α-SLOPED GENERALIZED HEEGAARD SPLITTING OF 3-MANIFOLDS

Abstract. We generalize the definition of thin position of Scharlemann and Thomp-son 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 α-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. 1