Complex hyperbolic geometry of knot complements

This morning there was a paper which caught my eye:

Deraux, M. & Falbel, E. 2015 Complex hyperbolic geometry of the figure-eight knot.
Geometry & Topology 19, 237–293.

In it, the authors study a very different geometric structure for the figure-eight knot complement, as the manifold at infinity of a complex hyperbolic orbifold.

Regina 4.96 and hyperbolic manifolds

For those of you who aren’t on regina-announce: Regina 4.96 came out last weekend.

There’s several new features, such as:

1. rigorous certification of hyperbolicity (using angle structures and linear programming);
2. fast and automatic census lookup over much larger databases;
3. much stronger simplification and recognition of fundamental groups;
4. new constructions, operations and decompositions for triangulations;
5. and more—see the Regina website for details.

You will find (1) and (2) on the Recognition tab, (3) on the Algebra tab, and (4) in the Triangulation menu.

If you work with hyperbolic manifolds then you may be happy to know that Regina now integrates more closely with SnapPy / SnapPea.  In particular, if you import a SnapPea triangulation then Regina will now preserve SnapPea-specific data such as fillings and peripheral curves, and you can use this data with Regina’s own functions (e.g., for computing boundary slopes for spun-normal surfaces) as well as with the in-built SnapPea kernel (e.g., to fill cusps or view tetrahedron shapes).  Try File -> Open Example -> Introductory Examples, and take a look at the figure eight knot complement or the Whitehead link complement for examples.

Finally, a note for Debian and Ubuntu users: the repositories have moved, and you will need to set them up again as per the installation instructions (follow the relevant Install link from the GNU/Linux downloads table).

Enjoy!

– Ben, on behalf of the developers.

When are two hyperbolic 3-manifolds homeomorphic?

A preprint of Lins and Lins appeared on the arXiv today, posing a challenge [LL].  In this post, I’m going to discuss that challenge, and describe a recent algorithm of Scott–Short [SS] which may point towards an answer.

The Lins–Lins challenge

The theory of 3-manifolds is now very advanced, and we can even say in a certain sense that we understand ‘all’ 3-manifolds (as I discussed in an earlier post).  But that understanding is very theoretical; the Lins–Lins challenge is to put this theory into practice.

They ask: ‘Are the two closed, hyperbolic 3-manifolds given by Dehn surgery on the following two framed links homeomorphic?’

(I’ve taken the liberty of copying the diagrams from their paper.)

Homogeneous hyperbolic groups

I just got back from an extremely enjoyable meeting in Montreal, where I learned some nice new results about homogeneity, a natural logical property of groups. Now, I realise that logic may seem like a distant and irrelevant subject to many topologists, but I hope you’ll bear with me, as in this case I think there’s a very interesting relationship between logic and geometric group theory. If you need to be further convinced, perhaps it would help if I told you that this circle of ideas is intimately connected to Sela’s solution to the homeomorphism problem for hyperbolic manifolds. (Perhaps I’ll write some more about that on another occasion.)

SnapPy: Computing with hyperbolic 3-manifolds for fun and profit

This post is so Marc Culler and I can announce SnapPy, a computer program for studying hyperbolic structures on 3-manifolds. It is based on Jeff Weeks’ SnapPea kernel from his very influential program of the same name, written in the early 1990s for Macintosh computers. While Jeff’s program doesn’t work (except in emulation) on any computer you can buy today, SnapPy runs on Mac OS X, Linux, and Windows. SnapPy combines a link editor and 3D-graphics for Dirichlet domains and cusp neighborhoods with a powerful command-line interface based on the Python programming language.

You can download it here, and we put some effort in making it trivial to install for OS X and Windows, and reasonably easy on Linux. There are screenshots of it in action, or you can watch an 11-minute tutorial on YouTube. Unlike previous Python interfaces to the SnapPea, this one has decent documentation and also useful graphics. SnapPy can also be used within my favorite general purpose mathematical software package Sage and has some extra features there for dealing with finite covers.

SnapPy was written by Marc Culler and myself, using Jeff’s kernel code, and today we released version 1.0 (superseding 1.0a and 1.0b). If you successfully install SnapPy, or installed it earlier this summer, leave a note in the comments mentioning what type of system you’re using. (We test it on seven or eight different setups, but you never know what happens out there in the wild.) Enjoy!

Surfaces in hyperbolic groups

Not long ago, Jesse was kind enough to blog about my recent work with Cameron Gordon. Now that the preprint has finally appeared on the arXiv, I thought I’d say a few more words about what we do, and why.

The context is Gromov’s famous question about surface subgroups.

