Symplectic Khovanov Homology

Over the last few days Ivan Smith and Mohammed Abouzaid each gave a talk on symplectic Khovanov homology (their joint work with Paul Seidel also). This theory is built from elements of Lagrangian intersection Floer homology on a particular symplectic manifold, and they expect it to be isomorphic to Khovanov’s homology of links. The definition is pretty involved so please fill in details I don’t understand yet if you can.

Part 1

Chapter 1: Khovanov homology

First, Smith discussed the structure of Khovanov homology that they were trying to emulate through symplectic definitions. While Khovanov’s original definition of the homology theory for links was diagrammatic, he also has a more algebraic reformulation that applies to tangles.

To describe this, first Smith defined the arc algebra. Consider a category whose objects consist of crossingless matchings of 2n points. In other words, put 2n points on a straight line in a plane and choose n arcs connecting the points such that the arcs do not intersect each other. Given two such crossingless matchings, A and B, we can put them together along the 2n endpoints with the arcs of A above the endpoints and the arcs of B below the endpoints. This gives a set of d circles in the plane, and we associate to this diagram [H^*(S^2)]^{\otimes d}. Thus we define the set of morphisms in this category by Mor(A,B)=[H^*(S^2)]^{\otimes d}. We define the arc algebra by H_n=\oplus_{A,B} Mor(A,B) over all crossingless matchings A and B.

Next we want to understand the derived category of H_n modules. This basically means the objects are chain complexes of integer graded projective modules, considered up to quasi-isomorphism, and the morphisms are chain maps. This category carries an action of the braid group, and a distinguished module P_n such that for an element \beta \in Br_{2n}, the Khovanov homology of the link obtained by the braid closure of \beta is given by Ext^*(P_n,\beta(P_n)). There are some basic bimodules that we can use to build up everything needed to compute Khovanov homology. The i^{th} cap, \cap_i is a (H_{n-1},H_n) bi-module which adds a cap between two new points inserted in the i^{th} place. Similarly the i^{th} cup, \cup_i is a (H_n,H_{n-1}) bimodule which cups together two strands in the i and i+1 places to eliminate two endpoints.

Given a knot, there is a projection that can be cut into simple pieces as in the below picture,


A knot in a simple form that can be broken into basic slices by horizontal lines.

so that there are finitely many levels, each containing cups, caps and crossings. Then it is possible to compute the Khovanov homology by using the cup and cap bimodules for each cup/cap in the diagram, plus a bimodule associated to a crossing defined by Cone(\cup_i\otimes_{H_{n-1}} \cap_i \to id_{H_n}) or Cone(id_{H_n} \to \cup_i\otimes_{H_{n-1}} \cap_i), depending on which strand crosses over the other. I’m not really sure what these maps to and from the identity are, if someone else has an explanation that would be greatly appreciated.

Chapter 2: Symplectic Khovanov homology

The goal of Seidel and Smith was to find a symplectic/Floer theoretic reformulation of Khovanov homology. To do this, they looked for these arc algebras, in geometric spaces. The center of the arc algebra H_n is the cohomology H^*(Y_n). To define this space Y_n, let M be the space of matrices of the form
\left(\begin{array}{cccc}A_1&I&\cdots & 0\\ \vdots & 0& \ddots & \\ & & &I\\ A_n & &&0\end{array}\right)
where each entry represents a 2×2 block and A_1 is trace free. This is a transverse slice to the (n,n) nilpotent matrices. Let \chi: M\to \mathbb{C}^{2n-1} take each matrix to the coefficients of its characteristic polynomial. The define Y_n to be the preimage of a generic point of \chi. We can map \mathbb{C}^{2n-1} to Sym^{2n}_0(\mathbb{C}), the space of unordered 2m tuples of complex numbers whose sum is 0, by sending the corresponding characteristic polynomial of the traceless matrix to its eigenvalues. For each point \tau \in Conf^{2n}_0(\mathbb{C}) \subset Sym^{2n}_0(\mathbb{C}) where all of the eigenvalues are distinct, there is a corresponding fiber \chi^{-1}(\tau)=Y_n^{\tau}. Parallel transport around loops in Conf^{2n}_0(\mathbb{C}) defines a representation from the braid group to \pi_0(Symplectomorphisms(Y_n^{\tau_0})).

Next, look at the Lagrangian intersection Floer homology in this space Y_n^{\tau_0}. It is a theorem of Seidel and Smith that there exists a Lagrangian submanifold L\cong(S^2)^n\subset Y_n^{\tau_0} such that HF^*(L,(\beta \times id)(L)) is an integer graded link invariant for a braid, \beta\in Br_n. They expect this to agree with Khovanov homology, where the integer grading coming from the Floer homology agrees with the difference between the Alexander and Maslov gradings in Khovanov homology.

Here is a way to understand the Lagrangian L. Manolescu constructed an open embedding from Y_n into Hilb^n(X_n), where X_n is the Milnor fiber X_n=\{x^2+y^2+z^{2n}=1\}\subset \mathbb{C}^3, and Hilb^n(X_n) is a resolution of Sym^n(X_n) at the singularities along the diagonal. Since X^n\subset \mathbb{C}^3, there is a projection \pi_z: X_n\to \mathbb{C} projecting onto the last complex coordinate. This projection has some critical values at the roots of z^{2n}-1, above which the fibers are singular cones. Above the regular values, the fibers are cylinders. If one draws a path between two critical values in \mathbb{C}, and looks at the vanishing cycles in the corresponding fibers you see a sphere as in the picture below.

