Titre de la thèse: Motivic knot theory
Résumé:
We introduce a counterpart in algebraic geometry to knot theory. Since this new theory uses motivic homotopy theory (specifically, quadratic intersection theory), we name it motivic knot theory. We focus on motivic linking, which means that we study how two disjoint closed F-subschemes of an ambient F-scheme can be intertwined, i.e. linked together (where F is a perfect field). In knot theory, the linking number of an oriented link with two components (i.e. of two disjoint oriented knots) is an integer which counts how many times one of the components turns around the other component. We define counterparts in algebraic geometry to oriented links with two components and to the linking number; we call these latter counterparts quadratic linking degrees. Our quadratic linking degrees are not necessarily integers; the ones we study the most take values in the Witt group of the ground field F, which is a group of equivalence classes of symmetric bilinear forms over F (or equivalently, of quadratic forms over F, when the characteristic of F is different from 2). After answering questions which naturally arise from these quadratic linking degrees, we devise methods to compute them. These methods rely on explicit formulas for the residue morphisms of Milnor-Witt K-theory (from which boundary maps for the Rost-Schmid complexes are constructed) and for the intersection product of the Rost-Schmid ring (and in particular of the Chow-Witt ring). Using these methods, we explicitly compute our quadratic linking degrees on examples. Some of these examples are inspired by knot theory, specifically by torus links (including the Hopf and Solomon links).
