Mini-courses: Second Week

Lenny Ng: Knot Contact Homology

The general philosophy behind Knot Contact Homology is to use the symplectic geometry of cotangent bundles to study the smooth topology of manifolds.  It is well known that cotangent bundles T^*M admit canonical symplectic structures and given a metric, there is an induced contact structure on the unit cotangent bundle ST^*M.  Given a knot K is \mathbb{R}^3, its conormal lift \Lambda_K is a Legendrian torus of ST^*M:

\Lambda_K = \{(q,p) \in ST^*M| q \in K, <p,v> = 0 \; \forall v \in T_qK\}

Legendrian Contact Homology is an invariant of Legendrian submanifolds L in contact manifolds P \times \mathbb{R}, which is a differential graded algebra  generated by the Reeb chords of L and whose differential counts punctured holomorphic disks in the symplectization P \times \mathbb{R}^2 with boundary on L \times \mathbb{R} such that the punctures tend asymptotically to Reeb chords.  The Knot Contact Homology of a knot K is the LCH of \Lambda_K in ST^*M = T^*S^2 \times \mathbb{R}.

This geometric interpretation of the invariant can be recast in a completely combinatorial manner.  If we braid our knot around an unknot then we get two sets of “short chords” \{a_{i,j}, b_{i,j}\}, with 1 \leq i,j \leq n, i \neq j of degree 0 and 1, respectively, and four sets of “long chords” \{c_{i,j}\}, \{ d_{i,j}\}, \{e_{i,j}\}, \{ f_{i,j}\}, with 1 \leq i,j \leq n of degree 1,1,2 and 2, respectively.  Moreover, the base ring of the DGA is R = \mathbb{Z}[\lambda^{\pm}, \mu^{\pm}, U^{\pm}] .  The differential can be defined in terms of an action of the braid group on the degree 0 generators.

We can think of the braid group B_n as the mapping class group of the unit disk with n points along the real axis removed.  Then an element a_{i,j} corresponds to an oriented arc in the upper half plane from the i-th point to the j-th point and let A_n denote the tensor algebra over \mathbb{Z} generated by these arcs, modulo a skein relation that identifies arcs passing through the lower half of the disk with arcs above the real axis(see Ng’s notes for more details).  There is an action of B_n on this algebra given by the mapping class group acting on the arcs.  The differential can then be defined on the DGA of a knot in terms of the action of the braid corresponding to that knot on the algebra A_n.

This approach gives a sequence of knot invariants, in decreasing strength:

  • Legendrian isotopy class of \Lambda_k
  • Knot DGA
  • The degree 0 component HC_0(K) of the knot contact homology.
  • Augmentation Polynomial: an augmentation of a DGA (A, \partial) is a ring homomorphism \epsilon: A \rightarrow S of unital rings whose kernel contains the image of \partial and all elements of positive degree.  Each augmentation can be used to construct a linearized version of the DGA.  The augmentation variety of a knot is algebraic set in (\mathbb{C}^*)^3 consisting of the images of \lambda, \mu, U under some augmentation to the complex numbers:

\{(\epsilon(\lambda),\epsilon(\mu), \epsilon(U) \in (\mathbb{C}*)^3 \epsilon: A \rightarrow \mathbb{C}\}

If the maximal dimensional component of the Zariski closure of the augmentation variety has codimension one, it is cut out by a reduced polynomial called the augmentation polynomial of K.  Some arguments from physics indicate that the augmentation polynomial is related to the HOMFLY-PT polynomial.  There is also a two-variable augmentation polynomial obtained by setting U = 1 and it is known that the A-polynomial of the knot divides this two-variable augmentation polynomial.

  • Cord Algebra: The cord algebra is obtained by setting U = 1 in HC_0(K), the degree 0 part of the DGA.  We can equivalently think of the cord algebra pictorially as a tensor algebra over the ring R_0 = \mathbb{Z}[\lambda^{\pm}, \mu^{\pm}] generated by homotopy classes of continuous paths in S^3 - K that start and end on K and that miss some specified base point *, up to some spatial relations.  A third interpretation of the cord algebra is as a tensor algebra over R_0 generated by elements of the knot group \pi_1 ( S^3 - K) modulo some relations.

Transverse Invariants:  Knot contact homology can also be used to obtain invariants of transverse knots.  The contact structure itself has a conormal lift \hat{\Xi} and for transverse knots, it is disjoint from \Lambda_k.  In the symplectization, we can chose an almost complex structure so that \hat{\Xi} \times \mathbb{R} is a holomorphic surface.  For any holomorphic disk contributing to the differential, we can count intersections of this disk with the holomorphic surface and  holomorphicity implies that these intersections are positive.  Including this data in our differential defines a filtration on the DGA which gives an invariant of the transverse isotopy class of the knot, not just the smooth isotopy class.  In a manner similar to Heegaard Floer theory, there are “hat” and “infinity” versions of this invariant as well.

Michael Hutchings: Embedded Contact Homology

Since Michael has a series of notes on his own blog about this lecture series, I think I’ll direct everyone over there if they want to discuss it or find the details.  But I’ll try to give a basic summary of his mini-course here as well.

