by | August 15, 2014 · 2:03 pm
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Some open computational problems in link homology and contact geometry

I’m thrilled to join everyone at the best-named math blog.

I am just home from Combinatorial Link Homology Theories, Braids, and Contact Geometry at ICERM in Providence, Rhode Island.  The conference was aimed at students and non-experts with a focus on introducing open problems and computational techniques.  Videos of many of the talks are available at ICERM’s site.  (Look under “Programs and Workshops,” then “Summer 2014”.)

One of the highlights of the workshop was the ‘Computational Problem Session’ MC’d by John Baldwin with contributions from Rachel Roberts, Nathan Dunfield, Joanna Mangahas, John Etnyre, Sucharit Sarkar, and András Stipsicz.  Each spoke for a few minutes about open problems with a computational bent.

I’ve done my best to relate all the problems in order with references and some background.  Any errors are mine.  Corrections and additions are welcome!

Rachel Roberts

Contact structures and foliations

Eliashberg and Thurston showed that a C^2 one-dimensional foliation of a three-manifold can be C^0-approximated by a contact structure (as long as it is not the product foliation on S^1 \times S^2).  Vogel showed that, with a few other restrictions, any two approximating contact structures lie in the same isotopy class.  In other words, there is a map \Phi from C^2, taut, oriented foliations to contact structures modulo isotopy for any closed, oriented three-manifold.

Geography: What is the image of \Phi?

Botany: What do the fibers of \Phi look like?

The image of \Phi is known to be contained within the space of weakly symplectically fillable and universally tight contact structures.  Etnyre showed that if one removes “taut”, then \Phi is surjective.  Etnyre and Baldwin showed that \Phi doesn’t “see” universal tightness.

L-spaces and foliations

A priori the rank of the Heegaard Floer homology groups associated to a rational homology three-sphere Y are bounded by the first ordinary homology group: \text{rank}(\hat{HF}(Y)) \geq |H_1(Y; \mathbb{Z})|. An L-space is a rational homology three-sphere for which equality holds.

Conjecture: Y is an L-space if and only if it does not contain a taut, oriented, C^2 foliation.

Ozsváth and Szabó showed that L-spaces do not contain such foliations.  Kazez and Roberts proved that the theorem applies to a class of C^0 foliations and perhaps all C^0 foliations.  The classification of L-spaces is incomplete and we are led to the following:

Question: How can one prove the (non-)existence of such a foliation?

Existing methods are either ad hoc or difficult (e.g. show that the manifold does not act non-trivially on a simply-connected (but not necessarily Hausdorff!) one-manifold). Roberts suggested that Agol and Li’s algorithm for detecting “Reebless” foliations via laminar branched surfaces may be useful here, although the algorithm is currently impractical.

Nathan Dunfield

What do random three-manifolds look like?

First of all, how does one pick a random three-manifold?  There are countably many compact three-manifolds (because there are countably many finite simplicial complexes, or because there are countably many rational surgeries on the countably many links in S^3, or because…) so there is no uniform probability distribution on the set of compact orientable three-manifolds.

To dodge this issue, we first consider random objects of bounded complexity, then study what happens as we relax the bound.  (A cute, more modest example: the probability that two random integers are relatively prime is $6/\pi^2$.1).  Fix a genus g and write G for the mapping class group of the oriented surface of genus g.  Pick some generators of G. Let \phi be a random word of length N in the chosen generators.   We can associate a unique closed, orientable three-manifold to \phi by identifying the boundaries of two genus g handlebodies via \phi.

Metaquestion: How is your favorite invariant distributed for random 3-manifolds of genus g?  How does it behave as g \to \infty?  Experiment! (Ditto for knots, links, and their invariants.)

Challenge: Show that your favorite conjecture about some class of three-manifolds or links holds with positive probability. For example:

Conjecture: a random three-manifold is not an L-space, has left-orderable fundamental group, admit a taut foliation, and admit a tight contact structure.

These methods can also be used to prove more traditional-sounding existence theorems. Perhaps you’d like to show that there is a three-manifold of every genus satisfying some condition. It suffices to show that a random three-manifold of fixed genus satisfies the condition with non-negative probability! For example,

Theorem: (Lubotzky-Maher-Wu, 2014): For any integers k and g with g \geq 2, there exist infinitely many closed hyperbolic three-manifolds which are integral homology spheres with Casson invariant k and Heegaard genus g.

