alpha-sloped Generalized Heegaard splittings of 3-manifolds

Abstract

We generalize the definition of thin position of Scharlemann and Thompson for compact orientable 3-manifolds with torus boundary components and introduce $α$-sloped generalized Heegaard splittings. We examine its relationship to generalized Heegaard splittings of manifolds resulting from Dehn filling. We compare alpha-sloped thin position of 3-manifolds to other types of thin position for knots and 3-manifolds and discuss how this kind of decomposition gives an organic picture of M and allows the structure of the manifold to dictate the most natural slope(s) on the boundary. Additionally, we provide illustrative examples and questions motivating the study of alpha-sloped thin position.

21 pages, 7 figures; Revisions to theorem 3.4. The revisions do not affect theorem 1.1