High Distance Heegaard Splittings via Dehn Twists

TL;DR

This paper demonstrates that high-distance Heegaard splittings can be achieved using high powers of Dehn twists, extending previous results that used pseudo-Anosov maps, and in some cases, determining the exact distance.

Contribution

It introduces a method to obtain lower bounds on the distance of Heegaard splittings using Dehn twists, expanding the understanding of their geometric complexity.

Findings

Lower bounds on Heegaard distance via Dehn twists

Exact distance determination in certain cases

Extension of prior work on strongly irreducible splittings

Abstract

In 2001, J. Hempel proved the existence of Heegaard splittings of arbitrarily high distance by using a high power of a pseudo-Anosov map as the gluing map between two handlebodies. We show that lower bounds on distance can also be obtained when using a high power of a suitably chosen Dehn twist. In certain cases, we can then determine the exact distance of the resulting splitting. These results can be seen as a natural extension of work by A. Casson and C. Gordon in 1987 regarding strongly irreducible Heegaard splittings.