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Non-triviality of the A-polynomial for knots in S^3

Dunfield, Nathan M. and Garoufalidis, Stavros (2004) Non-triviality of the A-polynomial for knots in S^3. Algebraic and Geometric Topology, 4 (2004). pp. 1145-1153. ISSN 1472-2747.

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The A-polynomial of a knot in S^3 defines a complex plane curve associated to the set of representations of the fundamental group of the knot exterior into SL(2,C). Here, we show that a non-trivial knot in S^3 has a non-trivial A-polynomial. We deduce this from the gauge-theoretic work of Kronheimer and Mrowka on SU_2-representations of Dehn surgeries on knots in S^3. As a corollary, we show that if a conjecture connecting the colored Jones polynomials to the A-polynomial holds, then the colored Jones polynomials distinguish the unknot.

Item Type:Article
Additional Information:Submitted: 13 June 2004. Accepted: 16 September 2004. Published: 1 December 2004. Both authors were partially supported by the U.S. National Science Foundation, and Dunfield was also partially supported by the Sloan Foundation.
Subject Keywords:Knot, A-polynomial, character variety, Jones polynomial
Record Number:CaltechAUTHORS:DUNagt04
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:1013
Deposited By: Archive Administrator
Deposited On:29 Nov 2005
Last Modified:26 Dec 2012 08:42

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