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.

Notes on file formats

Nathan M. Dunfield, Stavros Garoufalidis
Mathematics 253-37, California Institute of Technology
Pasadena, CA 91125, USA
and
School of Mathematics, Georgia Institute of Technology
Atlanta, GA 30332-0160, USA
Email: dunfield@caltech.edu, stavros@math.gatech.edu
URL: http://www.its.caltech.edu/~dunfield, http://www.math.gatech.edu/~stavros

AGT home page

EMIS/ELibM Electronic Journals

Outdated Archival Version

These pages are not updated anymore. They reflect the state of 21 Apr 2006. For the current production of this journal, please refer to http://msp.warwick.ac.uk/.