Right-angled links in thickened surfaces

Rose Kaplan-Kelly (Temple)

Abstract: Traditionally, alternating links are studied with alternating diagrams on $S^2$ in $S^3$. In this talk, we will consider links which are alternating on higher genus surfaces $S_g$ in $S_g \times I$. We will define what it means for such a link to be right-angled generalized completely realizable (RGCR) and show that this property 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 classify RGCR links according to the polygons in their checkerboard surfaces and provide a bound on the number of RGCR links for a given surface of genus $g$. Along the way, we will answer a question posed by Champanerkar, Kofman, and Purcell about links with alternating projections on the torus.