Bill Thurston passed away yesterday at 8pm, succumbing to the cancer that he had been battling for the past two years. I don’t think it’s possible to overstate the revolutionary impact that he had on the study of geometry and topology. Almost everything we blog about here has the imprint of his amazing mathematics. Bill was always very generous with his ideas, and his presence in the community will be horribly missed. Perhaps I will have something more coherent to say later, but for now here are some links to remember him by:
August 22, 2012
August 21, 2009
The basic tenet of Thurston’s approach to 3-manifolds is the idea that the topology of a 3-manifold should determine a fairly canonical geometric structure on the manifold (modulo some technicalities). This suggests that there should be a dictionary for translating topological features of the manifold into geometric features, an idea that has been called WYSIWYG (what you see is what you get) topology. (I think Steve Kerckhoff coined the term.) While the Theorems relating the topology of Heegaard splittings to geometry are still a bit coarse, they give a very nice intuitive picture that I think is useful, and I would like to describe this picture below.
August 20, 2009
Hyperbolic 3-manifolds with a single cusp have canonical ideal triangulations constructed by Epstein and Penner via a certain convex hull construction involving the light-cone in the hyperboloid model of hyperbolic space. (Occasionally, these “triangulations” are really cellulations more complicated cells.) When such a 3-manifold fibers over the circle, there is another type of natural triangulation, called a layered triangulation. Roughly, one starts with a certain ideal triangulation of the fiber surface, looks at the image of this triangulation under the bundle monodromy, interpolates between these by a series of Pachner moves which can then be realized geometrically by layering on tetrahedra.
When the fiber is a once-punctured torus, these two type of triangulations coincide. This was shown by Marc Lackenby using a remarkably soft and elegant argument. It’s natural to wonder whether this phenomena occurs more broadly. For instance Sakuma suggested considering the following:
Conjecture: Canonical triangulations of punctured surface bundles are always layered.
Saul Schleimer and I have discovered that this is false in general. In particular, the manifold v1348 from the SnapPea census is fibered by a once-punctured surface of genus 5 yet has a canonical triangulation which is not layered. Precisely, the canonical triangulation does not admit one of Lackenby’s taut structures so that the resulting branched surface carries something with positive weights. Note that v1349 is in fact the complement of a certain knot in the 3-sphere [CFP].
Technical details: The canonical triangulation of v1348 is in fact just the triangulation encoded in the SnapPea census (and it is a triangulation, not a celluation). It’s easy to check that it fibers using the BNS invariant (cf. [DT]), and compute the genus of the fiber from the Alexander polynomial. One then checks that there are no taut structures of this type using Marc Culler and I’s t3m Python package.
January 11, 2008
I mentioned in a previous post recent work by Cho and McCullough about the tree of unknotting tunnels. At the AIM a few weeks ago, Martin Scharlemann discussed their work and some of the earlier work that led to it, in particular his paper with Goda and Thompson . They showed that given a tunnel-number-one knot in minimal bridge position, one can always slide the tunnel around so that it sits in a level sphere. While sliding the tunnel around, one may need to pass the end points of the tunnel past each other along the knot. If we think of the theta graph (the knot union its tunnel) as the spine of a genus two handlebody, sliding the endpoints past each other corresponds to picking new meridians for the handlebody. (The meridian dual to the tunnel is uniquely determined, but the other two meridians are not.) By making the tunnel level, we choose a pair of meridians for the knot relative to the tunnel.
In Cho and McCullough’s work , one finds a sequence of theta graphs, starting with a standardly embedded graph, related by simple moves such that the final theta graph is isotopic to the knot union tunnel that you were looking for. Their theorem states that this sequence (the way they define it) is unique. In his talk at the AIM, Scharlemann pointed out that because the sequence is unique, the final pair of meridians for the knot relative to the tunnel are uniquely determined. Moreover, it turns out that this pair of meridians is the same as the pair that one gets by leveling the tunnel a la Goda-Scharlemann-Thompson.
In my previous post about Cho and McCullough’s result, I mentioned the (poorly defined) problem of trying to understand how the cabling moves they define affect the knot complement. David Futer has suggested one way to interpret the moves might be to put a cone hyperbolic metric on the knot complement. Then when one carries out the cabling move, one deforms the metric around the tunnel into a cone, then continues deforming until another edge of the theta graph becomes non-cone points. This might also be useful in answering the question of whether every unknotting tunnel in a hyperbolic tunnel-number-one knot is isotopic to a geodesic. More immediately, though, it suggests the following question: If we have a tunnel in a hyperbolic knot that happens to be a geodesic in the knot complement, does it define a unique pair of meridians for the knot relative to the tunnel, and if so are these the same as those defined by leveling the tunnel with respect to a bridge sphere?
The question of whether the geodesic determines a unique pair of meridians really has to do with how the hyperbolic manifold is embedded in the knot complement. It shouldn’t be too hard to work out, but I’ll leave that for the geometers to work out. The second part of the question above is probably much harder.
 H. Goda, M. Scharlemann, and A. Thompson. Leveling an unknotting tunnel. Geometry and Topology, 4:243–275, 2000. ArXiv:math.GT/9910099.
