Cusp Volumes of Alternating Knots on Surfaces

Brandon Bavier (MSU)

Abstract: We study the geometry of hyperbolic knots that admit alternating projections on embedded surfaces in closed 3-manifolds. By examining certain essential spanning surfaces, 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.