A-Polynomial Seminar

Reading List: (subject to change)

Representation Theory and the A-Polynomial (Cooper, Long)

Summary: This paper surveys various aspects of the two variable polynomial of knots defined by consideration of the representations of the fundamental group of the complement into SL(2, C)

On the AJ Conjecture for Knots (Le, Tran)

Summary: The AJ Conjecture confirmed for some classes of two-bridge knots and pretzel knots.

A DIAGRAMMATIC APPROACH TO THE AJ CONJECTURE (Detcherry, Garoufalidis)

Summary: We introduce a version of the Aˆ-polynomial that depends on a planar diagram of a knot (that conjecturally agrees with the Aˆ-polynomial) and we prove that it satisfies one direction of the AJ Conjecture. Our proof uses the octahedral decomposition of a knot complement obtained from a planar projection of a knot, the R-matrix state sum formula for the colored Jones polynomial, and its certificate.

NON-TRIVIALITY OF THE A-POLYNOMIAL FOR KNOTS IN S3 (Dunfield, Garoufalidis)

Summary: 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.

Other Resources:

• "Remarks on the A-Polynomial of a Knot" - Cooper, Long

• "Plane curves associated to character varieties of 3-manifolds" - Cooper, Culler, Gillet, Long, & Shalen