su}_2)$.}    (For our conventions for these polynomials, in particular their normalizations, see Section \ref{conventions}.)

These formulas should appear somewhat mysterious, and much of this paper is concerned with discovering the explanations for them. It turns out that each of these knot invariant identities comes from a coincidence of small modular categories involving the even part of one of the $\mathcal{D}_{2n}$. Just as families of finite groups have coincidences for small values (for example, the isomorphism between the finite groups $\text{Alt}_5$ and $\mathbf{PSL}_2(\mathbb{F}_5)$ or the outer automorphism of $S_6$), modular categories also have small coincidences.  Explicitly, we prove the following coincidences, where $\frac{1}{2} \mathcal{D}_{2n}$ denotes the even part of $\mathcal{D}_{2n}$ (the first of these coincidences is well-known).

\begin{itemize}
\item $\frac{1}{2} \mathcal{D}_{4} \cong \Rep{\mathbb{Z}/3}$, sending $P$ to $\chi_{\exp({\frac{2 \pi i}{3}})}$ (but an unusual braiding on $\Rep{\mathbb{Z}/3}$!).
\item $\frac{1}{2} \mathcal{D}_{6} \cong \uRep{U_{s = \exp({ 2 \pi i \frac{7}{10}})}(\mathfrak{sl}_2 \oplus \mathfrak{sl}_2)}^{modularize}$, sending $P$ to  $V_{1} \boxtimes V_{0}$. (See Theorem \ref{thm:D6-coincidence}, and \S \ref{sec:ribbonfunctors}.)
\item $\frac{1}{2} \mathcal{D}_{8} \cong \uRep{U_{s=\exp({2 \pi i \frac{5}{14} })}(\mathfrak{sl}_4)}^{modularize}$, sending $P$ to $V_{(100)}$. (See Theorem \ref{thm:D8-coincidence}.)
\item $\frac{1}{2} \mathcal{D}_{10}$ has an order $3$ automorphism: $\baselinetrick{\xymatrix@C-8mm@R-3mm{P \ar@{|->}@/^/[rr] & & Q \ar@{|->}@/^/[dl] \\ & f^{(2)} \ar@{|->}@/^/[ul]&}}$. (See Theorem \ref{thm:D10-coincidence}.)
\item $\frac{1}{2} \mathcal{D}_{14} \cong \Rep{U_{\exp({2 \pi i\frac{23}{26}})}(\mathfrak{g}_2)}$ sending $P$ to $V_{(10)}$. (See Theorem \ref{thm:G2}.)
\end{itemize}

To interpret the right hand sides of these equivalences, recall that the definition of the braiding (although not of the quantum group itself) depends on a choice of $s = q^{\frac{1}{L}}$, where $L$ is the index of the root lattice in the weight lattice.  Furthermore, the ribbon structure on the category of representations depends on a choice of a certain square root. In particular, besides the usual pivotal structure there's also another pivotal structure, which is called ``unimodal'' by Turaev \cite{MR1292673} and discussed in \S \ref{sec:quantumgroups} below.  By ``modularize" we mean take the modular quotient described by Brugui\`eres \cite{MR1741269} and M{\"{u}}ger \cite{MR1749250} and recalled in \S \ref{sec:modularization} below.

We first prove the knot polynomial identities directly, and later we give more conceptual explanations of the coincidences using
\begin{itemize}
\item coincidences of small Dynkin diagrams,
\item level-rank duality, and 
\item Kirby-Melvin symmetry.
\end{itemize}
These conceptual explanations do not suffice for the equivalence between the even part of $\mathcal{D}_{14}$ and $\Rep{U_{\exp({2 \pi i\frac{\ell}{26}})}(\mathfrak{g}_2)}$, $\ell=-3$ or $10$, which deserves further exploration.  Nonetheless we can prove this equivalence using direct methods (see Section \ref{sec:recognizeD}), and it answers a conjecture of Rowell's  \cite{MR2414692} concerning the unitarity of  $(G_2)_{\frac{1}{3}}$.

We illustrate each of these coincidences of tensor categories with diagrams of the appropriate quantum group Weyl alcoves; see in particular Figures \ref{fig:SO4}, \ref{fig:SO6}, \ref{fig:SO8} and \ref{fig:SO8-4fold} at the end of the paper. An ambitious reader might jump to those diagrams and try to understand them and their captions, then work back through the paper to pick up needed background or details.

