Knot polynomial identities and quantum group coincidences

Details Knot polynomial identities and quantum group coincidences Knot polynomial identities and quantum group coincidences

2010-02-26

arxiv.org/abs/1003.0022v1

Mathematics

Quantum Algebra

50 pages, many figures

Scientific paper

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.

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