The Thurston Norm, Fibered Manifolds and Twisted Alexander Polynomials

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 elegant form when interpreted as the degrees of the Reidemeister torsion of a certain twisted chain complex. The bounds are very powerful and can be easily implemented with a computer. We show that these lower bounds determine the genus of all knots with 12 crossings or less, including the Conway knot and the Kinoshita–Terasaka knot which have trivial Alexander polynomial. We also give many examples of closed manifolds and link complements where twisted Alexander polynomials detect the correct Thurston norm. 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 obstructions also extend work of Cha to the case of closed manifolds.