Long Disks and Lopsided Bigons
I’ve been working in a collaboration with Gordon and Luecke where we’ve been looking at thin positions of knots in manifolds that arise from non-integral, non-half-integral surgeries on knots in S^3 (with some other conditions). Fat vertexed graphs of intersection hijinks ensue.
A particular situation arises where we have a knot K that is 2–bridge with respect to a Heegaard surface F and on the graph from S^3 we see at a vertex a forked extended Scharlemann cycle (with a certain one of two possible labelings) with its trigon followed by a bigon or another trigon. These are on the left-hand sides of the figures below. The black and white (dark and light gray) faces are on opposite sides of the Heegaard surface F, the edges between them are curves on F. The “corners” are arcs that run along the boundary of the neighborhood of K from one intersection of K with F to another.
It so happens that the 41–edges of faces f and g cobound a disk d (disjoint from the rest of these faces and the knot). On the right-hand sides of the above images we reassemble the faces to form disks. The purple colored things run along the boundary of a neighborhood of K, between corners. The red shows the original K and these disks guide an isotopy of it to the salmonish colored arc.
The one on top is a “long disk” because it is like an extended bridge disk. Rather than guiding just one arc of K – F onto F, it guides three arcs. Consequentially this thins the knot to be 1–bridge.
The one on the bottom is a “lopsided bigon” because it has two edges on F and two corners along K, but one corner is made of three arcs and the other is made of one. This too gives a thinning since we can isotop three arcs of K down to two arcs on F and one arc on K.
It is perhaps not so obvious that these actually give isotopies. Fortunately the complex determined by these faces and the knot is, up to homeomorphism, unique in the top case and almost unique in the bottom case. Moreover these complexes can be embedded in R^3 and one can wrap the long disk or lopsided bigon around it to explicitly see the isotopy. I made a SketchUp model (get it here) to illustrate all this, though I cut it open leaving some identifications undone for better viewing.
Start with the Scharlemann cycle (SC) and attach it to the knot. When the corners of the SC are glued together, it forms a Mobius band. In the picture we’ve cut this face from edge to edge, so the two ends need to be glued together with a half twist. Also the top and the bottom 1s get identified to form the neighborhood of K.
Now we successively attach f, then g, then h, and finally d. A choice comes up when attaching g, but the alternatives are homeomorphic. In the final figure the green “J” shaped edges on the left and right sides get identified with a twist.
We can now wrap the long bigon around this complex. (Hmm. Not the best pictures.)
Here’s the wrapped disk on its own.
The red curve is the original knot. The last picture shows the knot after isotopy. Perhaps it’s not so clear on its own, but in the context of the others you can see it as an arc on F and the original arc of K from 1 to 2.