“Does every one-ended word-hyperbolic group contain a subgroup isomorphic to the fundamental group of a closed surface?”

Heegaard splittings and Hyperbolic knots

(Reprinted from my old ldt blog)

There are two gaping holes in our understanding of Heegaard splittings that, from a moral standpoint, really shouldn’t be there: First, there are no examples of closed hyperbolic 3-manifolds for which the minimal genus Heegaard splittings (let alone all Heegaard splittings) are classified. Second, there are no examples of closed three manifolds that are known to have irreducible, weakly reducible Heegaard splittings that are not minimal genus.

Moriah and Sedgwick’s recent paper [1] on twisted torus knots in the 3-sphere takes the first step towards solving the second of these problems, and perhaps indirectly towards the first one. A twisted torus knot is the image of a torus knot after integral Dehn surgery along an unknot that goes around some number of parallel strands. In other words, start with a torus knot, grab a few consecutive strands, then twist these around each other. Moriah and Sedgwick show that if the number of twists is high enough, the complement of the knot admits a unique isotopy class of genus two Heegaard splittings. (The Heegaard splitting is constructed from the one Heegaard splitting of the torus knot that is preserved by the construction that turns it into a twisted torus knot.)

The proof uses a nice result that was originally proved by Moriah and Rubinstein [2], using geometric methods, then proved again by Reick and Sedgwick [3] using purely combinatorial methods. The result says that with the exception of a certain set of slopes, every bounded genus Heegaard splitting for the result of Dehn surgery on a link is the image of a Heegaard splitting for the link complement. In the case that Moriah and Sedgwick consider, the link is the union of the torus knot and the unknot. The genus two Heegaard splittings of the link complement can be classified fairly easily. Moriah and Rubinstein’s Theorem then implies that for most Dehn surgeries on the unknot component, the only genus two Heegaard splitting will be the image of the unique genus two Heegaard splitting of the link complement.

Now, I know what your saying: What does this have to do with finding an irreducible, weakly reducible, non-minimal genus Heegaard splitting? Well, given a Heegaard splitting for a knot complement, there are two ways to get a higher genus Heegaard splitting: One is to stabilize your original splitting, but this always produces a reducible Heegaard splitting. The second way is to take a boundary parallel torus and tube it to the original Heegaard surface by a vertical arc, producing a new Heegaard surface. In many cases, this Heegaard splitting is reducible, but Moriah and Sedgwick conjecture that it may not always be. In particular, in the case they look at if this new Heegaard splitting is reducible then it is a stabilization of the original (unique) genus two Heegaard splitting. Thus to show that it is irreducible, one would only need to show that it is not isotopic to a stabilization of the genus two Heegaard splitting. It’s not obvious that this is significantly easier, but it does seem like a reasonable conjecture that this genus three Heegaard splitting might be irreducible. And a conjectured example is better than no example at all.

[1] Follow link to arXiv.
[2] Y. Moriah, H. Rubinstein, Heegaard structure of negatively curved 3-manifolds, Comm. in Ann. and Geom. 5 (1997), 375 – 412.
[3] Y. Reick , E. Sedgwick, Persistence of Heegaard structures under Dehn filling, Topology and its application 109 (2001), 41 – 53.

Hyperbolic geometry and normal surfaces

(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 [1] 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 [2] that every incompressible surface is isotopic to a normal surface and used this to prove some finiteness results. More recently, Stocking [3] and Rubinstein [4] 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.

[2] W. Haken, Theorie der Normalflachen: Ein Isotopickriterium fur der Kreisknoten, Acta Math. 105 (1961) 245–375.
[3] Stocking, Michelle, Almost normal surfaces in $3$-manifolds., Trans. Amer. Math. Soc. 352 (2000), no. 1, 171–207.
[4] 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

WYSIWYG Hyperbolic knots

(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 [1] 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 [2] 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 [3] 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.

[1] C. Adams. Unknotting tunnels in hyperbolic 3-manifolds. Math. Ann. 302 (1995), 177–195.

[2] and [3] available on the ArXiv via the links above.

The Topology of Neural Networks, Part 2: Compositions and Dimensions

In Part 1 of this series, I gave an abstract description of one of the main problems in Machine Learning, the Generalization Problem, in which one uses the values of a function at a finite number of points to infer the entire function. The typical approach to this problem is to choose a finite-dimensional subset of the space of all possible functions, then choose the function from this family that minimizes something called cost function, defined by how accurate each function is on the sampled points. In this post, I will describe how the regression example from the last post generalizes to a family of models called Neural Networks, then describe how I recently used some fairly basic topology to demonstrate restrictions on the types of functions certain neural networks can produce.