Twisted Alexander polynomials of hyperbolic knots

Abstract

We study a twisted Alexander polynomial naturally associated to a hyperbolic knot in an integer homology 3-sphere via a lift of the holonomy representation to SL(2, C). It is an unambiguous symmetric Laurent polynomial whose coefficients lie in the trace field of the knot. It contains information about genus, fibering, and chirality, and moreover is powerful enough to sometimes detect mutation. We calculated this invariant numerically for all 313,209 hyperbolic knots in S^3 with at most 15 crossings, and found that in all cases it gave a sharp bound on the genus of the knot and determined both fibering and chirality. We also study how such twisted Alexander polynomials vary as one moves around in an irreducible component X_0 of the SL(2, C)-character variety of the knot group. We show how to understand all of these polynomials at once in terms of a polynomial whose coefficients lie in the function field of X_0. We use this to help explain some of the patterns observed for knots in S^3, and explore a potential relationship between this universal polynomial and the Culler-Shalen theory of surfaces associated to ideal points.

37 pages; V2: Added Section 6.6 with experimental results on related adjoint torsion polynomial. V3: Minor edits

References in corpus (12)

• The asymptotics of the Ray-Singer analytic torsion of hyperbolic 3-manifolds
• Poincaré duality and degrees of twisted Alexander polynomials
• Mutation and the colored Jones polynomial
• Higher-Order Polynomial Invariants of 3-Manifolds Giving Lower Bounds for the Thurston Norm
• Twisted Alexander polynomials and character varieties of 2-bridge knot groups
• Approximations to the volume of hyperbolic knots
• Evaluations of the twisted Alexander polynomials of 2-bridge knots at $\pm 1$
• Twisted Alexander norms give lower bounds on the Thurston norm
• Polynomial recurrences and cyclic resultants
• Dynamics of Twisted Alexander Invariants
• Thurston norm, fibered manifolds and twisted Alexander polynomials
• Reidemeister torsion, twisted Alexander polynomial and fibered knots

Cited by in corpus (3)

• Poincaré duality and degrees of twisted Alexander polynomials
• On the twisted Alexander polynomial for representations into SL_2(C)
• Character varieties of once-punctured torus bundles with tunnel number one