Hodge–type structures as link invariants

Abstract

Based on some analogies with the Hodge theory of isolated hypersurface sin- gularities, we define Hodge-type numerical invariants of any, not necessarily algebraic, link in S 3 . We call them H-numbers. They contain the same amount of information as the (normalized) real Seifert matrix. We study their basic properties, and we express the Tristram-Levine signatures and the higher order Alexander polynomial in terms of them. Motivated by singularity theory, we also introduce the spectrum of the link (determined from these H-numbers), and we establish some semicontinuity properties for it. These properties can be related with skein-type relations, although they are not so precise as the classical skein relations.