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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Cusp size bounds from singular surfaces in hyperbolic 3-manifolds


Authors: C. Adams, A. Colestock, J. Fowler, W. Gillam and E. Katerman
Journal: Trans. Amer. Math. Soc. 358 (2006), 727-741
MSC (2000): Primary 57M50
Published electronically: September 22, 2005
MathSciNet review: 2177038
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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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Additional Information

C. Adams
Affiliation: Department of Mathematics, Williams College, Williamstown, Massachusetts 01267
Email: Colin.Adams@williams.edu

A. Colestock
Affiliation: Francis W. Parker School, Chicago, Illinois 60614
Email: acolestock@fwparker.org

J. Fowler
Affiliation: Department of Mathematics, University of Chicago, Chicago, Illinois 60637-1538
Email: fowler@math.uchicago.edu

W. Gillam
Affiliation: Department of Mathematics, Columbia University, New York, New York 10027
Email: wgillam@math.columbia.edu

E. Katerman
Affiliation: Department of Mathematics, University of Texas, Austin, Texas 78712
Email: katerman@mail.utexas.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-05-03662-7
PII: S 0002-9947(05)03662-7
Received by editor(s): October 2, 2002
Received by editor(s) in revised form: March 1, 2004
Published electronically: September 22, 2005
Additional Notes: This research was supported by the National Science Foundation under grant numbers DMS-9820570 and DMS-9803362.
Article copyright: © Copyright 2005 American Mathematical Society