Refined Chern-Simons Theory in Genus Two

TL;DR

This paper introduces a refined Chern-Simons topological quantum field theory for genus two surfaces, extending previous genus one results, and constructs explicit representations of the mapping class group with new knot polynomial computations.

Contribution

It constructs a Macdonald q,t-deformation of TQFT operators for genus two, generalizing genus one results and enabling refined knot invariants in higher genus.

Findings

Refined Jones polynomial computed for genus 2 knots.

Construction of a genus 2 mapping class group representation.

Demonstration of a q,t-deformation preserving relations beyond genus 1.

Abstract

Reshetikhin-Turaev (a.k.a. Chern-Simons) TQFT is a functor that associates vector spaces to two-dimensional genus g surfaces and linear operators to automorphisms of surfaces. The purpose of this paper is to demonstrate that there exists a Macdonald q,t-deformation -- refinement -- of these operators that preserves the defining relations of the mapping class groups beyond genus 1. For this we explicitly construct the refined TQFT representation of the genus 2 mapping class group in the case of rank one TQFT. This is a direct generalization of the original genus 1 construction of arXiv:1105.5117, opening a question if it extends to any genus. Our construction is built upon a q,t-deformation of the square of q-6j symbol of U_q(sl_2), which we define using the Macdonald version of Fourier duality. This allows to compute the refined Jones polynomial for arbitrary knots in genus 2. In contrast with genus 1, the refined Jones polynomial in genus 2 does not appear to agree with the Poincare polynomial of the triply graded HOMFLY knot homology.