A Product Disk for the Hopf Band

We have the Hopf link.

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.

Hopf Band

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).

hopf band monodromy

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.

hopf band product disk 7

Here’s the disk on its own.

hopf band product disk 5

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.

hopf band monodromy

The monodromy of the positive Hopf link is right veering.

~ by Ken Baker on September 4, 2008.

One Response to “A Product Disk for the Hopf Band”

  1. […] http://benblogged.com/?p=233 – bookmarked by 1 members originally found by pinotubo on 2008-10-29 A Product Disk for the Hopf Band https://sketchesoftopology.wordpress.com/2008/09/04/a-product-disk-for-the-hopf-band/ – bookmarked […]

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