Geometry MMA Seminar: Right-angled links in thickened surfaces

Speaker:   Rose Kaplan-Kelly, George Mason University

Title:   Right-angled links in thickened surfaces

Abstract:  An overarching question in knot theory is that of how much we can learn about a link's complement (the space that remains when the link is removed) from the link's diagrams. Alternating links, links which admit diagrams in which crossings alternate between over and under, have been especially useful towards addressing this question. We traditionally study alternating links with diagrams on the 2-sphere and complements in the 3-sphere. In this talk, we will consider a generalization of alternating links and their complements in thickened surfaces. In particular, we will find a family of generalized alternating links that correspond to Euclidean and hyperbolic tilings and have a right-angled complete hyperbolic structure on their complements. We will show that a link having this type of right-angled structure is equivalent to the link having two totally geodesic checkerboard surfaces, and equivalent to a set of restrictions on the link's alternating projection diagram. We will then use these diagram restrictions to find an upper bound on the number of these links for each projection surface and to construct a right-angled knot in a thickened surface.

Time: Monday, October 30, 2023 – 10:30am-11:30am

Place: Exploratory Hall, Room 4106