(Slides from this talk)

Johanna Mangahas

What do generic mapping classes look like?

Here are two sensible ways to study random elements of bounded complexity in a finitely-generated group.

  • Fix a generating set. Look at all words of length N or less in those generators and their inverses. (word ball)
  • Fix a generating set and the associated Cayley graph. Look at all vertices within distance N of the identity. (Cayley ball)

A property of elements in a group is generic if a random element has the property with probability, so the meaning of “generic” differs with the meaning of “random.” For example, consider the group $G = \langle a, b \rangle \oplus \mathbb{Z}$ with generating set $\{(a,0), (b,0), (id,1)\}$.  The property “is zero in the second coordinate” is generic for the first notion but not the second.  So we are stuck/blessed with two different notions of genericity.

Recall that the mapping class group of a surface is the group of orientation-preserving homeomorphisms modulo isotopy. Thurston and Nielsen showed that a mapping class \phi falls into one of three categories:

  • Finite order: \phi^n = id for some n.
  • Reducible: \phi fixes some finite set of simple closed curves.
  • Pseudo-Anosov: there exists a transverse pair of measured foliations which \phi stretches by \lambda and 1/\lambda.

The first two classes are easier to define, but the third is generic.

Theorem: (Rivin and Maher, 2006) Pseudo-Anosov mapping classes are generic in the first sense.

Question: Are pseudo-Anosov mapping classes generic in the second sense?

The braid group on n strands can be understood as the mapping class group of the disk with n punctures. But the braid group is not just a mapping class group; it admits an invariant left-order and a Garside structure. Tetsuya Ito gave a great minicourse on both of these structures!

Question’: Can one leverage these additional structures to answer genericity questions about the braid group?

Fast algorithms for the Nielsen-Thurston classification

Question: Is there a polynomial-time algorithm for computing the Thurston-Nielsen classification of a mapping class?

Matthieu Calvez has described an algorithm to classify braids in O(\ell^2) where \ell is the length of the candidate braid. The algorithm is not yet implementable because it relies on knowledge of a function c(n) where n is the index of the braid. These numbers come from a theorem of Masur and Minsky and are thus difficult to compute. These difficulties, as well as the power of the Garside structure and other algorithmic approaches, are described in Calvez’s linked paper.

Challenge: Implement Calvez’s algorithm, perhaps partially, without knowing c(n).

Mark Bell is developing Flipper which implements a classification algorithm for mapping class groups of surfaces.

Question: How fast are such algorithms in practice?2

John Etnyre

Contactomorphism and isotopy of unit cotangent bundles

For background on all matters symplectic and contact see Etnyre’s notes.

Let M be a manifold of any (!) dimension.  The total space of the cotangent bundle E = T^*M is naturally symplectic:  the cotangent bundle of E supports the Liouville one-form \lambda characterized by \alpha^*(\lambda) = \alpha for any one-form \alpha \in T^*M; the pullback is along the canonical projection T^* T^* \to T^*M.  The form d\lambda is symplectic on T^*M.

Inside the cotangent bundle is the unit cotangent bundle S^*M = \{(p,v) \in T^*M : |v| = 1\}. (This is not a vector bundle!) The form d\lambda restricts to a contact structure on the S^*M.

Fact: If the manifolds M and N are diffeomorphic, then their unit cotangent bundles S^*M and S^*N are contactomorphic

Hard question: In which dimensions greater than two is the converse true?

This question is attributed to Arnol’d, perhaps incorrectly.  The converse is known to be true in dimensions one and to and also in the case that M is the three-sphere (exercise!).

Tractable (?) question: Does contactomorphism type of unit cotangent bundles distinguish lens spaces from each other?

Also intriguing is the relative version of this construction. Let K be an Legendrian embedded (or immersed with transverse self-intersections) submanifold of M. Define the unit cosphere bundle of K to be L_K = \{w \in T^*M : w(v) = 0, \forall v \in TK\}. You can think of it as the boundary of the normal bundle to K. It is a Legendrian submanifold of the unit cotangent bundle T^*M.

