Heegaard Splittings of Twisted Torus Knots

TL;DR

This paper proves that a broad class of hyperbolic knot exteriors, specifically certain twisted torus knots, have a unique minimal genus Heegaard splitting, advancing understanding of 3-manifold topology.

Contribution

It establishes the uniqueness of minimal genus Heegaard splittings for an infinite class of hyperbolic knot exteriors, expanding classification knowledge.

Findings

Infinite class of hyperbolic knot exteriors have unique genus two splittings.

Conjecture on existence of genus three irreducible yet weakly reducible splittings.

No prior examples of such genus three splittings are known.

Abstract

Little is known on the classification of Heegaard splittings for hyperbolic 3-manifolds. Although Kobayashi gave a complete classification of Heegaard splittings for the exteriors of 2-bridge knots, our knowledge of other classes is extremely limited. In particular, there are very few hyperbolic manifolds that are known to have a unique minimal genus splitting. Here we demonstrate that an infinite class of hyperbolic knot exteriors, namely exteriors of certain "twisted torus knots" originally studied by Morimoto, Sakuma and Yokota, have a unique minimal genus Heegaard splitting of genus two. We also conjecture that these manifolds possess irreducible yet weakly reducible splittings of genus three. There are no known examples of such Heegaard splittings.