G\'en\'eralisation de l'homologie d'Heegaard-Floer aux entrelacs singuliers & Raffinement de l'homologie de Khovanov aux entrelacs restreints

TL;DR

This paper extends homological invariants like Heegaard-Floer and Khovanov homology to singular and restricted links, exploring their relations and potential for richer geometric insights.

Contribution

It generalizes Ozsvath-Szabo invariants to singular links and refines Khovanov homology for restricted links, advancing categorification and link deformation analysis.

Findings

Extended grid presentation for singular links.

Proved invariants are acyclic under certain conditions.

Provided a new tool for studying knot deformations.

Abstract

A categorification of a polynomial link invariant is an homological invariant which contains the polynomial one as its graded Euler characteristic. This field has been initiated by Khovanov categorification of the Jones polynomial. Later, P. Ozsvath and Z. Szabo gave a categorification of Alexander polynomial. Besides their increased abilities for distinguishing knots, this new invariants seem to carry many geometrical informations. On the other hand, Vassiliev works gives another way to study link invariant, by generalizing them to singular links i.e. links with a finite number of rigid transverse double points. The first part of this thesis deals with a possible relation between these two approaches in the case of the Alexander polynomial. To this purpose, we extend grid presentation for links to singular links. Then we use it to generalize Ozsvath and Szabo invariant to singular links. Besides the consistency of its definition, we prove that this invariant is acyclic under some conditions which naturally make its Euler characteristic vanish. This work can be considered as a first step toward a categorification of Vassiliev theory. In a second part, we give a refinement of Khovanov homology to restricted links. Restricted links are link diagrams up to a restricted set of Reidemeister moves. In particular, closed braids can be seen as a subset of them. Such a refinement give then a new tool for studying knots and their deformations.