#### Volume 4, issue 2 (2004)

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

### Nathan M Dunfield and Stavros Garoufalidis

Algebraic & Geometric Topology 4 (2004) 1145–1153
 arXiv: math.GT/0405353
##### 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.

##### Keywords
knot, $A$–polynomial, character variety, Jones polynomial
##### Mathematical Subject Classification 2000
Primary: 57M25, 57M27
Secondary: 57M50