The Thurston norm, fibered manifolds and twisted Alexander polynomials

Abstract

Every element in the first cohomology group of a 3-manifold is dual to embedded surfaces. The Thurston norm measures the minimal ‘complexity’ of such surfaces. For instance the Thurston norm of a knot complement determines the genus of the knot in the 3-sphere. We show that the degrees of twisted Alexander polynomials give lower bounds on the Thurston norm, generalizing work of McMullen and Turaev. Our bounds attain their most concise form when interpreted as the degrees of the Reidemeister torsion of a certain twisted chain complex. We show that these lower bounds give the correct genus bounds for all knots with 12 crossings or less, including the Conway knot and the Kinoshita–Terasaka knot which have trivial Alexander polynomial.

We also give obstructions to fibering 3-manifolds using twisted Alexander polynomials and detect all knots with 12 crossings or less that are not fibered. For some of these it was unknown whether or not they are fibered. Our work in particular extends the fibering obstructions of Cha to the case of closed manifolds.