Heegaard Diagrams of $S^3$ and the Andrews-Curtis Conjecture

TL;DR

This paper proves that the Andrews-Curtis conjecture is valid for all balanced presentations of the trivial group that are associated with Heegaard diagrams of the 3-sphere, $S^3$.

Contribution

It establishes the conjecture's validity specifically for presentations derived from Heegaard diagrams of $S^3$, linking geometric topology with algebraic group presentations.

Findings

The conjecture holds for all such presentations.

Heegaard diagrams of $S^3$ correspond to trivial group presentations.

Supports the conjecture's broader applicability in topology.

Abstract

We show that the Andrews-Curtis conjecture holds for all balanced presentations of the trivial group corresponding to Heegaard diagrams of $S^3$.