Here’s an interesting question that may not be too hard, but I don’t know the answer to it: Is there a simple closed curve in the boundary of some handlebody that is the boundary of two distinct (i.e. non-isotopic) incompressible surfaces in the handlebody? This is a side issue in a paper I wrote with Terk Patel [1] about the set of non-separating loops that bound incompressible surfaces in handlebodies, as a subset of a curve complex. (This set turns out to be 2-dense, but is still sparse enough that every Heegaard splitting of a non-Haken 3-manifold determines a pair of disjoint such sets.)

No loop can bound both a disk and an incompressible (higher genus) surface because if it did, then the disk would imply a compressing disk for the surface. However, there doesn’t seem to be an simple reason to expect a loop can’t bound two incompressible surfaces of different genus above zero.

Addendum: Here’s a construction suggested by Richard Kent (see the comments below). Take a separating, loop in the boundary of a handlebody. This loop bounds two boundary parallel surfaces, each of which is parallel to one of the complementary components in the boundary. Compress these surfaces down as much as possible. In many cases, the resulting incompressible surfaces will not be isotopic.

If you start with a non-separating loop, you can take a single boundary parallel surface whose boundary is two loops parallel to the original loop. When you compress it down, you then have to compress along a separating disk at some point, so that you the result is two surfaces, each with one boundary component parallel to the original loop. (Note: the construction described here sounds different than the one described by Richard, but it yields the same surfaces, and as Saul noted, in Richard’s construction you get the separating and non-separating loops without having to do two cases.)

In light of this construction, I want to reformulate the question: Is there a loop in the boundary of a handlebody bounding two distinct incompressible surfaces that don’t both come from compressing boundary parallel surfaces?

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Sure, take a disk-busting curve in a handlebody of genus 3 that cuts of a punctured torus.

Comment by Richard Kent — March 20, 2008 @ 8:09 am |

You can add a 1-handle if you don’t want the surfaces to be boundary parallel.

Comment by Richard Kent — March 20, 2008 @ 8:13 am |

Ah, that’s a nice construction. So, then I will reformulate the question: Are those the only examples? In other words, given a loop that bounds two distinct incompressible surfaces, can you boundary compress the handlebody in the complement of the loop until the two surfaces are boundary parallel?

Comment by Jesse Johnson — March 20, 2008 @ 12:15 pm |

Adding a one-handle can also make the loop in question non-separating.

Comment by Saul — March 20, 2008 @ 2:05 pm |

Good point. I’ll fix that.

Comment by Jesse Johnson — March 20, 2008 @ 3:15 pm |

I think that Theorem 2.7 of Ruifeng Qiu’s paper “Incompressible surfaces in handlebodies and closed 3–manifolds of Heegaard genus 2” might be useful in finding a curve on the boundary of a handlebody which is spanned by three non-parallel, disjoint (on their interiors), incompressible surfaces. I don’t yet understand his construction, however.

Comment by Saul — March 24, 2008 @ 4:40 pm |