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Counting spatial configurations in odd dimension with the Reidemeister torsion

Abstract : In this thesis, long knots are embeddings of the Euclidean space R^n in an asymptotic homology R^{n+2} with a standard behaviour near infinity, for some odd integer n. We define invariants (Z_k)_{k>1} of these knots up to ambient diffeomorphisms that are standard outside a ball. These invariants are a generalization to this wider setting of an invariant (Z_k)_{k>1} of Bott, Cattaneo, and Rossi for long knots in mathbb R^{n+2}. Our definition also applies to long knots in R^{n+2} and it is more flexible than the original one.The Bott-Cattaneo-Rossi invariant Z_k is a linear combination of some integrals of differential forms over configuration spaces associated to some graphs with 2k vertices of two kinds, and 2k edges of two kinds.These forms are products of pullbacks of some (n+1)-forms (called external propagating forms) on the two-point configuration space of the ambient space and of some (n-1)-forms (called internal propagating forms) on the two-point configuration space of R^n.In a dual way, we define Z_k as a combination of algebraic intersections of preimages of some propagating chains in these two two-point configuration spaces.We prove a formula for Z_k in terms of linking numbers of some cycles in a surface bounded by the knot, for virtually rectifiable knots.The class of virtually rectifiable long knots contains at least the long ribbon knots, and all the long knots when n = 1 mod 4.Our formula yields an expression of the Reidemeister torsion in terms of the Z_k.
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David Leturcq. Counting spatial configurations in odd dimension with the Reidemeister torsion. K-Theory and Homology [math.KT]. Université Grenoble Alpes [2020-..], 2020. English. ⟨NNT : 2020GRALM025⟩. ⟨tel-03012456⟩

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