Nathan Dunfield and Stavros Garoufalidis
Abstract: The A-polynomial of a knot in S^3 is a complex plane curve which is 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 the 3-sphere 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 the a conjecture connecting the Jones polynomials to the A-polynomial holds, then the colored Jones polynomials distinguishes the unknot.
Key words: A-polynomial, SL_2(C) character variety, SU(2) representations, knots.
Notes: 6 pages.