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

Dunfield, Nathan M; Garoufalidis, Stavros

doi:10.2140/agt.2004.4.1145

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

Nathan M Dunfield and Stavros Garoufalidis

Algebraic & Geometric Topology 4 (2004) 1145–1153

DOI: 10.2140/agt.2004.4.1145

arXiv: math.GT/0405353

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

Mathematical Subject Classification 2000

Primary: 57M25, 57M27

Secondary: 57M50

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Publication

Received: 13 June 2004

Accepted: 16 September 2004

Published: 1 December 2004

Authors

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

Volume 4, issue 2 (2004)