Low Dimensional Topology

April 29, 2008

Primitive disks and lens spaces

Filed under: 3-manifolds,Curve complexes,Heegaard splittings — Jesse Johnson @ 11:16 am

Here’s an interesting example that Sangbum Cho (a student of Daryl McCullough at Oklahoma) showed me: If I did things correctly, there should be a picture below of a genus to handlebody with some simple closed curves drawn in its boundary surface. The two blue loops form a Heegaard diagram for the lens space L(3,1). The red, green and orange loops are the boundaries of disks in the handlebody.

A disk in a handlebody of a Heegaard splitting is called primitive if there is a disk in the other handlebody such that the two boundary loops intersect in a single point. The thing about a primitive disk is that compressing the Heegaard surface across a primitive disk produces a new, lower genus Heegaard splitting for the same 3-manifold. (The original Heegaard splitting is a stabilization of the new one.) Notice that each of red, green and orange loops intersects one of the blue loops in a single point (and the other blue loop in possible more points).

In the curve complex, one can consider the subset consisting of boundaries of primitive disks for each of the handlebodies in a Heegaard splitting. This comes up, for example, in Cho and McCullough’s work on the tree of unknotting tunnels and Cho’s work on the Goeritz group. In the example above, the three primitive disks form a pair of pants decomposition for the surface, corresponding to a maximal (two) dimensional simplex in the curve complex.

The interesting thing is that the dimension of the set of primitive disks for genus two Heegaard splittings of lens spaces depends on the lens space. If you try to generalize the diagram above, you can find a family of lens spaces (with criteria in terms of the continued fraction expansion of p/q) that have two dimensional primitive sets. Sangbum has a nice proof (though I can’t reproduce it here) that these lens spaces are the only ones that have a two dimensional primitive set. All other lens spaces have a one dimensional primitive set.

Although the primitive set has important connections (especially for someone like me who’s obsessed with stabilization), I don’t know of any direct applications of knowing the dimension of the set. But it is pretty interesting that it can vary within a class of such similar seeming manifolds. I think it would be interesting to see what this set can look like for general 3-manifolds. For example, does every 3-manifold have a Heegaard splitting for which the primitive set has maximal dimension?

January 16, 2008

Algorithms to find geodesics in the curve complex

Filed under: Curve complexes — Jesse Johnson @ 3:16 pm

John Hempel has suggested a simple method for finding reasonably efficient paths between loops in the curve complex. Recall that the curve complex is the simplicial complex whose vertices are isotopy classes of essential simple closed curves in a given surface and whose simplices bound collections of pairwise disjoint (pairwise non-isotopic) loops. A path in this complex is a sequence of loops in the given surface such that consecutive loops are disjoint. It’s not too hard to show that this complex is connected. A geodesic between two vertices is a path of minimal length.

This algorithm might generalize to surfaces with boundary, but I’m going to assume we have a closed surface. We start with two loops, say a and z that have been isotoped to intersect minimally and whose complement is a collection of disks. (If a component of the complement contains a non-trivial loop then they are distance one or two and finding a geodesic is trivial.) Each component of the complement is a polygon such that the edges of the polygon alternate between arcs of a and arcs of z. Put a vertex at the center of each such polygon and draw an edge from the vertex to each edge of the polygon that comes from an arc of z. We connect the edges along arcs of z to form a graph.

The vertices in squares in the complement are valence two so we can erase these vertices and think of only the vertices in the larger polygons. The resulting graph turns out to be a spine for the complement in our surface of the loop a. Any simple closed edge path determines an essential loop in the complement of a and Hempel suggests we let b be any such loop. I want to be a little more cautious. Since there are a finite number of simple closed edge paths, we might as well pick one that intersects z minimally. In fact, if we do this then we will have chosen a loop in the complement of a that minimizes the intersection number with z over all loops in the complement of a. (This is a reasonably straightforward exercise.) We can then repeat the process with b and z and so on until we get to a loop that is distance two from z.

On principle, this process should give us a very efficient path. The question is, does it yield a geodesic? Hempel has shown that it does for very short paths (I think he said it works for distance three or four). Now, there is already an algorithm for computing geodesics in the curve complex, due to work of Kenneth Shakleton. However, Hempel’s algorithm is extremely simple. The algorithm points in the direction that locally seems best, always choosing a loop that is distance one from a and intersects z minimally. It would be very interesting to see if such a simple idea could be effective for understanding an object as complex and unforgiving as the curve complex.

