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

2004/05/31 by Nathan M. Dunfield, Stavros Garoufalidis · 1 citation
Mathematics · #math.GT #math.QA #msc:57M25 #msc:57M27 #msc:57M50

paper · pdf · doi:10.2140/agt.2004.4.1145

published as Algebr. Geom. Topol. 4 (2004) 1145-1153 · Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-50.abs.html

arxiv created 2004/12/03 · arxiv updated 2014/09/30

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 SL(2,C). 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

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