On the disk complexes of weakly reducible, unstabilized Heegaard splittings of genus three I - the Structure Theorem

Abstract

Let (V,W;F) be a weakly reducible, unstabilized, genus three Heegaard splitting in an orientable, irreducible 3-manifold M and DVW(F) the subset of the disk complex D(F) consisting of simplices having at least one vertex from V and at least one vertex from W. In this article, we describe the shape of DVW(F) and prove that there is a function from the components of DVW(F) to the isotopy classes of the generalized Heegaard splittings obtained by weak reductions from (V,W;F), where this function describes how the thick levels are embedded in the relevant compression bodies.

22 pages, 9 figures. arXiv admin note: text overlap with arXiv:1402.4253

References in corpus (4)

• On critical Heegaard splittings of tunnel number two composite knot exteriors
• Normalizing Topologically Minimal Surfaces II: Disks
• Normalizing Topologically Minimal Surfaces I: Global to Local Index
• Normalizing Topologically Minimal Surfaces III: Bounded Combinatorics

Cited by in corpus (1)

• On the disk complexes of weakly reducible, unstabilized Heegaard splittings of genus three III - Generalized Heegaard splittings and mapping classes