An algorithmic method to compute plat slide moves in 3-manifolds of Heegaard genus two

Abstract

This article is motivated by the equivalence problem of links embedded in 3-manifolds of Heegaard genus two. Casali and Grasselli introduced a way to encode every such 3-manifold M by an admissible 6-tuples of integers, that also determines a genus two Heegaard splitting of the manifold M. We construct an algorithm (implemented in C++) which, starting from an admissible 6-tuple of integers, allows to find the words in B2,2n, the braid group on 2n strands on a genus two surface, that realize the plat slide moves in that manifold. This extends the work of Cattabriga and Gabrovšek, who carried out this computation for 3-manifolds of Heegaard genus one. As an application of our algorithm, we compute the words that represent the plat slide moves for all orientable prime 3-manifolds of Heegaard genus two that can be triangulated with at most 42 colored tetrahedra.