Low Dimensional Topology

July 25, 2011

Cut Stabilization

Filed under: 3-manifolds,Heegaard splittings,Knot theory,Thin position — Jesse Johnson @ 8:10 am

A few months ago, I wrote about a generalization of thin position for bridge surfaces of knots and links (introduced by Maggy Tomova [1]) that comes from considering cut disks, i.e. compression disks that intersect the link in a single point in their interior. One implication of this new definition is that it suggests a generalized concept of stabilization. A stabilization of a Heegaard surface is defined by a pair of compressing disks on opposite sides of the Heegaard surface whose boundaries intersect in a single point. Compressing along either of these disks produces a new Heegaard surface with lower genus. A cut stabilization occurs when one of those disks is a cut disks, and this situation comes up, for example, in Taylor and Tomova’s version of the Casson-Gordon Theorem [2]. As I will describe below the fold, cut compressions have very interesting implications for studying bridge surfaces of different genera.

A bridge surface for a link L in a 3-manifold M is a Heegaard surface \Sigma for M such that the intersection of L with each of the handlebodies bounded by \Sigma is a collection of boundary parallel arcs. The classical case is when \Sigma is the genus zero Heegaard splitting for S^3 i.e. a sphere that bounds a ball on either side.

Let D_1 be a cut disk on one side of \Sigma and D_2 a compression disk on the other side such that the boundaries intersect in a single point. If we ignore the link, then these are both compression disks for \Sigma. It’s a relatively simple exercise to see that compressing along either disk produces the same (isotopy class of) surface in M (ignoring L) which is a new Heegaard surface with genus one less than the original.

If we now pay attention to the link, we notice that the two compressions no longer produce the same surface: Compressing along D_2 does not change the intersection number between \Sigma and L, but compressing along D_1 increases the intersection number by two. For either disk, the key question is whether the new surface is a bridge surface for L.

Before we compress along D_1, lets shrink a neighborhood (in \Sigma) of its boundary to a small tube that runs along the knot. The boundary of D_2 consists of an arc that runs along this tube and a second arc in the rest of \Sigma. If we then compress \Sigma along D_1, we can extend D_2 to a disk whose boundary consists of an arc in L and an arc in the new surface. This is a bridge disk, so compressing \Sigma across D_1 creates a new bridge (i.e. an arc parallel into the Heegaard surface), making the new surface a bridge surface with genus one less than the original.

The inverse of this compression is also an important construction: Given a bridge surface for L, we can always add a tube along any one of the bridges. The new surface is also a bridge surface for L and the disks defined by a meridian of the tube and a bridge disk for the arc along which the tube runs form the disks D_1, D_2 that define a cut stabilization.

Cutting along D_2, on the other hand, does not create a bridge surface in general. For example, if we start with a 2-bridge knot, we can tube along one of the bridges to find a genus-one, one-bridge surface for the knot. By construction, there is a cut stabilization pair, but if we compress the bridge disk along the non-cut disk, the resulting sphere will intersect L in exactly two points. The only genus-zero, one-bridge knot is the unknot, so in general this will not be a bridge sphere for L.



  1. One other type of stabilization that came up in our paper (in the section on sweepouts) is what we call “bimeridional stabilization”. (We call “cut stabilization” meridional stabilization) A bridge surface is “bimeridionally stabilized” if there are two cut discs on opposite sides intersecting (transversally) in a single point. We didn’t actually write down a proof (b/c we didn’t need the result) but it should be true that if a bridge surface is bimeridionally stabilized then it can be destabilized to get a bridge surface of lower complexity (and genus). As with meridional (cut) stabilization if a bridge surface is bimeridionally stabilized then the underlying Heegaard surface is stabilized. — Scott

    Comment by Scott Taylor — July 27, 2011 @ 10:32 am | Reply

    • I find that surprising that you always get a new bridge surface from a bimeridional stabilization. I didn’t include it in the post because it corresponded to anything like that. (I also forgot to check the paper to see if I was remembering the terminology correctly. Oops!)

      Comment by Jesse Johnson — July 29, 2011 @ 4:17 pm | Reply

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