In a post from a couple of years ago, I mentioned a paper by Yeonhee Jang showing that there are 3-bridge knots with infinitely many distinct 3-bridge spheres. This paper has now been published in Top. App., and you can find the link on Jang’s web page. She recently posted to the arXiv a second paper on the subject [1], showing that every (non-split) link with infinitely many three-bridge spheres has a relatively simple form consisting of a two-bridge knot with a second one-bridge component that wraps around the two-bridge knot in a very simple way that you can see below the fold.

The link consists of the green two-bridge knot and an unknot such as the one shown in blue in the image on the left. (The rectangles with the word “braid” in them are braids that I didn’t draw.) The three-bridge sphere isn’t drawn either, but you can imagine it there, as a horizontal line through the middle of the picture. Drawn in red is an incompressible torus that separates the two components and intersects the bridge sphere in two loops. If we spin the bridge surface around this torus, like we did with the incompressible torus in my last post, we get a new sphere which, in general won’t be isotopic to the original surface (though the bridge surfaces will be related by a homeomorphism of the complement.) This sort of construction has been used to find distinct Heegaard surfaces (for example in [2]), but Jang appears to be the first person to study this phenomenon for bridge surfaces. I like the fact that you can see the torus so clearly in these examples.

Spinning around the torus corresponds to taking a Haken sum of the original bridge surface and two copies of the incompressible torus. For Heegaard surfaces, Haken sums have also been used to create irreducible Heegaard surfaces of arbitrarily high genus. (For example [3] interprets a construction of Casson and Gordon in terms of Haken sums.) In the example above, Haken summing with a torus ensures that the new bridge surface is a sphere (since a torus has Euler characteristic zero). Using a torus disjoint from the knot ensures that the number of intersections stays the same, so the new bridge sphere is still three-bridge. If one wanted to use Haken sums to create irreducible bridge spheres of arbitrarily high bridge number, they would need to use a meridianal (probably incompressible) torus so the genus stays bounded but the number of intersection points increase. As far as I know, no such examples have been worked out.

In the 3-manifold setting, unlike the knot setting, you don’t get to really see what the incompressible torus looks like all at once. So as I mentioned above, getting to see the torus is (for me) a real treat. Based on how simple the incompressible torus is in the above example, I would be very curious to see what a meridianal torus that produces arbitrarily high bridge number bridge surfaces looks like.

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