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Non-triviality of the A-polynomial for knots in S³

Dunfield, Nathan M. and Garoufalidis, Stavros (2004) Non-triviality of the A-polynomial for knots in S³. Algebraic and Geometric Topology, 4 (2004). pp. 1145-1153. ISSN 1472-2747. doi:10.2140/agt.2004.4.1145.

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The A-polynomial of a knot in S³ defines a complex plane curve associated to the set of representations of the fundamental group of the knot exterior into SL₂C. Here, we show that a non-trivial knot in S³ has a non-trivial A-polynomial. We deduce this from the gauge-theoretic work of Kronheimer and Mrowka on SU₂-representations of Dehn surgeries on knots in S³. 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
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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.
Funding AgencyGrant Number
Alfred P. Sloan FoundationUNSPECIFIED
Subject Keywords:Knot, A-polynomial, character variety, Jones polynomial
Issue or Number:2004
Classification Code:AMS subject classification. Primary: 57M25, 57M27. Secondary: 57M50.
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:08 Nov 2021 19:06

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