A rational noncommutative invariant of boundary links.
A Dehn sphere Σ in a closed 3-manifold M is a 2-sphere immersed in M with only double curve and triple point singularities. The Dehn sphere Σ fills M if it defines a cell decomposition of M. The inverse image in S² of the double curves of Σ is the Johansson diagram of Σ and if Σ fills M it is possible to reconstruct M from the diagram. A Johansson representation of M is the Johansson diagram of a filling Dehn sphere of M. Montesinos proved that every closed 3-manifold has a Johansson representation...
In this paper we show that given any 3-manifold and any non-fibered class in there exists a representation such that the corresponding twisted Alexander polynomial is zero. We obtain this result by extending earlier work of ours and by combining this with recent results of Agol and Wise on separability of 3-manifold groups. This result allows us to completely classify symplectic 4-manifolds with a free circle action, and to determine their symplectic cones.