Certifying the Thurston norm via SL(2, C)-twisted homology

TL;DR

This paper investigates how twisted Alexander polynomials associated with SL(2, C) representations can detect the Thurston norm and genus of hyperbolic knots, extending the understanding of knot invariants and 3-manifold topology.

Contribution

It demonstrates that hyperbolic torsion polynomials determine the genus for a broad class of hyperbolic knots, including all special arborescent knots, via SL(2, C)-twisted homology.

Findings

Hyperbolic torsion polynomial determines genus for many hyperbolic knots.

Tautness of sutured manifolds can be certified through SL(2, C)-twisted homology.

The results include knots with trivial Alexander polynomial.

Abstract

We study when the Thurston norm is detected by twisted Alexander polynomials associated to representations of the 3-manifold group to SL(2, C). Specifically, we show that the hyperbolic torsion polynomial determines the genus for a large class of hyperbolic knots in the 3-sphere which includes all special arborescent knots and many knots whose ordinary Alexander polynomial is trivial. This theorem follows from results showing that the tautness of certain sutured manifolds can be certified by checking that they are a product from the point of view of homology with coefficients twisted by an SL(2, C)-representation.