Low Dimensional Topology

May 12, 2008

The Knot Wiki

Filed under: 3-manifolds,Heegaard splittings,Knot theory — Jesse Johnson @ 10:58 am

Connie Leidy recently pointed out to me the knot atlas, which is essentially a wikipedia for knots. One interesting thing about looking at knot tables (something that I hadn’t done in a few years) is seeing how the bridge number changes as you rotate the diagram. Often if the initial diagram is not in minimal bridge position, you can lower the bridge position by rotating it 90 degrees. (See, for example, 4_1, 5_2 and 6_2, just in the first row of Rolfsen’s table.)

If you rotate it 180 degrees around and keep track of the bridge number at each point, you get Hass, Thompson and Rubinstein’s 2-width for planar curves. They show that for any width, there are finitely many diagrams (and therefore finitely many knots) with that width. If you think of the rotation as an isotopy of the knot, then rotating the knot 180 degrees corresponds to flipping over the bridge surface. The smallest possible maximal bridge number during any such isotopy (we’ll call it the flip number) is analogous to the flip genus of a Heegaard splitting, discussed in a previous entry. So we can read from the diagram not only the bridge number (at least in many cases) but an upper bound on the flip number. This is also interestingly reminiscent of a proof of the stabilization theorem I wrote a year or so ago, but that’s another story.

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