Thin position for incompressible surfaces in 3-manifolds

TL;DR

This paper presents an algorithm for constructing and analyzing incompressible surfaces in 3-manifolds, linking thin position techniques with Heegaard splittings and curve complex properties.

Contribution

It introduces a method to build all relevant 3-manifolds with specific incompressible surfaces and relates these surfaces to thin position and Heegaard splitting properties.

Findings

Algorithm for constructing 3-manifolds with incompressible surfaces

Thin position method applied to surfaces in 3-manifolds

Relation between thin/thick levels and curve complex properties

Abstract

In this paper, we give an algorithm to build all compact orientable atoroidal Haken 3-manifolds with tori boundary or closed orientable Haken 3-manifolds, so that in both cases, there are embedded closed orientable separating incompressible surfaces which are not tori. Next, such incompressible surfaces are related to Heegaard splittings. For simplicity, we focus on the case of separating incompressible surfaces, since non-separating ones have been extensively studied. After putting the surfaces into Morse position relative to the height function associated to the Heegaard splittings, a thin position method is applied so that levels are thin or thick, depending on the side of the surface. The complete description of the surface in terms of these thin/thick levels gives a hierarchy. Also this thin/thick description can be related to properties of the curve complex for the Heegaard surface.