Title: The sheaf theory of classical and virtual knots
Speaker: Micah Chrisman (OSU)
Abstract: Virtual knots are typically defined combinatorially. They are collection of planar diagrams that are considered equivalent up to a finite sequence of moves. By contrast, knots in the 3-sphere can be defined geometrically. They are the points of a space of knots. The space has a topology so that equivalent knots lie in the same path component. Here we will give geometric construction of virtual knots using sheaf theory. We define a site $(\textbf{VK}, J_{\textbf{VK}})$ so that the category $\text{Sh}(\textbf{VK},J_{\textbf{VK}})$ of sheaves on this site can be naturally interpreted as the "space of virtual knots''. The points of this Grothendieck topos correspond exactly to virtual knots. An equivalence of virtual knots corresponds to a path in this space, or more precisely, a geometric morphism $\text{Sh}([0,1]) \to \text{Sh}(\textbf{VK},J_{\textbf{VK}})$. Many other combinatorial concepts in virtual knot theory can likewise be given a geometric reformulation using the language of sheaves.