The idea of stabilizing Heegaard splittings (which I’ve mentioned quite a few times) was discovered independently by Kurt Reidemeister and James Singer, each of whom published a paper about it in 1933. Singer’s paper was published in Transactions of the AMS and can be found on JSTOR. Reidemeister’s paper, on the other hand, appeared in a more obscure journal, Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg. This journal is so obscure that MathSciNet only lists its contents back to 1987, a good 54 years after Reidemeister’s paper. (I didn’t realize there were gaps in MathSciNet, but surprise!)
Needless to say, the contents from the 1933 issue aren’t available on line. However, it turns out Yale has the journal going back to the very first issue. Since stabilization is rather important to me, I though I’d scan in the paper. I then discovered I could run it through optical character recognition software to turn the ugly scanned PDFs into a fairly nice .tex file, which I then cleaned up and fixed the math symbols. The final product looks pretty nice (though it’s still in German). If you want to see which typos are original and which were introduced by me, you can also download the original scans. I know this probably violates the paper’s copyright, but I hope whoever owns the copyright won’t mind.
Also, while I’m at it, here’s a link to Waldhausen’s paper (also in German) on classifying Heegaard splittings of the 3-sphere. If you don’t want to learn German, you can get a good idea of the proof by reading this paper by Loretta Bartolini and Hyam Rubinstein, which uses a very similar proof to classify one sided Heegaard splittings of RP^3. Or (as Andy Putman has pointed out) you can read Saul Schleimer’s account of Waldhausen’s proof.
The layered triangulations I described in my last post reminded me of another construction that uses the same idea of layering tetrahedra onto a triangulated torus in order to change the boundary pattern. David Futer and Francois Gueritaud [1] use this idea to construct ideal triangulations of (most) punctured surface bundles.
Start with an ideal triangulation of a punctured torus (i.e. triangulate a torus with two triangles, then remove the vertex). Any two of the edges cut the torus onto a rectangle, and the third edge forms a diagonal of this rectangle. Take a tetrahedron, choose one of its edges and glue this edge and the two adjecent triangles to the diagonal edge and its two adjacent triangles. The result is a slightly bulging/thickened punctured torus. The triangulation on the top of the bulging torus comes from the bottom by removing the diagonal edge of the rectangle and replacing it with the other diagonal. (This is called a diagonal exchange move.) If we keep gluing in tetrahedra along different edges of the top, we eventually get a space homeomorphic to a punctured torus cross an interval. If we glue the top triangulation to the bottom, we get a punctured torus bundle. The order in which we glued in tetrahedra determines a sequence of diagonal exchange moves that defines the final gluing map and induces an ideal triangulation on the resulting manifold.
I think this construction is beautifully simple, but David and Francois had even more impressive things in mind. Because they understand the structure of the triangulation so well, they are able to apply angle-structure techniques to show that (for monodromies meeting appropriate combinatorial conditions) one can assign angles to the edges of the tetrahedra so that the angles around each edge add up to 360 and each tetrahedron looks like an ideal hyperbolic tetrahedron. With some more heavy machinery, this implies the existence of a hyperbolic structure on the torus bundle. As I understand it, they are currently working on extending the techniques (along with a third author who I can’t remember at the moment) to determine exactly which Dehn fillings on these punctured torus bundles produce hyperbolic manifolds. This doesn’t seem to be on the arXiv yet, but I’m looking forward to it.
Ken Baker has posted some pictures of the one tetrahedron triangulation of the solid torus to his blog. This is a surprisingly hard construction to visualize, given how simple it is. (Ken showed me the 3D models at the Georgia topology conference and after ten minutes of looking at them from different angles, I still couldn’t quite see it.) Start with a tetrahedron and choose two of the sides (it doesn’t matter which two.) There are three orientation reversing ways to glue the two sides together (an orientation preserving gluing would produce a non-orientable manifold) – You can just fold them over the common edge, or you can glue them by a 2pi/3 rotation clockwise or counter-clockwise. The first gluing gives you a 3-ball. The other two produce a solid torus, and one of these is shown in Ken’s pictures. (The blue triangles in the second picture show a Mobius band formed by the two glued triangles.)
This triangulation has a single vertex, which is in the boundary of the solid torus. The boundary is triangulated with two triangles (the two that aren’t glued to anything). One can produce other triangulations by taking another tetrahedron and gluing two of its faces to the two faces in the boundary. This produces a new solid torus with a different triangulation of its boundary. The process can be repeated to produce any desired triangulation of the boundary. The resulting triangulation of the solid torus is called a layered triangulation. Gluing the last two faces to each other produces a lens space, and the induced triangulation, also called a layered triangulation, is studied in a paper by Jaco, Rubinstein and Tillman that just hit the arXiv a few days ago [1]. They show that for a certain family of lens spaces, the minimal layered triangulation is in fact minimal among all triangulations of the lens space. They conjecture that this is true for all lens spaces, so I’m adding that to the open problem list as a question.
