A common problem in low-dimensional topology is to ask how the topology and geometry of a manifold changes if you glue a solid torus into one of its torus boundary components (also known as Dehn filling) or more generally, if you glue a handlebody into a higher genus boundary component. One topological version of this problem is to ask how the isotopy classes of Heegaard surfaces change. Every Heegaard surface for the unfilled manifold becomes a Heegaard surface for the filled manifold, but there may be other properly embedded non-Heegaard surfaces that also become Heegaard surfaces if you cap them off after the filling. In particular these new Heegaard surfaces may have lower genus, so the Heegaard genus of the manifold could drop after filling. The quintessential example of this is a knot complement in the 3-sphere: There are knot complements with arbitrarily high Heegaard genus, but if you Dehn fill to produce the 3-sphere, then the genus drops to zero.

Of course, for such a manifold there is exactly one filling that produces the 3-sphere and one can ask how much the genus can drop for the other fillings. There are examples where Heegaard genus drops by one for a line of slopes, and the resulting Heegaard surfaces are often called *horizontal*. However, Moriah-Rubinstein [1] (and later Rieck-Sedgwick [2]) showed that there are only finitely many slopes for which the genus can drop by more than one (and only finitely many lines of slopes where it drops by one.) As far as I know there are no examples where there are two slopes for which the genus drops by more than one. So one can ask:

**Question:** Is there a 3-manifold with Heegaard genus , a torus boundary component and two slopes on such that Dehn filling along each slope produces a 3-manifold with Heegaard genus less than or equal to ?

Note that this is closely related to the Berge conjecture, which asks which knots in the 3-sphere have a Dehn surgery that produces a 3-manifold with Heegaard genus one. If you could prove the answer to this question is “no” then you would reduce the Berge conjecture to a question of which tunnel-number-one knots (i.e. those with Heegaard genus two) have lens space surgeries. However, I’m more inclined to bet on “yes” for this one. Perhaps it’s because I’ve been spending too much time trying to construct manifolds with multiple non-isotopic Heegaard splittings. But whichever answer turns out to be the case will be very interesting.

For gluing in higher genus handlebodies, things get much more complicated because there is no longer a single slope that determines the gluing. Tao Li [3] has generalized Moriah-Rubinstein’s result to higher genus, but rather than having a finite number of fillings in which the genus may drop, Li requires that the (infinitely many) meridian curves for the glued in handlebody have distance in the curve complex above some bound from a finite collection loops in the boundary. (The difficulty has to do with the fact that disjoint essential loops in the torus are parallel, but disjoint essential loops in high genus surfaces may not be.) It would be interesting to find examples where the Heegaard genus drops after gluing in handlebodies in two inequivalent ways, but I don’t know the best way to generalize the question above.

[1] Moriah, Yoav; Rubinstein, Hyam, Heegaard structures of negatively curved 3-manifolds. *Comm. Anal. Geom.* 5 (1997), no. 3, 375–412.

[2] Rieck, Yo’av; Sedgwick, Eric, Persistence of Heegaard structures under Dehn filling. *Topology Appl.* 109 (2001), no. 1, 41–53.

Here’s a vague suggestion. Suppose you have a manifold with torus boundary that has two surgeries in which the Heegaard genus decreases by 1. One could try to “amplify” this effect, by gluing in a 1-bridge braid in a solid torus which has two solid torus surgeries, inducing these two slopes. Hopefully one could show that the genus increases by 1 after performing this satellite operation, which would show that there would be two surgeries in which the genus decreases by 2. One caveat is that the genus decreasing surgeries would have to have intersection number >1 (I’m actually not sure which slopes are induced by pairs of solid torus surgeries).

Comment by Ian Agol — October 6, 2011 @ 10:19 am |

This seems like a plausible approach. It’s possible that cabling along such a not wouldn’t increase the tunnel number, but that seems unlikely. If this construction did work, it seems like it would give potential counter examples to the Berge conjecture, since you could cable a Berge knot in this way. It’s definitely worth looking into.

Comment by Jesse Johnson — October 6, 2011 @ 2:34 pm |

I don’t think it could give a counterexample to the Berge conjecture. By the cyclic surgery theorem, cyclic surgeries are distance 1 apart, whereas the 1-bridge braid solid torus fillings induce slopes of distance >1 apart. What I am contemplating is that there might be genus-reducing fillings of distance > 1, in which case one might be able to insert a cabling or 1-bridge braid.

Comment by Ian Agol — October 7, 2011 @ 4:18 pm