November 29, 2007
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 19, 2007
(Reprinted from my old ldt blog)
I’ve mentioned before the idea of WYSIWYG (what you see is what you get) topology, the idea that we should be able to use the combinatorial data in structures like trianglulations and Heegaard splittings in order to understand the (hyperbolic) geometry of the ambient manifold. Mark Lackenby’s recent preprint  does the converse, using the geometry of a hyperbolic knot complement to show that there is an algorithm to calculate its tunnel number (aka Heegaard genus minus one). The algorithm is based on normal surface theory, which has been around for a while, but uses the hyperbolic geometry to fix the main problem with normal surface theory: normal tori.
A normal or almost normal surface is one that intersects a triangulation in a simple way that allows one to write the surface as an integral solution to a certain collection of linear equations. Because triangulations adapt well to algorithms, many things about normal and almost normal surfaces can be calculated (for example by using the program Regina). Haken showed  that every incompressible surface is isotopic to a normal surface and used this to prove some finiteness results. More recently, Stocking  and Rubinstein  showed independently that every strongly irreducible Heegaard surface is isotopic to an almost normal surface. Thus to study the Heegaard splittings of a 3-manifold (in particular to calculate the Heegaard genus/tunnel number), one can try to understand the almost normal surfaces for some triangulation.
The sum of two solutions to a set of linear equations is also a solution, so the correspondence between normal/almost normal surfaces and solutions to the equations mentioned above suggests that there should be some way to “add” surfaces. It turns out adding the solutions corresponds to a construction called a Haken sum, in which the loops of intersection between two embedded surfaces are “resolved” to form a single (possibly disconnected) embedded surface. Haken showed that there is a finite collection of solutions (called fundamental solutions) that generate all solutions to the equations. Thus any normal/almost normal surface is a Haken sum of elements from a finite collection of normal/almost normal surfaces.
The Euler characteristic of a Haken sum is the sum of the Euler characteristics of the two surfaces. Thus Rubinstein suggested the following way to look for an algorithm to calculate the Heegaard genus of a 3-manifold: For each Euler characteristic starting from zero, look at all the Haken sums of the fundamental solutions that sum to that number. If one of them is a Heegaard splitting then you’re done. Otherwise, move on to the next Euler characteristic. The problem is that if one of the fundamental solutions has Euler characteristic zero then taking a Haken sum with it does not change the Euler characteristic. There are then an infinite number of solutions one needs to check.
In the paper mentioned above, Lackenby offers the following solution to this problem: He shows that every hyperbolic 3-manifold with toroidal boundary admits a “partially flat, angled ideal triangulation”. This is an ideal triangulation (the interior of the 3-manifold is identified with a simplicial complex minus its vertices) with an angle assigned to each edge of each tetrahedron, with certain conditions (in particular, the angles around each edge must sum to 2\pi). A normal/almost normal surface in this triangulation is cut into polygons whose edges inherit angles from the edges of the tetrahedra. Lackenby shows that the conditions on the angles of the tetrahedra imply that the only normal torus is boundary parallel. This eliminates the problem of having infinitely many normal surfaces of the same Euler characteristic, leading to an algorithm to calculate the Heegaard genus of the 3-manifold.
 W. Haken, Theorie der Normalflachen: Ein Isotopickriterium fur der Kreisknoten, Acta Math. 105 (1961) 245–375.
 Stocking, Michelle, Almost normal surfaces in $3$-manifolds., Trans. Amer. Math. Soc. 352 (2000), no. 1, 171–207.
 J.H. Rubinstein, Polyhedral minimal surfaces, Heegaard splittings and decision problems for 3-dimensional manifolds, in ‘Geometric Topology (Athens, GA, 1993)’, Volume 2 of AMS/IP Stud. Adv. Math., 1–20, Amer. Math. Soc., Providence, RI, 1997
November 18, 2007
(Reprinted from my old ldt blog)
For my first blog entry I decided to write about a problem that combines a topic about which I know something (unknotting tunnels for knots) with a topic about which I should know more (hyperbolic geometry). Here’s the question: is every unknotting tunnel for a hyperbolic knot isotopic to a geodesic in the hyperbolic structure on the knot complement?
Colin Adams  showed that this is true for tunnel-number-one links (with two components) using the following trick: Given a one tunnel link in the three sphere, there is an automorphism that is orientation preserving on S^3, reverses the orientation of the knot, and preserves the unknotting tunnel pointwise. This automorphism is isotopic to a hyperbolic isometry, and it just so happens that any set that is preserved by a hyperbolic isometry is convex. Thus the isotopy that makes the automorphism into an isometry takes the unknotting tunnel to a geodesic.
It turns out the same trick works for two bridge knots. However, David Futer showed  that these are the only one-tunnel knots that admit such an inversion, so the problem remains open for higher bridge number knots. Incidentally, Cho and McCullough’s work  on the tree of unknotting tunnels gives a new proof of Futer’s result.
The problem falls under what Steve Kerckhoff calls WYSIWYG (what you see is what you get) topology. The idea is that even though the recently proved geometrization conjecture tells that all 3-manifolds are composed of geometric pieces, it doesn’t tell us how the combinatorial structure corresponds to the geometry. The goal of WYSIWYG topology is to see exactly how the combinatorial picture corresponds to the geometric picture. Here’s another closely related question: Given a minimal crossing diagram for a hyperbolic knot, draw an arc from the upper arc at a crossing straight down to the lower arc. Is this arc isotopic to a geodesic in the hyperbolic structure? Both this and the original question ask whether a certain combinatorially simple arc corresponds to a geometrically simple arc.
 C. Adams. Unknotting tunnels in hyperbolic 3-manifolds. Math. Ann. 302 (1995), 177–195.
 and  available on the ArXiv via the links above.