Ce jeudi 29 juin à 14h, en salle de séminaire du 2e étage de la FRUMAM, le séminaire de Géométrie et Topologie accueille Micah Chrisman, de Ohio State University.

A geometric foundation for virtual knot theory

The theory of knots in R^3 has both a geometric model and a diagrammatic model. The geometric model views a knot as point in a space K of knots, where two knots are equivalent if they lie in the same path component of K. Alternatively, two knot diagrams are equivalent if they are related by a finite sequence of Reidemeister moves. Although the two models produce the same set of equivalence classes (i.e. the knot types), one cannot safely dispense with one and rely exclusively on the other. The geometric model is needed for Chern-Simons theory and in the derivation of finite-type invariants from the cohomology of the knot space. Practical calculation of quantum invariants, however, requires the diagrammatics of skein theory and quantum groups. In the mid 1990s, Kauffman introduced a general framework for the study of knot diagrammatics. It investigates an expanded set of non-planar knot diagrams, called virtual knot diagrams, which are considered equivalent up to an expanded set of Reidemeister moves. In this talk, we will use sheaf theory to give a complementary geometric model for virtual knot theory. We define a site (VK,J_VK) so that the Grothendieck topos Sh(VK,J_VK) of sheaves on this site can be naturally interpreted as the “space of virtual knots”. A point of the space of virtual knots, that is a geometric morphism Sets → Sh(VK,J_VK), is exactly a virtual knot. The virtual isotopy relation and all virtual knot invariants are likewise realizable as geometric morphisms. This gives a model for virtual knot theory which is geometric in the same logical sense that classical knot theory is geometric.