Different Heegaard splittings of a 3-manifold

I want to study same 3-manifolds with different Heegaard splitings. Of course one has stabilization, but even with the same genus, we have different Heegaard splittings.

If we encode a 3-manifolds by a genus $g$, a set of curves $c_1,c_2,\ldots$ on the Heegaard surface of genus $g$, which are images of the meridian discs of one of the handlebodies, do we have an explicit set of "rules" (similar perhaps to Reidemeister moves) that tell us when two such encodings represent the same 3-manifold?

• The answers to this question are relevant here mathoverflow.net/questions/57232/… – j.c. Dec 30 '17 at 16:54
• Sort of. There's the stabilization theorem. And then there's tools like the Rubinstein-Scharlemann graphic. With this tool you are studying how the two Heegaard surfaces can intersect. There's some lovely combinatorics that describes various possibilities. – Ryan Budney Dec 31 '17 at 0:24

Moriah gives examples of Seifert fibered spaces over $$S^2$$ with three exceptional fibers with the property that there are multiple genus 2 Heegaard splittings. Obviously to show that these splittings are not invariant one needs an invariant. The invariant Moriah uses is special to the case of genus 2 splittings, each one admits a hyper elliptic involution. Moriah is able to show that the fixed point sets of these involutions are distinct knots in $$S^3$$.