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Formules combinatoires pour les invariants d'objets noués

Abstract : This thesis focuses on low-dimensional topology, and more specifically on the invariants of various knotted objects : knots and links, 3-manifolds, welded objects, surfaces in dimension 4. The purpose of this work is the study of the various interactions between these invariants, whose constructions are very diverse, from a mainly combinatorial point of view. The thesis is made up of two completely independent chapters.In the first chapter, we give a surgery formula for the Casson-Walker-Lescop invariant of closed 3-manifolds, using combinatorial tools from the theory of finite type invariants. For this, we use the Konstevich-LMO universal invariants and the combinatorics of Jacobi diagrams. We obtain several low-degree intermediate results on the Kontsevich integral, by determining which combination extracted from diagrams recognizes classical invariants of knot theory. We also show how the Conway coefficients are read in this integral, and we give diagrammatic results of factorization of the Kontsevich coefficients.The second chapter gives, in low degrees, characterization results for finite type invariants of ribbon surfaces in dimension 4. For this, we classify ribbon knotted annuli modulo an equivalence relation called RC-equivalence, which is declined according to a degree parameter: for degrees 1, 2 and 3, this classification is obtained by classical invariants of ribbon surfaces (Alexander, Milnor). A preparatory work for degree 4 is also presented, showing that the tools used for the lower degrees are no longer sufficient. These topological results are obtained as corollaries of analogous results for the diagrammatic theory of long welded links. Indeed, the latter are linked to the ribbon surfaces via a surjective map, the Tube map, which preserves a certain number of structure and equivalence relations. The major part of the work therefore consists in studying the combinatorics of these diagrams, by means of arrow calculus, which is a welded analogue of Habiro's theory of claspers. Among several intermediate results, we study the group structure of welded long links up to w-equivalence.
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Adrien Casejuane. Formules combinatoires pour les invariants d'objets noués. Combinatoire [math.CO]. Université Grenoble Alpes [2020-..], 2021. Français. ⟨NNT : 2021GRALM013⟩. ⟨tel-03327714⟩

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