Low Dimensional Topology

January 31, 2011

Unknotting tunnels for (1,1) knots

Filed under: 3-manifolds,Heegaard splittings,Knot theory — Jesse Johnson @ 4:24 pm

An unknotting tunnel for a knot K in a 3-manifold is an arc \tau whose endpoints are in K such that the complement of K \cup \tau is an open genus two handlebody.  Or equivalently, if we let H^- be the closure of a regular neighborhood of K \cup \tau then the closure of its complement will be a second handlebody H^+ and their common boundary will be a Heegaard surface.  The problem of classifying isotopy classes of genus-two Heegaard surfaces for the complement of K  is equivalent to classifying unknotting tunnels for K, up to isotopies in which the endpoints of \tau are allowed to pass each other along K. Unknotting tunnels have been classified for two-bridge knots [1] and satellite knots [2] (the ones that allow unknotting tunnels.)  A few months ago, Goda and Hayashi posted two papers to the arXiv [3], [4] that present a nice classification for a class of knots called (1,1) knots, which I will describe below the fold.

A knot K in the 3-sphere is (g, n) if there is a genus g Heegaard splitting for S^3 such that the intersection of K with each of the handlebodies in the splitting is a collection of n boundary-parallel arcs. (And this definition can also be used for other 3-manifolds.)  This is a generalization of n-bridge knots, in which we replace the bridge sphere with a higher genus surface.  So for a (1,1) knot K, there is an unknotted torus T that hits K in two points such that the arc of K on either side of T is parallel into T.  However, we can’t necessarily isotope both of them into the surface at the same time.  In particular a (1,1) knot is defined by a braid in the torus, and this is what makes (1,1) knots more general than, say, two-bridge knots.  (Here’s a fun exercise: Show that every two-bridge knot is a (1,1) knot.)

Every (1,1) knot has two unknotting tunnels that come from its one-bridge position:  In one of the solid tori bounded by T, one can attach \tau with its endpoints on the arc in the solid torus so that \tau and a piece of the arc in K form a core of the solid torus.  The remainder of K is parallel into the boundary of this solid torus, which implies that the complement of K \cup \tau is a handlebody, i.e. that \tau is an unknotting tunnel.  These unknotting tunnels are called meridionally stabilized because there is a pair of stabilizing disks for the induced Heegaard splitting such that one of the disks intersects K in a single point (i.e. a meridian.)

As I mentioned above, every two-bridge knot is a (1,1) knot, and it is shown by Kobayashi [1] and Morimoto-Sakuma [2] that most two-bridge knots will have six distinct unknotting tunnels.  That means that from the (1,1) perspective, there are at least four other tunnels hiding somewhere.  So where do they come from?  Goda and Hayashi prove that there are three situations in which a (1,1) knot can have a Heegaard splitting that is not meridionally stabilized.

In the one situation, the unknotting tunnel can be isotoped into the bridge torus, but it cannot be made to define the core of either torus. Goda and Hayashi show that in this situation, one of the bridge arcs can be simultaneously isotoped into the bridge torus so that the two arcs form a torus knot. The remaining arc of K forms an unknotting tunnel for this torus knot.  In other words, we get K from the torus knot by replacing one of the arcs of the torus knot with the unknotting tunnel.  In Cho-McCullough’s tree of unknotting tunnels [5], this construction corresponds to moving one edge from a torus knot.  (However, this construction does not work for all unknotting tunnels for a torus knot!)

In another situation, the (1,1) knot K does not take advantage of the entire torus.  In particular, there is a loop \ell \subset T that is a longitude of one of the solid tori such that each arc of the knot K can be isotoped into T so that it is disjoint from \ell.  This class of knots includes 2-bridge knots, though it appears to be much more general.  It also does not appear that they can determine what the alternate unknotting tunnels are in this situation (I haven’t finished reading the paper yet.) so this might be an interesting area for further exploration.

The final situation is the most complicated and is the sole focus of the second of the two papers.  In fact, the authors mention that there are no known examples that meet this criteria, but they were unable to rule it out.  I won’t try to describe it here, because I don’t think I could write it up at all coherently.  But it does suggest a nice open problem: Find an example of a (1,1) knot with an unknotting tunnel that only fits into this category, or show that no such knot exists.

[1], [3], [4] and [5] are linked to preprints on the arXiv.

[2] Morimoto, Kanji; Sakuma, Makoto, On unknotting tunnels for knots. Math. Ann. 289 (1991), no. 1, 143–167.


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