December 3, 2007

Special Session at the AMS San Diego Conference

I want to mention the upcoming special session at the Joint Mathematics Meetings in San Diego for which I’m one of the organizers. (Abby Thompson and Robin Wilson are the other two.) We’ve got a whole day of talks ranging from old pros like John Hempel and Hyam Rubinstein to grad students just entering the field. (See the official program.) The title of the session, “Heegaard splittings, bridge positions and low dimensional topology”, refers to the fraternal relationship between Heegaard splittings of 3-manifolds and bridge positions (or bridge surfaces) for knots and links. A bridge surface for a link (which doesn’t appear to have a wikipedia entry) is a Heegaard surface for the ambient 3-manifold such that the link intersects each handlebody of the Heegaard splitting in boundary parallel arcs. (This definition is a generalization of the classical definition, in which the Heegaard surface is a sphere in S^3.)

The connection between Heegaard surfaces and bridge surfaces is initially motivated by the fact that in a branched cover over a link, a bridge surface for the link lifts to a Heegaard surface for the cover. In a few simple cases, this yields a one-to-one correspondence, but in most cases things aren’t so nice. (It breaks down because one has to consider equivariance.) There is still a strong analogy between the two setting, first demonstrated by Gabai’s generalization of bridge position to thin position for links. Scharlemann and Thompson showed that translating this definition to thin position for Heegaard splittings led to a more intuitive proof of an important theorem of Casson and Gordon. Thompson later translated Casson and Gordon’s theorem into the framework of bridge surfaces.

Since then, there have been a number of theorems that have been translated from one setting to the other. The goal of the special session is to find more ways in which our knowledge of one of the settings can be used to improve our knowledge of the other. The session runs a full day, starting with talks about knots and link in the morning, then a couple of talks about stabilizations of Heegaard splittings right before lunch. After lunch, we have talks about the curve complex, beginning with their applications to Heegaard splittings and then with their applications to tunnel-number-one knots. Following that, it’s two talks about surface automorphisms and handlebodies/Heegaard splittings, then a couple of miscellaneous talks (degree one mappings and hyperbolic geometry, respectively) and then two talks related to the Berge conjecture.

November 29, 2007

WYSIWYG geometry from Heegaard gluing maps

Recall that WYSIWIG stands for “What you see is what you get”. (I believe this term was first applied to topology by Steve Kerckhoff during his talk at Peter Scott’s birthday conference, but I haven’t been able to track down a reference.) WYSIWYG topology is the idea that there should be a more direct connection between the combinatorics of a topological object and its geometry.

The title of this entry refers a project that was started by Hossein Namazi and Juan Souto (which has since been expanded to include Jeff Brock and Yair Minsky), in which they use the gluing map of a Heegaard splitting to directly calculate (almost) hyperbolic metrics on the ambient manifolds. After waiting a number of years for a preprint to appear on the front, I found out last fall that the preprint had been available the whole time on Namazi’s home page. I will have to get into the habit of looking at peoples web pages for papers that haven’t made it to the front yet.

The first incarnation of the project, a joint paper between Namazi and Souto, considers Heegaard splittings in which the gluing maps are higher and higher powers of pseudo-Anosov surface automorphisms. Given such a Heegaard splitting, they construct a pinched negatively curved metric on the 3-manifold that is pinched less and less for higher powers. The second incarnation, by all four authors (and still in preparation) replaces the high power pseudo-Anosov automorphism with a criteria based on the curve complex, which they call high distance, bounded combinatorics.

Both incarnations begin with a construction related to the model manifold construction developed by Minsky for the ending laminations conjecture. (It is worth noting that Namazi was Minsky’s student.) By constructing what’s called a path hierarchy (due to Masur and Minsky), they construct a metric on a surface cross an interval that looks like a piece of a cusp of a hyperbolic 3-manifold and such that near the boundary of this manifold, the metric is (close to) compatible with a metric constructed on each handlebody. Brock and Souto have a related result (which they announced a few years ago but haven’t written up yet) showing that the distance in the pants complex defined by the Heegaard splitting is (assymptotically) related to the volume of the ambient hyperbolic 3-manifold. My understanding is that they use similar methods (a path hierarchy is very closely related to a path in the pants complex).

