# 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 $L$.