Seifert surfaces of dual knots

If you do surgery on a knot, you get a dual knot in the resulting manifold. The original knot and the dual knot have the same complement. A Seifert surface for the original knot is then a (rational) Seifert surface for the dual knot. We can see this dual knot and corresponding Seifert surface before we do the surgery.

Take a knot with orange meridian and purple longitude.

For a null homologous knot, we may take the purple longitude to be the boundary of a Seifert surface. The orange meridian is the boundary of a meridional disk.

Let’s look at +4 surgery on the curve. The green curve will now bound a meridional disk, but we can’t see this disk fully until we reembed.

In the surgered manifold, the green curve now bounds a meridional disk. Since the surgery was integral, the orange curve is now a longitude. Since it was a +4 surgery, the purple boundary of our Seifert surface runs 4 times longitudinally.

Let’s push a copy of our new “dual” knot out of the surgery solid torus. We can make this copy parallel to the longitudinal orange curve.

We got this copy of the dual by a pushoff isotopy, so drag the purple Seifert surface along. The original dual we got from surgery now intersects the surface transversally once. The surface intersects the surgery solid torus in a single meridional disk. I shrunk the surgery solid torus.

Now let’s look back at the copy of the dual before we did surgery. Since it’s parallel to the orange curve, the dual is parallel to the meridian.

The Seifert surface of the dual can now be seen. It’s punctured once by the surgery solid torus, but that gets capped off by the surgery.

There’s a solid torus neighborhood of the surgery solid torus, the thickened dual knot and the parallelism of the dual knot to the orange meridian. Outside this, nothing has changed.

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~ by Ken Baker on April 21, 2011.