Quantum Topology Seminar
Micah Chrisman
The Ohio State University
A geometric foundation of virtual knot theory (Episode II)
Abstract: Last time, we reinterpreted the language of (classical) geometric knot theory in terms of sheaf theory. The space of knots in R^3 was reinterpreted as the Grothendieck topos Sh(K) of sheaves on the knot space. Knots themselves can be recovered from the geometric morphisms from the category of Sets to Sh(K). These are called the points of the Grothendieck topos Sh(K). The isotopy relation is generated by geometric morphisms Sh([0,1]) to Sh(K) and knot invariants are geometric morphisms from Sh(K) to Sh(G), where G is a discrete topological space. This motivated the definition of the "virtual knot space" as a Grothendieck topos Sh(VK,J_{VK}) of sheaves on a site. In Episode II, we will review the basics of this construction and show how to recover virtual knots as points of the "virtual knot space". Similarly, we will show that the virtual isotopy relation is generated by geometric morphisms Sh([0,1]) to Sh(VK,J_{VK}). All virtual knot invariants are realizable as geometric morphisms from Sh(VK,J_{VK}) to Sh(G). In this way, we obtain a model of virtual knot theory that is geometric in the same sense that classical knot theory is geometric.
Thursday February 16, 2023 at 12:00 PM in Zoom