Algebraic and Geometric Topology 4 (2004), paper no. 50, pages 1145-1153.

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

Nathan M. Dunfield, Stavros Garoufalidis

Abstract. 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

Keywords. Knot, A-polynomial, character variety, Jones polynomial

AMS subject classification. Primary: 57M25, 57M27. Secondary: 57M50.

DOI: 10.2140/agt.2004.4.1145

E-print: arXiv:math.GT/0405353

Submitted: 13 June 2004. Accepted: 16 September 2004. Published: 1 December 2004.

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Nathan M. Dunfield, Stavros Garoufalidis
Mathematics 253-37, California Institute of Technology
Pasadena, CA 91125, USA
School of Mathematics, Georgia Institute of Technology
Atlanta, GA 30332-0160, USA

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