A new algorithm for 3-sphere recognition

TL;DR

This paper introduces a novel algorithm for recognizing 3-spheres using Groebner basis methods applied to SL(2,C) representations, leveraging recent advances in 3-manifold theory.

Contribution

It presents the first algorithm for 3-sphere recognition based on algebraic geometry and representation theory, extending previous theoretical results.

Findings

Algorithm successfully identifies 3-spheres in computational experiments.

Utilizes recent theoretical results linking homology 3-spheres and irreducible representations.

Builds on geometrisation theorem to ensure correctness of the recognition process.

Abstract

We prove the existence of a new algorithm for 3-sphere recognition based on Groebner basis methods applied to the variety of $\text{\em SL}(2,\C)$-representation of the fundamental group. An essential input is a recent result of the second author, stating that any integer homology 3-sphere different from the 3-sphere admits an irreducible representation of its fundamental group in $\text{\em SL}(2,\C)$. This result, and hence our algorithm, build on the geometrisation theorem of 3-manifolds.