Generalized Seifert surfaces and signatures of colored links

Abstract

In this paper, we use `generalized Seifert surfaces' to extend the Levine-Tristram signature to colored links in S^3. This yields an integral valued function on the m-dimensional torus, where m is the number of colors of the link. The case m=1 corresponds to the Levine-Tristram signature. We show that many remarkable properties of the latter invariant extend to this m-variable generalization: it vanishes for achiral colored links, it is `piecewise continuous', and the places of the jumps are determined by the Alexander invariants of the colored link. Using a 4-dimensional interpretation and the Atiyah-Singer G-signature theorem, we also prove that this signature is invariant by colored concordance, and that it provides a lower bound for the `slice genus' of the colored link.