Heegaard Splittings of Branched Coverings of S 3

This paper concerns itself with the relationship between two seemingly different methods for representing a closed, orientable 3-manifold: on the one hand as a Heegaard splitting, and on the other hand as a branched covering of the 3-sphere. The ability to pass back and forth between these two representations will be applied in several different ways: 1. It will be established that there is an effective algorithm to decide whether a 3-manifold of Heegaard genus 2 is a 3-sphere. 2. We will show that the natural map from 6-plat representations of knots and links to genus 2 closed oriented 3-manifolds is injective and surjective. This relates the question of whether or not Heegaard splittings of closed, oriented 3manifolds are "unique" to the question of whether plat representations of knots and links are "unique". 3. We will give a counterexample to a conjecture (unpublished) of W. Haken, which would have implied that S could be identified (in the class of all simply-connected 3-manifolds) by the property that certain canonical presentations for 7TjS3 are always "nice". The final section of the paper studies a special class of genus 2 Heegaard splittings: the 2-fold covers of S which are branched over closed 3-braids. It is established that no counterexamples to the "genus 2 Poincare conjecture" occur in this class of 3-manifolds.