%\begin{figure}[!ht]
%\begin{equation*}
%\mathfig{0.5}{coincidences/SO4-chamber}
%\end{equation*}
%\caption{This figure gives the correspondence between simple objects in $\frac{1}{2} \mathcal{D}_{6}$ and $\vRep{U_{q = -\exp({-\frac{1}{5} 2 \pi i})}(\mathfrak{so}_4)} // (V_{3e_1})$ given by level-rank duality.  Since the Dynkin diagrams $\mathcal{D}_2$  and $A_1 \times A_1$ coincide we can replace $\mathfrak{so}_4$ by $\mathfrak{sl}_2 \oplus \mathfrak{sl}_2$; the four labelled corner vertices in the diagram show the names of the corresponding representations of $\mathfrak{sl}_2 \oplus \mathfrak{sl}_2$. In particular the representation $V_{3e_1}$ of $\mathfrak{so}_4$ is sent to $V_3 \boxtimes V_3$.
%The vector representations are circled. The region that is not grayed out is a fundamental domain for the deequivariantization by the involution $\tensor (V_3 \boxtimes V_3)$, which acts by a rotation by $\pi$.   The arrows represent the additional Kirby-Melvin symmetry which can be realized as an additional deequivariantization in $\uRep{U_{s = \exp({\frac{7}{10} 2 \pi i})}(\mathfrak{sl}_2 \oplus \mathfrak{sl}_2)}$ (note here we enlarge to the full representation category, not just the vector representations).  Combining all of this we see that $\frac{1}{2} \mathcal{D}_{6} \cong \uRep{U_{s = \exp({\frac{7}{10} 2 \pi i})}(\mathfrak{sl}_2 \oplus \mathfrak{sl}_2)}^{modularize}$, and that the functor sends $P \mapsto V_{1} \boxtimes V_{0}$.
%}
%\label{fig:SO4}
%\end{figure}

In more detail the outline of the paper is as follows.  In the background section we recall some important facts about planar algebras, tensor categories, quantum groups, knot invariants and their relationships.  We fix our conventions for knot polynomials.  We also briefly recall several key concepts like semisimplification, deequivariantization, and modularization.

In Section \ref{sec:invariants} we use the skein theoretic description of $\mathcal{D}_{2n}$ to show that the Kauffman bracket gives a braiding up to sign for $\mathcal{D}_{2n}$, and in particular gives a braiding on the even part (this was already known; see for example the description of $\mathrm{Rep}^0 A$ in \cite[p. 33]{MR1936496}).  Using this, we define and discuss some new invariants of links which are the $\mathcal{D}_{2n}$ analogues of the colored Jones polynomials.  We also define some refinements of these invariants for multi-component links.

In Section \ref{sec:identities}, we discuss some identities relating the $\mathcal{D}_{2n}$ link invariants at small values of $n$ to other link polynomials.  This allows us to prove the above identities between quantum group invariants of knots.  The main technique is to apply the following schema to an object $X$ in a ribbon category (where $A$ and $B$ always denote simple objects),
\begin{itemize}
\item if $X \otimes X=A$ then the knot invariant coming from $X$ is trivial,
\item if $X \otimes X = 1\oplus A$ then the knot invariant coming from $X$ is a specialization of the Jones polynomial,
\item if $X \otimes X = A\oplus B$ then the knot invariant coming from $X$ is a specialization of the HOMFLYPT polynomial,
\item if $X \otimes X = 1\oplus A \oplus B$ then the knot invariant coming from $X$ is a specialization of the Kauffman polynomial or the Dubrovnik polynomial,
\end{itemize}
Furthermore we give formulas that identify which specialization occurs.  This technique is due to Kauffman, Kazhdan, Tuba, Wenzl, and others \cite{MR2132671, MR1237835, MR958895}, and is well-known to experts. We also use a result of Kuperberg's which gives a similar condition for specializations of the $G_2$ knot polynomial.

In Section \ref{sec:coincidences}, we reprove the results of the previous section using coincidences of Dynkin diagrams, generalized Kirby-Melvin symmetry, and level-rank duality.  In particular, we give a new simple proof of Kirby-Melvin symmetry which applies very generally, and we use a result of Wenzl and Tuba to strengthen Beliakova and Blanchet's statement of $SO$ level-rank duality.

We'd like to thank Stephen Bigelow, Vaughan Jones, Greg Kuperberg, Nicolai Resh\-e\-ti\-khin, Kevin Walker, and Feng Xu for helpful conversations.  During our work on this paper, Scott Morrison was at Microsoft Station Q, Emily Peters was supported in part by NSF grant DMS0401734 and Noah Snyder was supported in part by RTG grant DMS-0354321 and by an NSF postdoctoral grant.
