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.
- Publication:
-
arXiv e-prints
- Pub Date:
- February 2010
- DOI:
- 10.48550/arXiv.1003.0022
- arXiv:
- arXiv:1003.0022
- Bibcode:
- 2010arXiv1003.0022M
- Keywords:
-
- Mathematics - Quantum Algebra;
- Mathematics - Geometric Topology;
- 18D10;
- 57M27 17B10 81R05 57R56
- E-Print:
- 50 pages, many figures (this version corrects a sign error in the G_2 braiding)