Cusp size bounds from singular surfaces in hyperbolic 3-manifolds

Adams, C.; Colestock, A.; Fowler, J.; Gillam, W.; Katerman, E.

doi:10.1090/S0002-9947-05-03662-7

Cusp size bounds from singular surfaces in hyperbolic 3-manifolds
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by C. Adams, A. Colestock, J. Fowler, W. Gillam and E. Katerman PDF

Trans. Amer. Math. Soc. 358 (2006), 727-741 Request permission

Abstract:

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 $\mathbb {S}^3$ depending on crossing number. Particular improved bounds are obtained for alternating knots.

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