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

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

2004-05-18

arxiv.org/abs/math/0405353v3

Algebr. Geom. Topol. 4 (2004) 1145-1153

Mathematics

Geometric Topology

Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-50.abs.html

Scientific paper

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

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