Presentations of Roger and Yang's Kauffman bracket arc algebra

Abstract

The Jones polynomial for knots and links was a breakthrough discovery in the early 1980s. Since then, it’s been generalized in many ways; in particular, by considering knots and links which live in thickened surfaces and by allowing arcs between punctures or marked points on the boundary of the surface. One such generalization was recently introduced by Roger and Yang and has connections with hyperbolic geometry. We provide generators and relations for Roger and Yang’s Kauffman bracket arc algebra of the torus with one puncture and the sphere with three or fewer punctures.