Knot polynomial identities and quantum group coincidences

TL;DR

This paper constructs link invariants from $D_{2n}$ subfactor planar algebras, revealing new identities among quantum knot polynomials and uncovering coincidences between modular categories related to Dynkin diagrams.

Contribution

It introduces novel identities connecting colored Jones polynomials and other quantum invariants via subfactor planar algebras and explains their origins through category coincidences and dualities.

Findings

New identities relating quantum knot polynomials

Coincidences between small modular categories involving $D_{2n}$

An exceptional coincidence involving $G_2$

Abstract

We construct link invariants using the $D_{2n}$ subfactor planar algebras, and use these to prove new identities relating certain specializations of colored Jones polynomials to specializations of other quantum knot polynomials. These identities can also be explained by coincidences between small modular categories involving the even parts of the $D_{2n}$ planar algebras. We discuss the origins of these coincidences, explaining the role of $SO$ level-rank duality, Kirby-Melvin symmetry, and properties of small Dynkin diagrams. One of these coincidences involves $G_2$ and does not appear to be related to level-rank duality.