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 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.

Keywords

knot, A–polynomial, character variety, Jones polynomial

Mathematical Subject Classification

Primary: 57M25, 57M27

Secondary: 57M50

References

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Publication

Received: 13 June 2004
Accepted: 16 September 2004
Published: 1 December 2004

Authors