A Generalized Banding
The preprint Band Surgeries and Crossing Changes between Fibered Links by Buck-Ishihara-Rathbun-Shimokawa caught my eye this morning. They describe a describe a generalization of the plumbing of a Hopf band. Like Hopf plumbing, this operation preserves fiberedness. But unlike Hopf plumbing which occurs in a neighborhood of a disk, it is non-local occurring in a neighborhood of an annulus.
I thought I’d show the product disk associated to the band. This lets one verify the persistence of fiberedness and work out the resulting monodromy (which I haven’t done myself yet).
Let’s talk through the construction.
The generalized banding is defined by an arc in a Seifert surface with one transverse self intersection and meeting the boundary of the surface in its endpoints. A neighborhood of this blue arc is an annular chunk of the surface. The white tubes are the pieces of the link (the boundary of the surface). The rest of the boundary of this annulus continues on doing whatever it was already doing in the surface.
A spanning arc of the band gives rise to a product disk. If the side of the original surface we saw was “up” then here we have a product disk going from the red arc to the green one.
Taking the sutured manifold coming from this new banded surface and decomposing it along this product disk leaves us with the sutured manifold coming from the old surface. (Yeah, we could see explicit pictures of this… maybe in a future update.) So if the original Seifert surface was a fiber, then the banded surface will be a fiber too.
A couple of interesting things to note:
1) If the “hole” the original blue arc went around actually bounded a disk in the surface, then this gives us the standard Hopf banding. As I’ve drawn it here, this would result in the positive Hopf band.
2) If the hole doesn’t bound a disk, then it is not a Hopf band as the monodromy it offers is neither right-veering nor left-veering.