6 may 2015 -- 16:00
Sala Seminari, DM, Pisa
Seminari di Geometria, Pisa
Abstract.
The splice of two links is an operation defined by Eisenbund and Neumann which generalizes several other operations on links, such as the connected sum, the cabling or the disjoint union. The precise definition will be given in the talk but the rough idea goes as follows: the splice of the links L’ and L'' along the components K' and K'' is the link (L' \ K') U (L''\ K'') obtained by identifying the exterior of K' with the exterior of K''. There has been much interest to understand the behavior of different link invariants under the splice operation (genus, fiberability, Conway polynomial, Heegaard-Floer homology among others) and the goal of this talk is to present a formula relating the colored signature of the splice of two oriented links to the colored signatures of its two constituent links. As an immediate consequence, we have that the conventional univariate Levine-Tristram signature of a splice depends, in general, on the colored (or multivariate) signatures of the summands. There is one case for which we do not have yet a closed formula. If time permits, we will discuss this case whose study leads to a generalization of Kojima's eta-function for links.This is a joint work, still in progress, with Alex Degtyarev and Vincent Florens.