Cusp shape and tunnel number

TL;DR

This paper proves that the set of cusp shapes of hyperbolic tunnel number one manifolds is dense in the Teichmuller space, and extends this to tunnel number n manifolds, revealing diverse geometric structures.

Contribution

It demonstrates the density of cusp shapes in Teichmuller space for hyperbolic tunnel number manifolds and explores implications for Dehn fillings.

Findings

Cusp shapes of hyperbolic tunnel number one manifolds are dense in Teichmuller space.

For fixed n, infinitely many hyperbolic tunnel number n manifolds have at most one exceptional Dehn filling.

Contrast with Berge knots, whose cusp shapes converge to a single point in Teichmuller space.

Abstract

We show that the set of cusp shapes of hyperbolic tunnel number one manifolds is dense in the Teichmuller space of the torus. A similar result holds for tunnel number n manifolds. As a consequence, for fixed n, there are infinitely many hyperbolic tunnel number n manifolds with at most one exceptional Dehn filling. This is in contrast to large volume Berge knots, which are tunnel number one manifolds, but with cusp shapes converging to a single point in Teichmuller space.