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).

banddiskcurves


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.
initialsurfacewithbandingarc

Now run a band across the surface following the arc, going over itself, and joining the boundary of the old fiber.
bandedsurface

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.
banddiskcurves
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.

Here’s a few more pics of this disk on its own and with parts of the surface.
diskalone
diskalone2
diskcurves
bandboundarydiskcurves
banddiskcurves
There’s a few more on the Flickr. Clicking any of these pics should take you there.

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.

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~ by Ken Baker on April 26, 2013.

2 Responses to “A Generalized Banding”

  1. These wonderful experiments and excursions in topology are amazing material for architecture. This is one of those moments where 3d printers could make wonders with your designs.

    Would be great to post up tutorials and the programs you use to make these images or how to export them to other programs. My background is in mathematics and physics while my interest is architecture since I study it.
    There are thousands of entries here with lots of potential applications such as diagrams for architectural program, the use of graphs and of course topology.

    The job in these pages is simply amazing and intriguing.

  2. Hi,

    I am a Physics major and an artist from Goa, India. I stumbled upon your blog while looking for some ideas regarding my final year thesis project. I have been interested in topological visualisations and am thinking about a project with which I can render interesting surface models in painting and sculpture form. My issue though, is the lack of a specific problem statement which I can study and work upon, since topology is too vast a field for something like this. I was hoping you’d be able to help me out a bit by giving me a sense of direction.

    Cheers.

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