Heegaard Splittings with Boundary and Almost Normal Surfaces

Details Heegaard Splittings with Boundary and Almost Normal Surfaces Heegaard Splittings with Boundary and Almost Normal Surfaces

1998-06-04

arxiv.org/abs/math/9806019v3

Topology and its Applications 116 (2001) 153-184

Mathematics

Geometric Topology

33 pages, 13 figures; New 13 page erratum giving a complete proof of Theorem 6.3 has been added

Scientific paper

This paper generalizes the definition of a Heegaard splitting to unify Scharlemann and Thomspon's concept of thin position for 3-manifolds, Gabai's thin position for knots, and Rubinstein's almost normal surface theory. This gives generalizations of theorems of Scharlemann, Thompson, Rubinstein, and Stocking. In the final section, we use this machinery to produce an algorithm to determine the bridge number of a knot, provided thin position for the knot coincides with bridge position. We also present several finiteness and algorithmic results about Dehn fillings with "small" Heegaard genus.

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