Cusp volumes of alternating knots on surfaces

Abstract

We study the geometry of hyperbolic knots that admit alternating projections on embedded surfaces in closed $3$–manifolds. We show that, under mild hypothesis, their cusp area admits two-sided bounds in terms of the twist number of the alternating projection and the genus of the projection surface. As a result, we derive diagrammatic estimates of slope lengths and give applications to Dehn surgery. These generalize results of Lackenby and Purcell about alternating knots in the $3$–sphere.

Using a result of Kalfagianni and Purcell, we point out that alternating knots on surfaces of higher genus can have arbitrarily small cusp density, in contrast to alternating knots on spheres whose cusp densities are bounded away from zero due to Lackenby and Purcell.