Knot polynomial identities and quantum group coincidences

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.

50 pages, many figures (this version corrects a sign error in the G_2 braiding)

Cited by in corpus (6)

• Webs and quantum skew Howe duality
• Skein theory for the D_{2n} planar algebras
• Quadratic Tangles in Planar Algebras
• On constraints for knots to admit chirally cosmetic surgeries and their calculations
• Drinfel'd doubles of the $n$-rank Taft algebras and a generalization of the Jones polynomial
• Interpolation categories for Conformal Embeddings