# Waist size for cusps in hyperbolic 3-manifolds II

## TL;DR

This paper investigates the minimal waist sizes of cusps in hyperbolic 3-manifolds, identifying specific manifolds that realize the second and third smallest sizes, and applies these results to improve bounds on unknotting tunnels.

## Contribution

It proves the uniqueness of manifolds realizing the second and third smallest waist sizes in hyperbolic 3-manifolds, extending previous results on the minimal waist size.

## Key findings

- The second smallest waist size is realized by the 5_2 knot complement.
- The third smallest waist size is realized by a manifold from (2,1)-surgery on the Whitehead link.
- An improved universal upper bound for unknotting tunnel length is established.

## Abstract

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.