A few months ago I asked whether it was possible to embed a theta graph in the 3-sphere so that all the edge loops were isotopic to the trefoil knot. Well it turns out that the answer was already known (and moreover, my intutition was way off.) A much stronger result was proved by Kouki Taniyama and Akira Yasuhara [1]. Given a graph, they ask you to associate to each edge loop an isotopy class of knots. They call a graph adaptable if for any choice of knots, the graph can be embedded in the 3-sphere so that each edge loop is sent to a knot in the associated isotopy class. They not only find a reasonably large class of adaptable knots (all of which contain theta graphs as minors) but they find all graphs that are minor-minimal non-adaptable. (Such a graph is not adaptable, but each of its minors is.) Thus a theta graph can be embedded to contain three trefoils, or any choice of three knots. I haven’t read the paper carefully enough to figure out how to construct such a graph, but I’m curious how complicated a three-trefoil embedding would look.
[1] Realization of knots and links in a spatial graph. (English summary) Topology Appl. 112 (2001), no. 1, 87–109.
I just got back from the Georgia topology conference. While I was there, I talked to a number of contact topologists, and the conversations mostly revolved around open book decompositions. I’ve mentioned the links between Heegaard splittings, open books and contact structures previously, but I wanted to reiterate my feeling that his should be an interesting area to study. It’s very easy to go from an open book to a Heegaard splitting or from an open book to a contact structure. Conversely, every contact structure induces a whole family of open books, but there is no good way (so far) to get from a Heegaard splitting to an open book. Many Heegaard splittings (in particular, high distance ones) don’t come from any open book, while others could come from lots of unrelated open books.
I don’t have anything else to say about possible connections, other than to suggest studying the way a sweep-out for a Heegaard splitting passes through the contact structure. One could, for example, consider the family of foliations of the surfaces that come from intersecting the sweep-out surfaces with the contact planes. There are probably better ways to analyze the intersection of the sweep-out surfaces and the contact planes as well, but I don’t know what they are. I’m going to add the following question to the open problems page, which I think roughly captures the problem: Is there a connection between the existence of a tight/fillable/etc. contact structure and the existence of a high distance Heegaard splitting? In other words, if a 3-manifold admits one of these types of contact structure, does that imply the existence, or perhaps rule out the existence of a high distance Heegaard splitting?
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.
I’ve been meaning to write about a recent preprint of Kobayashi and Rieck [1] that improves a result of Saul Schleimer’s [2]. Saul showed that for every 3-manifold, there is a value k such that every Heegaard splitting for that manifold of genus greater than k has the disjoint curve property (i.e. Hempel distance at most 2). Schleimer’s bound is an exponential function in terms of the number of tetrahedra in a minimal triangulation for the 3-manifold. Kobayashi and Rieck have improved the bound to a linear function of the number of tetrahedra.
These results are most interesting in the context of Tao Li’s work [3] [4] on branched surfaces and Heegaard splittings. Li showed (roughly) that every 3-manifold has a finite family of Heegaard splittings such that every irreducible Heegaard splitting for the manifold comes from Haken summing a surface from this finite family with a collection of incompressible surfaces. Combinging this with Schleimer’s result implies that in an atoroidal 3-manifold there are finitely many high distance splittings. The splittings that come from repeated Haken summing (and therefore have high genus) must all have the disjoint curve property. This suggests that there is a sort of fundamental distinction between the finitely many high distance splittings and the possibly infinitley many low distance ones.
Both Schleimer’s proof and Kobayahsi-Rieck’s proof use normal surface theory. Recall that a surface in a 3-manifold is normal with respect to a given triangulation if it intersects each tetrahedron in a collection of (normal) triangles and quadrelaterals. A surface is almost normal if it intersects each tetrahedron in a collection of triangles and quadrelaterals plus its intersection with exactly one tetrahedron also contains an octagon or an annulus whose boundary loops each intersect three or four edges.
Notice that there are two types of almost normal surfaces: those with octagon pieces and those with annulus pieces. If you will allow me to descend into sheer speculation, I’d like to suggest that there should be some sort of connection between the octagon/annulus dichotomy and the high distance/low distance dichotomy.
This speculation is motivated by the fact that in a tube almost normal surface, compressing along the tube produces a normal surface that bounds a handlebody on one side. If this handlebody is not a regular neighborhood of a subcomplex of the triangulation then the normal structure on the surface induces a two dimensional spine for the handlebody that has no order one edges. This two dimensional complex is homtotopy equivalent to a graph, but it is not collapsible so it’s a higher genus version of something like the house with two rooms. In a reasonable triangulation, one would hope to be able to avoid this sort of pathological behavior. (Note that for an octagon normal surface, the induced spines on both handlebodies have order one edges, so they could easily be collapsible onto graphs.)
I don’t know enough about normal surfaces to suggest a specific conjecture or question that would sum up what I’m trying to get at. I think it’s very unlikely that every octagon normal surface has the disjoint curve property and I know that every tube almost normal surface will not have high distance. I will thus leave it as a vague suggestion that there should be a more subtle connection lurking just in the background of all this normal surface/Hempel distance business.
