Jessica Purcell

The shape of cusps of hyperbolic knot complements

Abstract:

We show that the number of twists and twist regions in a prime diagram of an alternating knot with hyperbolic complement give estimates on the geometric shape of the cusp of the knot complement, including estimates on the lengths of slopes on the cusp. Here by the geometric shape of the cusp we mean the shape of the Euclidean similarity class of structures on horoball neighborhoods of the cusp in the hyperbolic structure on the knot complement. We use this to show that a large class of alternating knots have the property that every non-trivial Dehn filling is hyperbolic. This generalizes work of Lackenby.

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