A Product Disk for the Hopf Band

We have the Hopf link.

With an annular Seifert surface, we have a Hopf band. Choosing the surface as shown orients the link to give the positive Hopf link.

Now we take an arc (blue) on the band, push it off in the positive direction (sequence of yellows), and lay it back down on the backside of the band (red).

The endpoints of these arcs sweep out a meridian (gray) on each component of the Hopf link. But we’ve offset them a bit to emphasize the direction of sweep and that these arcs sweep out a product disk.

Here’s the disk on its own.

Ignore the disk and look at the initial blue arc and final red arc on the surface. With the positive side of the surface up, we see that the final red arc veers to the right from the endpoints.

The monodromy of the positive Hopf link is right veering.

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~ by Ken Baker on September 4, 2008.