Volume 4, issue 2 (2004)

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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

arXiv: math.GT/0405353

Abstract

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 SL2. 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.

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/