Non-commutative multivariable Reidemester torsion and the Thurston norm

Abstract

Given a 3–manifold the second author defined functions ${\delta }_n:H^1\left(M;\mathbb{Z}\right)\to \mathbb{N}$, generalizing McMullen’s Alexander norm, which give lower bounds on the Thurston norm. We reformulate these invariants in terms of Reidemeister torsion over a non-commutative multivariable Laurent polynomial ring. This allows us to show that these functions are semi-norms.