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.
Adams, C. and Colestock, A. and Fowler, J. and Gillam, W. and Katerman, E. (2006). Cusp size bounds from singular surfaces in hyperbolic 3-manifolds. Trans. Amer. Math. Soc., Volume 358, 727–741 (electronic).