ON DUAL UNIT BALLS OF THURSTON NORMS

Abstract

Thurston norms are invariants of 3-manifolds defined on their second homology groups, and understanding the shape of theirs dual unit ball is a (widely) open problem. W. Thurston showed that every symmetric polygon in Z^2 whose vertices have the same parity is the dual unit ball of a Thurston norm on a 3-manifold. However, it is not known if the parity condition on the vertices of polyhedra is a sufficient condition in higher dimension or if their are polyhedra with mod 2 congruent vertices which are not dual unit ball of any Thurston norm. In this article, we provide a family of polytopes in Z^2g which can be realized as dual unit balls of Thurston norms on 3-manifolds. Those polytopes come from intersection norms on oriented closed surfaces and this article widens the bridge between those two norms.