A few months ago, I wrote about a generalization of thin position for bridge surfaces of knots and links (introduced by Maggy Tomova ) 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 . 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 in a 3-manifold is a Heegaard surface for such that the intersection of with each of the handlebodies bounded by is a collection of boundary parallel arcs. The classical case is when is the genus zero Heegaard splitting for i.e. a sphere that bounds a ball on either side.
Let be a cut disk on one side of and 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 . It’s a relatively simple exercise to see that compressing along either disk produces the same (isotopy class of) surface in (ignoring ) 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 does not change the intersection number between and , but compressing along increases the intersection number by two. For either disk, the key question is whether the new surface is a bridge surface for .
Before we compress along , lets shrink a neighborhood (in ) of its boundary to a small tube that runs along the knot. The boundary of consists of an arc that runs along this tube and a second arc in the rest of . If we then compress along , we can extend to a disk whose boundary consists of an arc in and an arc in the new surface. This is a bridge disk, so compressing across 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 , we can always add a tube along any one of the bridges. The new surface is also a bridge surface for 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 that define a cut stabilization.
Cutting along , 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 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 .