# 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

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

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 $\Delta_1$.

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.