Berge's distance 3 pairs of genus 2 Heegaard splittings

TL;DR

This paper constructs a specific 4-component link in a connected sum of two S^1 x S^2 manifolds, demonstrating that Dehn surgery on this link yields 3-manifolds with two distinct genus 2 Heegaard splittings, each having Hempel distance 3.

Contribution

It provides an explicit example of a link in a particular 3-manifold whose Dehn surgeries produce manifolds with multiple inequivalent genus 2 Heegaard splittings of Hempel distance 3.

Findings

Existence of a 4-component link with specified properties.

Dehn surgery results in 3-manifolds with two inequivalent genus 2 splittings.

Each splitting has Hempel distance 3.

Abstract

Following an example discovered by John Berge, we show that there is a 4-component link L \subset (S^1 x S^2)#(S^1 x S^2) so that, generically, the result of Dehn surgery on L is a 3-manifold with two inequivalent genus 2 Heegaard splittings, and each of these Heegaard splittings is of Hempel distance 3.