An algorithmic method to compute plat-like Markov moves for genus two 3-manifolds

TL;DR

This paper presents an algorithmic approach to compute Markov moves for links in genus two 3-manifolds, extending previous genus one results, with implementation details and explicit examples for certain manifolds.

Contribution

It introduces a new algorithm to find braid words realizing plat-equivalence in genus two 3-manifolds, expanding the applicability of prior genus one methods.

Findings

Algorithm successfully computes braid words for genus two manifolds.

Implementation in C++ demonstrates practical feasibility.

Explicit words provided for notable manifold groups.

Abstract

This article deals with equivalence of links in 3-manifolds of Heegaard genus 2. Starting from a description of such a manifold introduced by Casali et al., that uses 6-tuples of integers and determines a Heegaard decomposition of the manifold, we construct an algorithm (implemented in c++) which allows to find the words in $B_{2, 2n}$, the braid group on 2n strands of a surface of genus 2, that realizes the plat-equivalence for links in that manifold. In this way we extend to the case of genus 2 the result obtained by Cattabriga et al. for genus 1 manifolds. We describe explicitly the words for a notable group of manifolds.