On the Turaev–Viro endomorphism and the colored Jones polynomial

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 the 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.