Heegaard distance of the link complements in $S^3$

TL;DR

This paper proves that for any specified genus and distance, there exists a link in the 3-sphere whose complement admits a Heegaard splitting with those parameters, demonstrating the diversity of link complements.

Contribution

It establishes the existence of links in $S^3$ with complements having prescribed genus and Heegaard distance, expanding understanding of 3-manifold topology.

Findings

Existence of links with arbitrary genus and distance in their complements

Construction method for such links

Implications for the topology of link complements

Abstract

We show that, for any integers, $g \geq 3$ and $n \geq 2$, there exists a link in $S^3$ such that its complement has a genus $g$ Heegaard splitting with distance $n$.