Fact: If K_1 is Legendrian isotopic to K_2 then L_{K_1} is Legendrian isotopic to L_{K_2}.

Relative question: Under what conditions is the converse true?

Etnyre noted that contact homology may be a useful tool here.  Lenny Ng’s “A Topological Introduction to Knot Contact Homology” has a nice introduction to this problem and the tools to potentially solve it.

Sucharit Sarkar

How many Szabó spectral sequences are there, really?

Ozsváth and Szabó constructed a spectral sequence from the Khovanov homology of a link to the Heegaard Floer homology of the branched double cover of S^3 over that link. (There are more adjectives in the proper statement.) This relates two homology theories which are defined very differently.

Challenge: Construct an algorithm to compute the Ozsváth-Szabó spectral sequence.

Sarkar suggested that bordered Heegaard Floer homology may be useful here. Alternatively, one could study another spectral sequence, combinatorially defined by Szabó, which also seems to converge to the Heegaard Floer homology of the branched double cover.

Question: Is Szabó’s spectral sequence isomorphic to the Ozsváth-Szabó spectral sequence?

Again, the bordered theory may be useful here. Lipshitz, Ozsváth, and D. Thurston have constructed a bordered version of the Ozsváth-Szabó spectral sequence which agrees with the original under a pairing theorem.

If the answer is “yes” then Szabó’s spectral sequence should have more structure. This was the part of Sarkar’s research talk which was unfortunately scheduled after the problem session. I hope to return to it in a future post (!).

Question: Can Szabó’s spectral sequence be defined over a two-variable polynomial ring? Is there an action of the dihedral group D_4 on the spectral sequence?

András Stipsicz

Knot Floer Smörgåsbord

Link Floer homology was spawned from Heegaard Floer homology but can also be defined combinatorially via grid diagrams. Lenny Ng explained this in the second part of his minicourse. However you define it, the theory assigns to a link L a bigraded \mathbb{Z}[U]-module $HFK^-(L)$. From this group one can extract the numerical concordance invariant $\tau(L)$. Defining HFK^- over \mathbb{Q}[U] or \mathbb{Z}/p\mathbb{Z}[U] one can define invariants \tau_0 and \tau_p.

Question: Are these invariants distinct from \tau?

Harder question: Does HFK^- have p-torsion for some p \in \mathbb{Z}? (From a purely algebraic perspective, a “no” to the first question suggests a “no” to this one.)

Stipsicz noted that there are complexes of \mathbb{Z}[U]-modules for which the answer is yes, but those complexes are not known to be HFK^-(L) of any link. Speaking of which,

“A shot in the dark:” Characterize those modules which appear as HFK^-.

In another direction, Stipsicz spoke earlier about a family of smooth concordance invariants \Upsilon_t. These were constructed from link Floer homology by Ozsváth, Stipsicz, and Szabó. Earlier, Hom constructed the smooth concordance invariant \epsilon. Both invariants can be used to show that the smooth concordance group contains a \mathbb{Z}^\infty summand, but their fibers are not the same: Hom produced a knot which has \Upsilon_t = 0 for all t and \epsilon \neq 0.

Conversely: Is there a knot with \epsilon = 0 by \Upsilon_t \neq 0?

Stipsicz closed the session by waxing philosophical: “When I was a child we would get these problems like ‘Jane has 6 pigs and Joe has 4 pigs’ and I used to think these were stupid. But now I don’t think so. Sit down, ask, do calculations, answer. That’s somehow the method I advise. Do some calculations, or whatever.”

1. An analogous result holds for arbitrary number fields — I make no claims about the cuteness of such generalizations. ↩
2. An old example: the simplex algorithm from linear programming runs in exponential time in the worst-case, but in

2 Comments

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2 responses to “Some open computational problems in link homology and contact geometry”

  1. Mirko
    January 12, 2015 at 7:27 am

    Hey,
    thanks for taking the time to write that up! Just stumbled over your blog, great stuff.

    In the section “John Etnyre – Contactomorphism and isotopy of unit cotangent bundles”. What do you mean by “Legendrian” in the section on the relative Version of the conjecture: “Let K be a Legendrian submanifold of M.? Shouldn’t that be left out?

    Take care

    Reply

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