Reidemeister torsion, the Thurston norm and Harvey’s invariants

Stefan Friedl

Vol. 230 (2007), No. 2, 271–296

DOI: 10.2140/pjm.2007.230.271

Abstract

Cochran introduced Alexander polynomials over noncommutative Laurent polynomial rings. Their degrees were studied by Cochran, Harvey and Turaev, who gave lower bounds on the Thurston norm. We extend Cochran’s definition to twisted Alexander polynomials, and show how Reidemeister torsion relates to these invariants, giving lower bounds on the Thurston norm in terms of the Reidemeister torsion. This yields a concise formulation of the bounds of Cochran, Harvey and Turaev. The Reidemeister torsion approach also provides a natural approach to proving and extending certain monotonicity results of Cochran and Harvey.

Keywords

Thurston norm, Reidemeister torsion, 3-manifolds, knot genus

Mathematical Subject Classification 2000

Primary: 57M27

Secondary: 57N10

Milestones

Received: 31 August 2005

Revised: 13 December 2005

Published: 1 April 2007