Chopping solid tori and genus 2 handlebodies along annuli.
Properly embedded surfaces in handlebodies are either compressible or boundary-compressible. If such a surface is compressible, it can be rather knotted; chopping the handlebody along the surface could yield manifolds that are not handlebodies. Incompressible surfaces are a bit nicer.
Let us begin with disks. A properly embedded disk in a handlebody is either boundary-parallel, separating but not boundary-parallel, or non-separating.
Now let us consider incompressible (properly embedded) annuli. A boundary-compression of an incompressible annulus forms a disk. Undoing the boundary-compression is equivalent to banding the disk. If this disk is boundary-parallel, then the annulus was. If this disk is separating but not boundary-parallel, then the annulus was separating and not boundary-parallel to the side of the boundary-compressing disk (though it might be boundary-parallel to the other side). If this disk is non-separating then the annulus was.
An incompressible annulus in a solid torus is either longitudinal or cabled. When you chop along the annulus it becomes two solid tori. On one solid torus, it leaves a longitudinal impression. On the other solid torus it leaves a longitudinal or cabled impression, whatever the annulus was in the original solid torus.
An incompressible annulus in a genus 2 handlebody is either boundary-parallel, separating but not boundary-parallel, or non-separating.
A boundary-parallel annulus can run around a genus 2 handlebody like any curve on the genus 2 surface. Chopping along it leaves a genus 2 handlebody and a solid torus. The annulus leaves a longitudinal impression on the solid torus.
The boundary compression of a separating but not boundary-parallel incompressible annulus forms a separating but not boundary-parallel disk. Such a disk splits a genus 2 handlebody into two solid tori. One of these solid tori contains the annulus. Since the annulus is not boundary-parallel in the handlebody, it must be cabled in the solid torus: the impression of the disk on this solid torus must obstruct the boundary parallelism of the annulus in the solid torus. Hence the annulus is cabled in the genus 2 handlebody. It chops the genus 2 handlebody into another genus 2 handlebody on which its impression is primitive and a solid torus on which its impression is cabled.
The boundary compression of a non-separating incompressible annulus forms a non-separating disk. Such a disk splits a genus 2 handlebody into one solid torus. Since the annulus is not separating in the handlebody, the two impressions of the disk on this solid torus must lie on opposite sides of the boundary of the annulus. The annulus chops the genus 2 handlebody into another genus 2 handlebody on which one impression is primitive and the other impression is either primitive or cabled. Thus in the original genus 2 handlebody, the annulus was either primitive or cabled respectively.
This was rather wordy. Perhaps later we’ll look at Mobius bands and thrice-punctured spheres.