On the Turaev-Viro endomorphism, and the colored Jones polynomial

TL;DR

This paper constructs a module endomorphism for knots using TQFT techniques and shows its trace encodes the colored Jones polynomial at roots of unity, linking quantum invariants to topological structures.

Contribution

It introduces a new construction of knot endomorphisms via TQFT and relates their traces to colored Jones polynomials, extending to infinite cyclic covers.

Findings

Endomorphisms are strong shift equivalent to Turaev-Viro endomorphisms.

Traces of colored meridian and longitude endomorphisms encode colored Jones polynomials.

Results apply to infinite cyclic covers of 3-manifolds.

Abstract

By applying a variant of the TQFT constructed by Blanchet, Habegger, Masbaum, and Vogel, and using a construction of Ohtsuki, we define a module endomorphism for each knot K by using a tangle obtained from a surgery presentation of K. We show that it is strong shift equivalent to the Turaev-Viro endomorphism associated to K. Following Viro, we consider the endomorphisms that one obtains after coloring the meridian and longitude of the knot. We show that the traces of these endomorphisms encode the same information as the colored Jones polynomials of K at a root of unity. Most of the discussion is carried out in the more general setting of infinite cyclic covers of 3-manifolds.