# Budapest Research Talks: Tuesday

Frederic Bourgeois: The Geography of Legendrian submanifolds
Frederic’s talk focused on Legendrian submanifolds in 1-jet spaces that arise from generating families of functions.  A simple way to create Legendrian submanifolds in $J^1(M) = T^*M \times \mathbb{R}$ is to look at the graph in $J^1(M)$ of a smooth function f on M.  However, this is a trivial example because all such Legendrians are isotopic to each other through Legendrians, including Legendrian isotopic to the 0-section.  A subtler construction is through generating families.  Here, we take a smooth function F on $M \times \mathbb{R}^n$, which we can think of as a family of smooth functions on M parametrized by $\mathbb{R}^n$.  If we let $\Sigma$ be the critical set of the F, then we can map $\Sigma$ into $J^1(M)$ as a Legendrian by $(x,e)\mapsto (x,\frac{\partial F}{\partial x}(x,e), F(x,e))$.

There are two operations for generating families F for a Legendrian L that define an equivalence relation.  We can stabilize, meaning we extend F to a function on $M \times \mathbb{R}^{n+1}$ in a trivial manner, and we can precompose F with a fiber diffeomorphism that maps each vector space $\mathbb{R}^n$ to itself.

We can also define an invariant of the pair (L,F), called generating family homology, as follows.  Take the difference function $\Delta(x,e,e') = F(x,e) - F(x,e')$, defined on $M \times \mathbb{R}^{2n}$.  There is a one-to-one correspondence between critical points at positive critical values and Reeb chords of L.  If we let $\Delta^a = \{(x,e,e')| \Delta(x,e,e') < a \}$ and choose some $\delta >0$ small than the length of any Reeb chord, we can define our invariant to be the relative homology $GFH(L,F) = H_*(\Delta^{\infty},\Delta^{\delta})$, which really is invariant up to the equivalence relation mentioned above.

This is starting to look like Legendrian Contact Homology, as GFH is a homology invariant whose generators are Reeb chords.  In fact, for Legendrian knots, GFH is equivalent to the linearized LCH of L for some augmentation of the dg-algebra.  An augmentation $\epsilon$ is a ring homomorphism from an algebra A to a field that nicely splits the differential into a graded morphism.  We obtain a new map $\partial^{\epsilon}$ that sends each word of k letters $b_1b_2 \dots b_k$ to some linear combination of words, each still with exactly k letters.  Let $m(a,b_1,\dots,b_2)$ be the sum over $\mathbb{Z}/2$ of punctured curves mapping the boundary of the disk to L and the punctures to $a, b_1,\dots,b_n$, as is standard in LCH.  Then the new map is $\partial^{\epsilon}(a) = \sum_{b_{i_1},\dots , b_{i_k}} \sum_{j=1}^k m(a,b_{i_1},\dots , b_{i_k}) \epsilon(b_{i_1}) \dots \epsilon(b_{i_{j-1}}) b_{i_j} \epsilon(b_{i_{j+1}}) \dots \epsilon(b_{i_k})$.

Moreover, it restricts to a differential on the 1-graded piece, the single letter words that correspond 1-1 with our Reeb chords.  The homology of this is much simpler than the homology of the whole dg-algebra and we can encode the ranks of our the homology groups of this chain complex in a Poincare polynomial.  Now, not every Legendrian knot admits an augmentation on is DGA, but if L is constructed from a generating family, then there exists a unique choice of augmentation and every augmentation comes from some generating family.  Fuchs and Rutherford have proved that GFH is in fact graded isomorphic to linearized DGA.

A more powerful invariant can be obtained by considering two augmentations at once, using the following linearization:

$\partial^{\epsilon_1, \epsilon_2}(a) = \sum_{b_{i_1},\dots, b_{i_k}} \sum_{j=1}^k m(a,b_{i_1},\dots, b_{i_k}) \epsilon_1(b_{i_1}) \dots \epsilon_1(b_{i_{j-1}}) b_{i_j} \epsilon_2(b_{i_{j+1}}) \dots \epsilon_2(b_{i_k})$.

This is more powerful because it keeps track of the order that Reeb chords appear at punctures for a disk.  Similarly, we could consider a difference function $\Delta' = F_1(x,e) = F_2(x,e')$ for two different generating families $F_1,F_2$ and obtain a homology $GFH(L,F_1,F_2)$.  They prove that these two homologies are in fact equivalent.

As a final question, he discussed what Poincare polynomials of linearized LCH can occur.  He cited a result due to Ekholm, Etnyre and Sabloff that established a form of Poincare duality for LCH and defined a double compatible Laurent polynomial as any polynomial that could be the Poincare polynomial of a complex whose homology satisfies the above condition.  He, Sabloff and Traynor have proved that for any given doubly compatible polynomial, there exists some Legendrian and an augmentation such that the Poincare polynomial of the linearized LCH is exactly the specified one.

Jacob Rasmussen:

Jacob’s talk investigated a conjecture that the Khovanov-Rozansky homology of a torus knot T(m,n) can be constructed using finite dimension representations of so-called rationals DAHAs (or double affine Hecke algebras; also referred to as Cherednik algebras).  I’m not an expert on Khovanov homology and the actual statement of the conjecture seems too subtle for me to correctly reproduce.  But I will try to state what seems to be the general idea.

The rational DAHAs are deformations of the familiar Weyl algebra.  The Weyl algebra is the ring of differential operators on the polynomial ring $R = \mathbb{C}[x_1,\dots,x_n]$ with polynomial coefficients.  It is generated by the operators $x_i$, corresponding to multiplication by $x_i$, and $\partial/\partial x_i$, corresponding to partial differentiation with respect to $x_i$.  These operators commute except for $[\partial/\partial x_i, x_i] = id$; this is just an elementary calculation in calculus and is a manifestation of the so-called canonical commutation relation.  This algebra is simple, meaning it has no nontrivial left or right ideals, and so it has no finite-dimensional representations.

However, we can deform the Weyl algebra by replacing the partial differential operators with Dunkl operators

$D_i f = \frac{\partial f}{\partial x_i} + c \sum_{j \neq i} \frac{s_{i,j} (f)-f}{x_i-x_j}$.

where c is some complex number and $s_{i,j}$ is the involution on $\mathbb{C}[x_1,\dots,x_n]$ that switches the two indeterminates $x_i \leftrightarrow x_j$.  Clearly, setting c equal to 0 gives us the original Weyl algebra.

Interestingly, these deformed algebras do admit finite dimensional representations if $c = m/n$ for some integer coprime to n.  The vector space $L_{m/n}$ on which it acts can be realized as a quotient of R by a homogeneous ideal of polynomials of degree less than or equal to m.

If we let V denote the linear polynomials in R, then the conjecture states that

$\bar{H}(T(m,n)) \cong (\Lambda*V \otimes_{Gr} L_{m/n})^{S_n}$

i.e. the $S_n$-invariant elements of the associated graded of the exterior algebra of V tensored with the representation $L_{m/n}$.  The a-grading should be sent to the grading in the exterior algebra, the q-grading should be sent to the degrees of polynomials in $L_{m/n}$ and the t-grading should be sent to the associated graded of a filtration on $L_{m/n}$.