Given a separating surface in a 3-manifold , a Heegaard surface is called an *amalgamation* along if it can be constructed from Heegaard splittings of components of the complement . I discussed this construction in my previous post here. Given a Heegaard surface in , a *stabilization* of is a Heegaard splitting that results from adding some number of trivial handles to . Given a Heegaard surface that is not an amalgamation along , one can always stabilize some number of times to find a Heegaard surface that is an amalgamation. (This follows from Reidemeister and Singer’s theorem that any two Heegaard surfaces are related by stabilization.) So, one might ask: Over all pairs of incompressible and Heegaard surfaces as above of given genera, what is the most number of times it will be necessary to stabilize the Heegaard surface to get an amalgamation?

Ryan Derby-Talbot showed that when is a graph manifold and is a torus, one needs to stabilize at most once [1]. In any toroidal 3-manifold, he showed that the number of stabilizations needed is bounded in terms of the number of loops of intersection between and after is isotoped to intersect essentially [2]. As for lower bounds, Jennifer Schltens and Richard Weidmann constructed a family of examples in [3] such that for every , there is an example where the Heegaard surface must be stabilized times before it becomes an amalgamation along an incompressible torus. (They actually describe these examples from a different angle, stating that the Heegaard genus of the glued manifold is less than the sum of the genera of the complementary pieces.)

So, here’s a slightly more complex question: Is there an example of a Heegaard surface that is not an amalgamation along an incompressible surface , but for its minimal stabilization that is an amalgamation along , one of the Heegaard surfaces of the complement of used to form is stabilized. In other words, if we stabilize to find an amalgamation along , we can then destabilize while keeping it an amalgamation.

Here, the answer is again yes, and technically these examples were known for longer. There are horizontal Heegaard splittings of Seifert fibered spaces that are strongly irreducible (so they are not amalgamations) and have much higher genus than the minimal genus, vertical Heegaard splittings for the manifold. Schultens [4] showed that after a single stabilization, such a Heegaard splitting becomes a vertical Heegaard splitting (which is an amalgamation along the vertical tori), and destabilizes all the way down to a minumal genus splitting.

At this point, we can either call it a day, or we can start to split hairs. How about an example of a Heegaard splitting that is not an amalgamation along a given incompressible torus, must be stabilized at least twice to become an amalgamation, and then after that it destabilizes at least twice while remaining an amalgamation? This, I think, is a really hard problem. For the known examples where one needs to stabilize more than once, then destabilize more than once to change from one Heegaard surface to another (ignoring orientation) the beginning and ending Heegaard splittings are already amalgamations. (This is true in both my examples [5] and in Dave Bachman’s examples [6].) So it seems like new methods would be needed to find such an example. But, of course, the better idea might have been to call it a day back at the beginning of this paragraph.

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