Understanding 3-manifolds in the context of permutations

TL;DR

This paper introduces a permutation-based approach to analyze 3-manifolds, providing an algorithm to determine if a given group presentation corresponds to a closed 3-manifold, thus offering a new invariant for group presentations.

Contribution

It develops a novel method linking permutations, group presentations, and 3-manifolds, enabling the detection of closed 3-manifolds from permutation data.

Findings

Algorithm can determine if permutation data represents a closed 3-manifold.

Permutation data provides an invariant for group presentations.

Finite class of 3-manifolds associated with permutation-based group presentations.

Abstract

We demonstrate how a 3-manifold, a Heegaard diagram, and a group presentation can each be interpreted as a pair of signed permutations in the symmetric group $S_d.$ We demonstrate the power of permutation data in programming and discuss an algorithm we have developed that takes the permutation data as input and determines whether the data represents a closed 3-manifold. We therefore have an invariant of groups, that is given any group presentation, we can determine if that fixed presentation presents a closed 3-manifold. (The proposed techniques begin with a pair of signed permutations and builds a finite group presentation. The finite group presentation results in a finite class of associated 3-manifolds. Notice that a negative answer only implies the fixed presentation does not result in a closed 3-manifold under this construction, but says nothing about an isomorphic form of the group presentation.)