Commensurability

Here’s an interesting construction that I recently encountered: It’s possible to find a knot in the 3-sphere whose complement has a finite cover that is also a 3-sphere knot complement. Let K be a knot with a Dehn surgery producing a lens space (for example a Berge knot). The lens space is finitely covered by the 3-sphere and the image of K lifted to the 3-sphere is a new knot K’ . The complement of K’ finitely covers the complement of K. It turns out this is the only way to build a knot complement covered by a knot complement, which is proved in [1]. It also shows up in Reid and Walsh’s paper on commensurability classes of 2-bridge knots [2]. (For the record, it was grad. student Neil Hoffman of UT Austin who told me about this construction.)

Two compact 3-manifolds are called commensurable if one has a finite cover that is homeomorphic to a finite cover of the other. Two groups are called commensurable if one has a finite index subgroup that is isomorphic to a finite index subgroup of the other. I don’t know which definition came first, but thanks to some basic algebraic topology, the definitions are more or less equivalent: A cover of a topological space is uniquely determined by (the conjugacy class of) a subgroup of the fundamental group. Thus if two 3-manifolds are commensurable then their fundamental groups are commensurable. Conversely, a closed (or cusped) hyperbolic 3-manifold is determined by its fundamental group. Thus two hyperbolic 3-manifolds have commensurable fundamental groups if and only if they’re commensurable. Note that commensurable groups are quasi-isometric, so these ideas are related to coarse geometry as well.

Walsh and Reid prove their result by showing that in the commensurability class for a knot complement, there is a unique minimal element (i.e. it is covered by every 3-manifold in the class) that is the quotient of the hyperbolic plane by the normalizer in Isom(H^3) of the group of isomitries that produce the knot complement. (The knot complement is a regular cover of this manifold, so they need to show that the knot complement doesn’t have any hidden symmetries, i.e. doesn’t irregularly cover any smaller 3-manifold.) They then show that this minimal element of the commensurability class covers exactly one knot complement.

[1] F. Gonzalez-Acuna and W. C. Whitten, Imbeddings of three-manifold groups,
Mem. Amer. Math. Soc. 474 (1992).

Triangulations of the solid torus

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.

Update: Spacial graphs

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.

Normal surfaces and the disjoint curve property.

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.

Writing math papers on a cell phone?

The New York Times has a story about a new literary trend in Japan: novels written on (and often read on) cell phones. I occasionally wonder if getting a cell phone with word processing capabilities would improve my productivity, but I’m pretty sure the answer is no. What I found interesting about the article is that writing on cell phones appears to have a distinct impact on the resulting literary style, characterized by shorter sentences and simpler words. In particular it mentions near the end that after one cell phone author switched to writing on a regular computer (because her thumb nails had begun cutting into her skin) her sentences became longer and her vocabulary increased.

While I don’t know of any mathematicians who have started composing their papers on cell phones, there has been an evolution of writing media from pens to typewriters to computers and I wonder what sort of effect this has had on the style of math papers. I tend to compose papers entirely on the computer from brainstorming to final revisions (though I do print them out to proof read them). Of course, I started writing math papers with computers so I have nothing of my own to compare it to. Have any of you, dear readers, noticed a change in your writing or in others’ writing that might be attributed to a shift from typewriters to computers?

Undergrad research problems?

Now that this blog is starting to get some traffic, I thought I’d take advantage of the comments box by asking if anyone had thoughts on open problems in low dimensional topology that would be suitable for undergrads, in particular something that might work for a summer REU. There seems to have been at least one successful undergraduate project on intrinsically knotted/linked graphs and I saw a presentation by Denise Halverson at last year’s Spring Topology and Dynamics conference about an undergrad. project to find the smallest connected set containing a given collection of points in different geometric surfaces. I’m trying to put together an undergraduate project for this summer having to do with normal loops in triangulated surfaces. (This seems like an approachable problem for someone with minimal background and it might yield some useful ideas for thinking about normal and almost normal surfaces.) Any thoughts on how to find a good undergraduate research project, or what areas of low dimensional topology might have good problems?