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

Nathan M. Dunfield; Stavros Garoufalidis

doi:10.2140/agt.2004.4.1145

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

Nathan M. Dunfield·Stavros Garoufalidis

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Abstract

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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Nathan M. Dunfield, Stavros Garoufalidis. 2004-05-18. Non-triviality of the A-polynomial for knots in S^3. https://doi.org/10.2140/agt.2004.4.1145

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