Proof of a stronger version of the AJ Conjecture for torus knots

For a knot $K$ in $S^{3}$ , the ${sl}_{2}$ –colored Jones function $J_{K} (n)$ is a sequence of Laurent polynomials in the variable $t$ that is known to satisfy non-trivial linear recurrence relations. The operator corresponding to the minimal linear recurrence relation is called the recurrence polynomial of $K$ . The AJ Conjecture (see Garoufalidis [Proceedings of the Casson Fest (2004) 291–309]) states that when reducing $t = - 1$ , the recurrence polynomial is essentially equal to the $A$ –polynomial of $K$ . In this paper we consider a stronger version of the AJ Conjecture, proposed by Sikora [arxiv:0807.0943], and confirm it for all torus knots.

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Keywords

colored Jones polynomial, $A$–polynomial, AJ Conjecture

Mathematical Subject Classification 2010

Primary: 57N10

Secondary: 57M25

References

Publication

Received: 22 November 2011

Accepted: 29 October 2012

Published: 23 March 2013

Authors

Anh T Tran
Department of Mathematics The Ohio State University 100 Math Tower 231 West 18th Avenue Columbus OH 43210 USA

Volume 13, issue 1 (2013)

Anh T Tran

Abstract