Embedded Contact Homology is an invariant of closed, oriented 3-manifolds Y that also encodes information about contact geometry in Y and symplectic geometry on compact 4-manifolds whose boundary is Y.

For a closed, oriented 3-manifold Y, a nondegenerate contact form \lambda on Y and homology class \Gamma \in H_0(Y;\mathbb{Z}), there is a chain complex ECC(Y, \lambda, \Gamma) generated by orbit sets of closed Reeb orbits \gamma  and whose differential counts holomorphic curves of a certain index in \mathbb{R} \times Y asymptotic to the orbit sets.  An orbit set is a finite collection of pairs (\gamma_i, m_i) such that in homology, \sum m_i [\gamma_i] = \Gamma .  The relevant index of holomorphic curves is called the ECH index and is somewhat tricky to define (see Michael’s notes).  This differential squares to 0 and the homology of the chain complex ECH(Y, \lambda, \Gamma) is well-defined up to choosing some other nondegenerate contact form \lambda` defining the same contact structure.

The construction exploits two facts about holomorphic curves in 4-dimensional symplectic manifolds:

  • Intersection Positivity: For any two somewhere injective, connected, distinct holomorphic curves u,v, any intersection point in their images is isolated and has positive multiplicity.
  • Adjunction formula: The familiar adjunction formula from complex geometry holds.  For a somewhere injective holomorphic curve u: \Sigma \rightarrow X

c_1(TX)( [u(\Sigma)]) = \chi(\Sigma) + [u(\Sigma)]* [u(\Sigma)] - 2 \delta(u(\Sigma))

where \delta counts singularities with positive, integer weights.

An inspiration for ECH is Taubes’s Gromov invariant, which connects the Seiberg-Witten invariants on a closed, connected symplectic 4-manifold X to counts of holomorphic curves of a certain index in X.

There is also some extra structure on this invariant:

  • U-maps: There is a map

U: ECH_*(Y, \lambda, \Gamma) \rightarrow ECH_*(Y, \lambda, \Gamma)

that counts index 0 holomorphic curves passing through a specified basepoint in \mathbb{R} \times Y.

  • Canonical class: There is an element [\phi] \in ECH_*(Y, \lambda, 0) that canonically represents the contact structure \Xi_{\lambda}.  It vanishes for overtwisted contact structures and is nontrivial for strongly fillable contact structures.
  • Filtrations: Each closed Reeb orbit has an action, given by integrating \lambda over an embedded orbit.  We can extend this action linearly to orbit sets and the differential necessarily decreases the action, yielding a filtration on the chain complex:

ECH^L(Y, \lambda, \Gamma)

is the homology of the chain complex whose generators have action less that L.

  • Cobordism Maps: Let X be a (weakly?) exact symplectic cobordism from Y_+ to Y_-.  That means X has an exact symplectic form \omega, concave boundary Y_+ and convex boundary Y_- such that d \lambda_+ = \omega on Y_+ and d \lambda_- = \omega on Y_-.  Then there is a map

\Phi^L: ECH^L(Y_+, \lambda_+,0) \rightarrow ECH^L(Y_-, \lambda_-,0)

That commutes with the U maps.

A major fact about ECH is that it is equivalent to the “hat” versions of two other invariants of 3-manifolds, Monopole Floer/Seiberg-Witten Floer homology and Heegaard Floer Homology:

Theorem: (Taubes, Kutluhan-Lee-Taubes, Colin-Ghiggini-Honda) There is an isomorphism of the following homologies:

ECH_*(Y, \lambda, \Gamma) \simeq \widehat{HM}^*(Y, \mathbb{S}_{\lambda} + \Gamma) \simeq \widehat{HF}(-Y, \mathbb{S}_{\lambda} + \Gamma)

Where \mathbb{S}_{\lambda} + \Gamma is a spin-\mathbb{C} structure determined by the \lambda, \Gamma.

Applications: Michael mentioned two applications of ECH to problems in contact and symplectic topology.

An ellipsoid in \mathbb{C}^2 is the subset

E(a,b) = \{ (z,w) : \frac{\pi z^2}{a} + \frac{\pi w^2}{b} \leq 1 \}

The boundary \partial E(a,b) is topologically S^3 and inherits a contact structure from the standard symplectic structure on \mathbb{C}^2.  A basic question to ask is when does E(a,b) embed symplectically in E(c,d)?  To answer this, one can define ECH capacities c_k from the filtered ECH on S^3 with the induced contact structure.

A second application is to the 3-dimensional Weinstein conjecture, which posits that every contact form on a closed, oriented 3-manifold has at least 1 Reeb orbit.  This was proved by Taubes in all cases and follows directly from the isomorphism between ECH and \widehat{HM}.  Kronheimer and Mrowka proved that Seiberg-Witten Floer is infinitely generated but a counter example to the Weinstein conjecture would have trivial ECH for some nontrivial \Gamma or ECH exactly one copy of \mathbb{Z} if \Gamma is trivial.

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