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(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

Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-50.abs.html

References in corpus (3)

• On the characteristic and deformation varieties of a knot
• The colored Jones function is q-holonomic
• Framed knot contact homology

Cited by in corpus (24)

• Knot contact homology
• Quantum Riemann Surfaces in Chern-Simons Theory
• Instantons and L-space surgeries
• Metabelian SL(n,C) representations of knot groups
• Detection of knots and a cabling formula for A-polynomials
• Conormal bundles, contact homology and knot invariants
• A topological introduction to knot contact homology
• SU(2)-cyclic surgeries and the pillowcase
• Mutation and the colored Jones polynomial
• Splicing and the SL(2,C) Casson invariant
• Rationality of the SL(2,C)-Reidemeister torsion in dimension 3
• Non-triviality of the $M$-degree of the $A$-polynomial
• Framed knot contact homology
• The AJ-Conjecture for Cables of Two Bridge Knots
• The A-polynomial 2-tuple of twisted Whitehead links
• The A-polynomial And Holonomy Perturbations
• Difference and differential equations for the colored Jones function
• Finiteness of a section of the $SL(2,\mathbb{C})$-character variety of knot groups
• A Slope invariant and the A-polynomial of knots
• Eigenvalue varieties of Brunnian links
• Augmentations and link group representations
• Torus knots, the A-polynomial, and SL(2,C)
• On the AJ conjecture for cables of the figure eight knot
• Winding numbers and SU(2)-representations of knot groups