Heegaard splittings of prime $3$-manifolds are not unique.

Abstract

The authors construct an infinite family of prime homology 3-spheres of Heegaard genus 2, satisfying the following two non-uniqueness properties: (1) Each of the manifolds can be structured as the 2-fold cyclic branched cover over each of two inequivalent knots, one of which is a torus knot. (2) Each of the manifolds admits at least two equivalence classes of genus 2 Heegaard splittings. All of the manifolds are Seifert fiber spaces, the properties of which are used to prove (1). The non-uniqueness of Heegaard splittings is based on the work of the first author and H. M. Hilden [Trans. Amer. Math. Soc. 213 (1975), 315–352], who proved that for Heegaard genus 2 splittings of the 2-fold branched cyclic cover of the knot K, the equivalence class of the Heegaard splitting determines uniquely the knot type K. The authors then show that if Σp,q is the 2-fold cyclic branched cover of the torus knot (p,q), then Σp,q is also the 2-fold cyclic branched cover of a knot different from (p,q), and that Σp,q admits a Heegaard splitting of genus 2.