| Résume||This work is motivated by our previous study of the behavior of the signature of colored links under the splice operation. The signature is mainly additive, with a regular correction term related to the generalized Hopf links. However, this almost additivity is lost along a certain ``singular locus,'' which is the subject of our current work. To describe the extra correction term (arising as a Maslov index in Wall's non-additivity theorem), we introduce a collection of invariants of colored links, called slopes. It turns out that the slope can be represented as the ratio of two sign-refined Alexander polynomials (or rather derivatives thereof), whenever this ratio makes sense. However, experiments with the link tables show that, when both polynomials in question vanish, the rational function obtained is independent of the higher Alexander polynomials, thus providing a new link invariant. (This invariant does distinguish some of the links in the tables.) Even isolated common zeroes of the two polynomials sometimes lead to surprises, as l'Hôpital's rule does not work. Should time permit, I will discuss further properties of the new invariants and outline several ways of computing them.
This is a joint work in progress with Vincent Florens and Ana G. Lecuona.