The waist size of a cusp in an orientable hyperbolic 3-manifold is the length of the shortest nontrivial curve generated by a parabolic isometry in the maximal cusp boundary. Previously, it was shown that the smallest possible waist size, which is 1, is realized only by the cusp in the figure-eight knot complement. In this paper, it is proved that the next two smallest waist sizes are realized uniquely for the cusps in the \(5_2\) knot complement and the manifold obtained by (2,1)-surgery on the Whitehead link. One application is an improvement on the universal upper bound for the length of an unknotting tunnel in a 2-cusped hyperbolic 3-manifold.
Hyperbolic 3-manifold Waist size Cusp
Mathematics Subject Classification (2010)
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Adams, C.: Hyperbolic structures on knot and link complements, Ph.D. thesis, University of Wisconsin (1983)Google Scholar