It is known that every 3-manifold with a Heegaard splitting of distance three or more is hyperbolic, but the only way to prove this is by showing that every Heegaard splitting of a toroidal of Seifert fibered 3-manifold has distance at most two, and then applying the geometrization conjecture/theorem. There’s no direct/constructive proof. In fact, the difficulty of getting even the results mentioned above shows how far the current knowledge is from getting a constructive proof for lower distance Heegaard splittings. Still, one may hope that the project mentioned above, and its future repercussions may eventually lead to a better understand of how even low distance gluing maps are related to the hyperbolic geometry of the resulting 3-manifold.

November 20, 2007

Quasi-isometries of the curve complex

Filed under: Curve complexes,Mapping class groups — Jesse Johnson @ 4:20 pm

A lot has been published in the last few years about the curve complex. This is a simplicial complex determined by a given surface, whose vertices are isotopy classes of simple closed curves in the surfaces and faces bounding sets of pairwise disjoint, non-parallel loops. Most of the motivation for studying it comes from its connections to mapping class groups, Teichmuller space and, recently, the ending laminations conjecture. It also makes good fodder for Gromov/coarse geometry techniques since its local structure is untenable. Of course, my interest in the curve complex is motivated by the fact that it’s very useful for understanding Heegaard splittings. But for this blog entry I want to talk about a result that may very well have no relevance to Heegaard splittings:

Kasra Rafi and Saul Schleimer recently posted a very nice, relatively short preprint [1] proving a number of results about quasi-isometries of curve complexes. A quasi-isometric map between two geometric objects is a map such that for any two points in the domain, the distance between their images is bounded above and below by a linear function, plus or minus a constant, of their distance in the original space. A quasi-isometry is a quasi-isometric map whose image is k-dense for some k. (In other words, every point in the range must be within distance k of the image of a point from the domain.) By allowing this sort of flexibility, one essentially ignores the local behavior of the map in favor of its large scale or asymptotic properties.

Every automorphism of a surface induces an isometry of its curve complex, and it is known [2] that except in a few trivial cases, there is a one-to-one correspondence between isometries of the curve complex and automorphisms of the surface. The isometries are completely determined by their large scale behavior, in the sense that the only isometry that moves each point a bounded distance is the identity isometry. (This is true for any finite bound.) Rafi and Schleimer have shown that for most surfaces (those whose curve complex has a connected Gromov boundary) every quasi-isometry of the curve complex is a bounded distance from an actual isometry.

It is a reasonably simple exercise to show that two curve complexes are isometric if and only if their underlying surfaces are homeomorphic. Rafi and Schleimer’s result imples that two curve complexes are quasi-isometric if and only if their underlying surfaces are homeomorphic. The proof uses a result of Ursula Hamenstaedt [2], which states that for any Cayley graph of the mapping class group of a surface, every quasi-isometry is a bounded distance from a true isometry.

November 19, 2007

Fibered knots and the tree of unknotting tunnels

Filed under: Curve complexes,Heegaard splittings,Knot theory — Jesse Johnson @ 5:33 pm

(Reposted from my old ldt blog)

Here’s the question: what are all the degree two, tunnel number one fibered knots? A knot in the 3-sphere is fibered if its complement is a surface bundle. An unknotting tunnel for a knot is an arc with its endpoints on the knot such that the complement of the knot and the arc is a genus two handlebody, and a knot has tunnel number one if it has an unknotting tunnel. Cho and McCullough [1] showed that every knot and unknotting tunnel is determined by a unique sequence of what they call cabling operations. (Finding the sequence of cablings is easy, showing that it’s unique is the hard part.) The degree of a knot and unknotting tunnel is the number of cabling operations necessary to produce the knot and tunnel.

The problem is that while the cabling move is very natural as a way of modifying the knot, it is unclear how to understand what it does to the homeomorphism type of the knot complement, in particular when it should produce a fibered complement. A sequence of cabling operations is described by a sequence of rational numbers and the long term motivation behind this question would be to list all sequences of a fixed length that describe fibered knots.

Degree one knots are precisely 2-bridge knots, and Gabai [2] classified the rational numbers that determine fibered two-bridge knots. The question above is really asking: When does cabling a 2-bridge knot and its upper or lower tunnel by a rational tangle produce a fibered knot? As a preliminary sub-question, one might ask if it is possible to get a fibered degree two knot by cabling a non-fibered 2-bridge knot. It is probably too optimistic to hope for a “no” answer to this second question.

[1] Follow link to the ArXiv.

[2] Genera of the arborescent links. Mem. Amer. Math. Soc. 59 (1986), no. 339, i–viii and 1–98.

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