2
$\begingroup$

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?

Improve this question
$\endgroup$
2
  • $\begingroup$ The answers to this question are relevant here mathoverflow.net/questions/57232/… $\endgroup$
    – j.c.
    Commented Dec 30, 2017 at 16:54
  • 2
    $\begingroup$ 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. $\endgroup$
    – Ryan Budney
    Commented Dec 31, 2017 at 0:24

1 Answer 1

Reset to default
2
$\begingroup$

This isn't quite an answer to your question, but it should help give examples of what you are looking for. In

Moriah, Yoav, Heegaard splittings of Seifert fibered spaces, Invent. Math. 91, No. 3, 465-481 (1988). ZBL0651.57012.Yoav

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 $S^3$.

Improve this answer
$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .