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.

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  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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