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

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

2012-02-08

arxiv.org/abs/1202.1815v2

Mathematics

Geometric Topology

26 pages, 18 figures, some typos corrected and other minor changes made

Scientific paper

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 first 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.

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