Finite type invariants of knots in homology 3-spheres
Also appears in collection : Representation spaces, Teichmüller theory, and their relationship with 3-manifolds from the classical and quantum viewpoints / Espaces de représentations, théorie de Teichmüller et leur rapport avec les 3-variétés d'un point de vue classique et quantique
For null-homologous knots in rational homology 3-spheres, there are two equivariant invariants obtained by universal constructions à la Kontsevich, one due to Kricker and defined as a lift of the Kontsevich integral, and the other constructed by Lescop by means of integrals in configuration spaces. In order to explicit their universality properties and to compare them, we study a theory of finite type invariants of null-homologous knots in rational homology 3-spheres. We give a partial combinatorial description of the space of finite type invariants, graded by the degree. This description is complete for knots with a trivial Alexander polynomial, providing explicit universality properties for the Kricker lift and the Lescop equivariant invariant and proving the equivalence of these two invariants for such knots.
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