Lefschetz Fibration

Lefschetz Fibration

To get (S^2)^n, you take n disjoint paths in \mathbb{C} between critical values, i.e. a crossingless matching of 2n points. Taking the preferred crossingless matching which matches the ith point to the (2n-(i-1))th point for 1\leq i \leq n, gives the Lagrangian L of the theorem. Note that this was also the preferred crossingless matching in Khovanov’s algebraic construction.

One can form the analog of the arc algebra in this symplectic setting in the following way. For crossingless matchings A and B, let L_A and L_B be the associated Lagrangians. Then define H_n^{symp}=\sum_{A,B}HF^*(L_A,L_B) where the Floer Homology is taken in Y_n. There is an expectation that H_n^{symp}=H_n, and this was proven over \mathbb{Z}/2 by Rezazadegan.

Next one would like to analyze what happens when you look at \chi^{-1}(\tau_{sing}) when \tau_{sing} is no longer a regular value of the characteristic polynomial map. Start out with the simplest kind of singularities when only two eigenvalues coincide. Seidel and Smith show that the singular locus of this fiber can be canonically identified with Y_{n-1} as two eigenvalues come together to one in Y_n. Transverse to the singular locus is n=1 Milnor fiber, which has a vanishing cycle giving rise to an S^2. Thus colliding (i,i+1) critical points give rise to a Lagrangian \Gamma_i\subset Y_{n-1}\times Y_n, where Y_{n-1}. (There are some holes in what I’ve said here, but I’m not sure yet how to fill them in.)

In the end, they obtain a Fukaya category from Y_n, \mathcal{F}(Y_n), and bimodules defined by the Lagrangians \Gamma_i between \mathcal{F}(Y_{n-1}) and \mathcal{F}(Y_n). They build up a symplectic cube of resolutions using long exact sequences in Floer theory for fibered Dehn twists, where the edges and diagonals are defined by the differential and higher products in the Fukaya category. To show that this is isomorphic to the original Khovanov homology, they want to show that these Fukaya categories are “formal” meaning equivalent to a minimal A_{\infty} algebra whose higher products vanish. Abouzaid explains this in more detail in part two of this talk, below.

Symplectic Khovanov Part 2

Chapter 3: Recovering the bigrading

Lagrangian Floer homology has a single grading, but Khovanov homology is bigraded. It requires some effort to recover the second grading on the symplectic Khovanov homology side. The first step is to partially compactify the space Y_n by adding in some divisor D. In the Milnor fiber X_n, you should add in two points at infinity to each fiber in the Lefschetz fibration so that the cylindrical fibers become spheres and the cone fibers become a wedge of two spheres. Use this and the embedding of Y_n into Hilb^n(X_n) to define the appropriate partial compactification of Y_n. Now we have a manifold \overline{Y} = Y_n\cup D. Choose a perturbation D_{\varepsilon} of D in \overline{Y}.

They define \Delta^0 \in HF^1(L,L) by counting points on disks with boundary along L, which intersect D and D_{\varepsilon} each in a unique point, as in this picture.

disks defining $\Delta_0$ and $\Delta_1$

Disks defining $\Delta_0$ and $\Delta_1$.

This is well defined when some Gromov-Witten invariant of \overline{Y} vanishes. In the case that \Delta^0 vanishes, they call L infinitesimally invariant. If (L_0,L_1) are infinitesimally invariant Lagrangians, it is possible to define a relative bigrading on HF^*(L_0,L_1). The first grading is just the homological grading, and the second grading is a weight determined by a certain map \Delta^1: HF^*(L_0,L^1)\to HF^*(L_0,L_1). For $x\in HF^*(L_0,L_1)$, which can be represented by an intersection point between the two Lagrangians, we define \Delta^1(x) by the picture above, by counting all disks with boundary along L_0 and L_1 containing x in the boundary and a summand of \Delta^1(x) at the other intersection of the Lagrangians on the boundary, with the condition that the interior of the disk intersects D and D_{\varepsilon} each in a unique point (see picture). To obtain the relative grading, decompose $HF^*(L_0,L_1)$ by the generalized eigenvalues of \Delta^1. Some bubbling issues prevent this from being an absolute grading without some additional choices, but this can be fixed by making some cohomology choices. Although the resulting absolute grading is not a priori integral, it is integral in practice.

There are similarly defined \Delta^i for each i\in \mathbb{N}, and these are needed to show that the weight grading is compatible with multiplication.

Chapter 4: Formality

An A_{\infty} algebra is a differential algebra whose multiplication is not quite associative, but is endowed with higher product operations which describe the homotopies that describe the failure of associativity of the lower products. The product operations are called m_1,m_2,m_3,\cdots, where m_1 is the differential, m_2 is a product, m_3 is the homotopy showing m_2 is associative on the level of homology, etc. An A_{\infty} algebra is called minimal if m_1\equiv 0. It is a theorem that every A_{\infty} algebra is equivalent to a minimal one. An A_{\infty} algebra is called formal if it is equivalent to a minimal algebra whose higher products, m_3,m_4,\cdots all identically vanish. They show that the symplectic arc algebra C is formal by using the class \Delta=\{\Delta_i\}\in HH^1(C,C). I am lacking some of the algebraic knowledge to say much more about what these objects are or how the proof of this part goes.

Once formality is established, the symplectic cube would correspond to Khovanov’s cube of resolutions, so the spectral sequence from Khovanov homology to Symplectic Khovanov homology would degenerate immediately, and the two theories would be isomorphic. This would provide and interesting link between symplectic geometry and Khovanov’s more combinatorial formulation of the link invariant. I would be interested to see what kinds of new information we can obtain about Khovanov homology from the symplectic version, or what we can learn about symplectic geometry from Khovanov homology.

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