It was shown by Bonahon–Otal and Hodgson–Rubinstein that any two genus–one
Heegaard splittings of the same 3–manifold (typically a lens space) are isotopic. On
the other hand, it was shown by Boileau, Collins and Zieschang that certain Seifert
manifolds have distinct genus–two Heegaard splittings. In an earlier paper, we
presented a technique for comparing Heegaard splittings of the same manifold and,
using this technique, derived the uniqueness theorem for lens space splittings as a
simple corollary. Here we use a similar technique to examine, in general, ways in
which two non-isotopic genus–two Heegard splittings of the same 3–manifold
compare, with a particular focus on how the corresponding hyperelliptic involutions
are related.
