22 feb 2018 -- 15:00
Aula N, Polo Fibonacci, Pisa
Seminari dei Baby-Geometri
Abstract.
A knot is a connected compact smooth sub-manifold of $\mathbb{S}^3$ (more in general in a three manifold, but we will be interested only in $\mathbb{S}^3$). Two knots are \emph{concordant} if they bound a properly embedded cylinder in $\mathbb{S}^3 \times I$. It turns out that the knots up to concordance form an Abelian group with respect to the connected sum. The study of the concordance group of knots is a central topic in knot theory, and low dimensional topology in general. There are quite some problems related to the study of this group; for instance, computating the slice genus of a knot or understanding if a given knot is concordant to a specific type of knot (e.g. alternating, quasi alternating etc.). Originally, these problems have been tackled with classical invariants (e.g. signatures), or reduced to a more algebraic framework (e.g. the study of the algebraic concordance group). These methods are still in use and provide useful information on the concordance group, however with the introduction of link homologies new invariants came up.
In this seminar, after a general introduction to the concordance group, and to the above mentioned problems, we will talk about a large family of concordance invariants defined from link homologies. In particular, I wish to highlight some features of these invariants, their properties and some of their "faults". Time permitting, I will also spend few words on the generalisation of these invariants to links (which is a part of joint work with Alberto Cavallo).