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

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 ${SL}_2ℂ$. 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.