#### Algebraic and Geometric Topology 4 (2004),
paper no. 50, pages 1145-1153.

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

### Nathan M. Dunfield, Stavros Garoufalidis

**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

**AMS subject classification**.
Primary: 57M25, 57M27.
Secondary: 57M50.

**DOI:** 10.2140/agt.2004.4.1145

**E-print:** `arXiv:math.GT/0405353`

Submitted: 13 June 2004.
Accepted: 16 September 2004.
Published: 1 December 2004.

Notes on file formats
Nathan M. Dunfield, Stavros Garoufalidis

Mathematics 253-37, California Institute of Technology

Pasadena, CA 91125, USA

and

School of Mathematics, Georgia Institute of Technology

Atlanta, GA 30332-0160, USA

Email: dunfield@caltech.edu, stavros@math.gatech.edu

URL: http://www.its.caltech.edu/~dunfield, http://www.math.gatech.edu/~stavros

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