Geodesic
Non-triviality of the A–polynomial for knots in S3
Dunfield, Nathan M1 ; Garoufalidis, Stavros2
1Mathematics 253-37, California Institute of Technology, Pasadena CA 91125, USA
2School of Mathematics, Georgia Institute of Technology, Atlanta GA 30332-0160, USA
Algebraic and Geometric Topology, Tome 4 (2004) no. 2, pp. 1145-1153
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The A–polynomial of a knot in S3 defines a complex plane curve associated to the set of representations of the fundamental group of the knot exterior into SL2ℂ. Here, we show that a non-trivial knot in S3 has a non-trivial A-polynomial. We deduce this from the gauge-theoretic work of Kronheimer and Mrowka on SU2–representations of Dehn surgeries on knots in S3. 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.

DOI : 10.2140/agt.2004.4.1145
Keywords: knot, $A$–polynomial, character variety, Jones polynomial

Dunfield, Nathan M  1   ; Garoufalidis, Stavros  2

1 Mathematics 253-37, California Institute of Technology, Pasadena CA 91125, USA
2 School of Mathematics, Georgia Institute of Technology, Atlanta GA 30332-0160, USA 
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Dunfield, Nathan M; Garoufalidis, Stavros. Non-triviality of the A–polynomial for knots in S3. Algebraic and Geometric Topology, Tome 4 (2004) no. 2, pp. 1145-1153. doi: 10.2140/agt.2004.4.1145

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