Scott Morrison, Emily Peters and I have just uploaded to the arxiv our paper Knot polynomial identities and quantum group coincidences. In this paper we prove several new strange identities between certain specializations of colored Jones polynomials and other classical knot polynomials. For example, we prove that for any knot (but not for links!) the 6th colored Jones polynomial at a 28th root of unity is twice the value of a certain specialization of the HOMFLYPT polynomial (for the exact formulas see the first page of the paper).

Each of these identities of knot polynomials comes from a coincidence of small tensor categories involving the even part of one of the subfactors. For example, the above identity comes from an equivalence

To recover the knot polynomial identity, one computes the Reshetikhin-Turaev invariant for a particular object on the left (getting half the relevant colored Jones polynomial for knots, but something worse for links) and for the corresponding object on the right (getting the specialization of HOMFLYPT).

(In that equation above there’s a lot of technical terms on the right side. “Uni” means we’re using Turaev’s unimodal pivotal structure instead of the usual pivotal structure, and “modularize” means take the Bruguieres-Mueger modularization where you add isomorphisms between the trivial object and all simple objects that “behave like the trivial object.”)

For most of these coincidences we give no fewer than three separate proofs, as well as an exciting diagram which explains the proof in pictures. After the jump I’ll sketch the flavor of these different arguments.

One of the biggest difficulties in writing this paper was getting all our conventions straight because we constantly need to jump between different quantum groups, and knot polynomials. Life becomes very messy if you’re using one convention for q in one place and a different one somewhere else. So one goal of this paper is to be “full of correctness.” We hope that if you ever wanted to know about the relationship between quantum groups and diagramatics you can just look at this paper and have nice consistent conventions for everything. In particular, if you notice even small errors we’d really love to hear about them so that they can be fixed.

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