We study the geometry of hyperbolic knots that admit alternating projections on embedded surfaces
in closed
–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
–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.
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