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2013 On the Turaev–Viro endomorphism and the colored Jones polynomial
Xuanting Cai, Patrick M Gilmer
Algebr. Geom. Topol. 13(1): 375-408 (2013). DOI: 10.2140/agt.2013.13.375


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.


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Xuanting Cai. Patrick M Gilmer. "On the Turaev–Viro endomorphism and the colored Jones polynomial." Algebr. Geom. Topol. 13 (1) 375 - 408, 2013.


Received: 20 February 2012; Revised: 9 September 2012; Accepted: 17 September 2012; Published: 2013
First available in Project Euclid: 19 December 2017

zbMATH: 1270.57016
MathSciNet: MR3031645
Digital Object Identifier: 10.2140/agt.2013.13.375

Primary: 57M25 , 57M27 , 57R56

Keywords: $3$–manifold , knot , quantum invariant , strong shift equivalence , surgery presentation , TQFT

Rights: Copyright © 2013 Mathematical Sciences Publishers


Vol.13 • No. 1 • 2013
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