Morse-Novikov theory for links

Abstract

For a compact 3-manifold W. Thurston introduced a norm on the first cohomology group of the manifold. The unit ball $B$ of this norm is a polyhedron and the set of cohomology classes that are representable by fibrations over a circle is a union of cones on some of the open faces of $B$. In the present paper we study the fibred faces of the Thurston polyhedra of exteriors of links in $S^3$. Our approach is based on the non-abelian Novikov homology associated with the universal covering of the exterior of the link. We prove in particular that for a 2-component 2-bridge link $L$ a cohomology class $\xi\in H^1(E(L))$ can be represented by a fibration over a circle if and only if its 2-variable Alexander polynomial is $\xi$-monic. We compute the Morse-Novikov numbers for a majority of 2-component prime links with number of crossings $\leq 8$.