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

We study the behavior of the signature of colored links [Flo05, CF08] under the splice operation. We extend the construction to colored links in integral homology spheres and show that the signature is almost additive, with a correction term independent of the links. We interpret this correction term as the signature of a generalized Hopf link and give a simple closed formula to compute it.

We contruct the signature of a µ-colored oriented link, as a locally constant integer valued function with domain (S 1 − {1}) µ . It restricts to the Tristram-Levine's signature on the diagonal and the discontinuities can occur only at the zeros of the colored Alexander polynomial. Moreover, the signature and the related nullity verify the Murasugi-Tristram inequality. This gives a new necessary condition for a link to bound a smoothly and properly embedded surface in B 4 , with given Betti numbers. As an application, we achieve the classification of the complex orientations of maximal plane non-singular projective algebraic curves of degree 7, up to isotopy.

No abstract