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 $\mu$-dimensional torus, where $\mu$ is the number of colors of the link. The case $\mu =1$ corresponds to the Levine-Tristram signature. We show that many remarkable properties of the latter invariant extend to this $\mu$-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.