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arxiv math/0405353 v3 pith:MMAF4H4V submitted 2004-05-18 math.GT math.QA

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

classification math.GT math.QA
keywords a-polynomialknotcoloredjonesknotsnon-trivialpolynomialsrepresentations
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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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