# Cusp size bounds from singular surfaces in hyperbolic 3-manifolds Joint work with Colin Adams, Adam Colestock, William Gillam, Eric Katerman

Singular maps of surfaces into a hyperbolic 3-manifold are utilized to find upper bounds on meridian length, $\ell$-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, $\ell$-curve length and maximal cusp volume for hyperbolic knots in $S^3$ depending on crossing number. Particular improved bounds are obtained for alternating knots.

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Adams, C., Colestock, A., Fowler, J., Gillam, W., & Katerman, E. (2006). Cusp size bounds from singular surfaces in hyperbolic 3-manifolds.*Trans. Amer. Math. Soc.*,

*358*(2), 727–741 (electronic).