Hodge-type structures as link invariants

Abstract

Based on some analogies with the Hodge theory of isolated hypersurface singularities, we define Hodge-type numerical invariants (called H-numbers) of any, not necessarily algebraic, link in $S^3$. They contain the same information as the (normalized) real Seifert matrix. We study their basic properties, 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.