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Benjamin Audoux
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Généralisation de l'homologie d'Heegaard-Floer
aux entrelacs singuliers
&
Raffinement de l'homologie de Khovanov
aux entrelacs restreints

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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. Ozsváth and Z. Szabó 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 Ozsváth and Szabó 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.
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Status

_- Ph.D. thesis of Université Toulouse III - Paul Sabatier
_- defended on december 5th, 2007 with thesis committee
_C. Blanchet Université Paris VII Referee
_V. Colin Université de Nantes Committee member
_T. Fiedler Université Toulouse III Supervisor
_S. Orevkov Université Toulouse III Committee member
_P. Ozsváth Columbia University Referee
_V. Vershinin Université Montpellier II Chairman
_- arXiv:0803.4478

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