Cusp size bounds from singular surfaces in hyperbolic 3-manifolds

Abstract

Singular maps of surfaces into a hyperbolic 3-manifold are utilized to find upper bounds on meridian length, ℓ \ell -curve length and maximal cusp volume for the manifold. This allows a proof of the fact that there exist hyperbolic knots with arbitrarily small cusp density and that every closed orientable 3-manifold contains a knot whose complement is hyperbolic with maximal cusp volume less than or equal to 9. We also find particular upper bounds on meridian length, ℓ \ell -curve length and maximal cusp volume for hyperbolic knots in S 3 \mathbb {S}^3 depending on crossing number. Particular improved bounds are obtained for alternating knots.