Cell Decompositions of Hecke Traces
and Link Polynomials



Minh-Tâm Trinh & Nathan Williams
FPSAC 2025



Knot Invariants

Gordian Knots

A knot is an embedding of $S^1$ into $\mathbb{R}^3$. A link is a disjoint collection of knots.

We wish to distinguish knots by finding knot invariants.
We wish to distinguish knots by finding knot invariants.
That is, we want to associate a number/polynomial/space to a knot that is invariant up to ambient isotropy or Reidemeister moves (on projections).
Example. Tricolorability (can you color with at least two of red, blue, and green, so that at each crossing either one color or all three colors appear) distinguishes the unknot from the trefoil.
Definition. (Freyd, Yetter; Hoste; Lickorish, Millett; Ocneanu; Przytycki, Traczyk)
The HOMFLY(PT) polynomial $\mathcal{P}(L)$ is defined by the skein relations:

Specializations recover the Jones and Alexander polynomials.

Theorem. (Shende, Treumann, Zaslow)
For positive links $L$, the lowest $a$-coefficient of HOMFLY counts the number of points in a certain braid variety over $\mathbb{F}_q$ constructed from $L$.
Example (Jones). For the torus knot $T_{n,p}$ (with $n$, $p$ relatively prime), $$\mathcal{P}(T_{n,p}) \propto \sum_k \frac{(-1)^k q^{\binom{k+1}{2}}}{[p]_q} \left[\!\begin{matrix} n-1 \\ k \end{matrix}\!\right]_q \left[\!\begin{matrix} n+p-k-1 \\ n \end{matrix}\!\right]_q a^{2k}.$$ For example, $\mathcal{P}(T_{n,n+1})$ gives the $f$-vector of the associahedron at $q=1$: $$\mathcal{P}(T_{4,5})\Big|_{q=1} \propto \color{blue}{14}\color{black} - 21a^2+9a^4-a^6.$$
Example (continued). The lowest $a$-coefficient of the HOMFLY polynomial for the torus knot $T_{n,n+1}$ is $$[\text{lowest $a$-coeff}]\mathcal{P}(T_{n,n+1})\propto \frac{1}{[n+1]_q} \left[\!\begin{matrix} 2n \\ n \end{matrix}\!\right]_q.$$
With Galashin and Lam, we explained this by showing that classical Catalan objects index certain cells of the Deodhar decomposition for the braid variety encoding the lowest $a$-coeff of $\mathcal{P}(T_{n,n+1})$.
        
We generalized to all finite Coxeter groups and parameters $p$ coprime to $h$.
Theorem. (Shende, Treumann, Zaslow)
For positive links $L$, the lowest $a$-coefficient of HOMFLY counts the number of points in a certain braid variety over $\mathbb{F}_q$ constructed from $L$.
$$\mathcal{P}(T_{4,5})\Big|_{q=1} \propto \color{blue}{14}\color{black} - 21a^2+9a^4-a^6.$$
Problem. Find braid varieties with combinatorial cell decompositions to describe the other HOMFLY coefficients (besides the lowest).
Notes:
  • Bottom coefficient is top coefficient after multiplication by full twist.
  • Standard method for all coefficients uses a trace on the Hecke algebra with values in $\mathbb{Z}[q^{\pm 1},a^{\pm 1}]$.
  • Bezrukavnikov-Tolmachov use $\mathbb{Z}[q^{\pm 1}]$-traces combined with Jucys-Murphy braids to recover all HOMFLY coefficents.

Results

For any Weyl group $W$ and positive braid $\beta \in B_W^+$, we define parabolic unipotent braid varieties $Z_\beta^J$ with
Theorem. (Trinh, W.)
Each $a$-degree of the HOMFLY polynomial of a (positive) link $L$ is the point count of certain cells of parabolic unipotent braid varieties over $\mathbb{F}_q$.




Flag Varieties

Let $G(\mathbb{F}_q)$ be a connected, smooth, reductive, algebraic group split over $\mathbb{F}_q$
with Borel subgroup $B(\mathbb{F}_q)$, rank $r$, and Weyl group $W$.

"Claim". The Weyl group $W$ is the flag variety $G(\mathbb{F}_q)/B(\mathbb{F}_q)$ at $q = 1$.
$$\begin{array}{ccccc} |G(\mathbb{F}_q)| &=& \underbrace{(q-1)^r q^N \vphantom{\prod_{i=1}^r}}_{|B|} &\cdot& \underbrace{\prod_{i=1}^r [d_i]_q}_{|G/B|} \\ & & & &\\ |W| &=& & & \prod\limits_{i=1}^r d_i \end{array}$$ Here $V$ is the reflection representation of $W$ and the degrees $d_1,d_2,\ldots,d_r$ are the degrees of homogenous generators of the polynomial ring $\mathbb{C}[V]^W$.
Example. Let $G=SL_3$, $B = \left\{\left[\begin{array}{ccc} * & * & * \\ 0 & * & * \\ 0 & 0 & *\end{array}\right]\right\}$.
Then $W=S_3$ and $$\begin{array}{ccc}|SL_3(\mathbb{F}_q)|/|B(\mathbb{F}_q)|&=&[3]_q [2]_q\\ & & \\ |S_3| &=& 3\cdot 2 = 6\end{array}$$
Coset representatives of $B$ are defined up to adding multiples of a column to columns to the right—so we can think of $G/B$ as flags of subspaces $$(0 \subset F_1 \subset F_2 \subset \mathbb{F}_q^3) \text{ \hspace{3em} } (\dim(F_i)=i),$$ with $F_i$ the span of the leftmost $i$ columns of any coset representative.
Some of the flags in $G/B$ contain (lifts of) permutation matrices...

$$( 0 \subset \langle e_1 \rangle \subset \langle e_1,e_2 \rangle \subset \mathbb{F}_2^3 ) = \left\{\left[\begin{array}{ccc} * & * & * \\ 0 & * & * \\ 0 & 0 & *\end{array}\right]\right\} \ni \left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]$$ $$( 0 \subset \langle e_2 \rangle \subset \langle e_1,e_2 \rangle \subset \mathbb{F}_2^3 ) = \left\{\left[\begin{array}{ccc} 0 & * & * \\ * & * & * \\ 0 & 0 & *\end{array}\right]\right\} \ni \left[\begin{array}{ccc} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1\end{array}\right]$$ $$ ( 0 \subset \langle e_1 \rangle \subset \langle e_1,e_3 \rangle \subset \mathbb{F}_2^3 ) = \left\{\left[\begin{array}{ccc} * & * & * \\ 0 & 0 & * \\ 0 & * & *\end{array}\right]\right\} \ni \left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0\end{array}\right]$$ $$\ldots$$
...and some flags don't:
$$( 0 \subset \langle e_1+e_3\rangle \subset \langle e_1+e_3,e_2 \rangle \subset \mathbb{F}_2^3 )=$$ $$\left\{\left[\begin{array}{ccc} 1 & 1 & 0\\ 0 & 1 & 1\\ 1 & 1 & 1\end{array}\right], \left[\begin{array}{ccc} 1 & 0 & 0\\ 0 & 1 & 0\\ 1 & 0 & 1\end{array}\right], \left[\begin{array}{ccc} 1 & 0 & 1\\ 0 & 1 & 1\\ 1 & 0 & 0\end{array}\right], \left[\begin{array}{ccc} 1 & 1 & 1\\ 0 & 1 & 1\\ 1 & 1 & 0\end{array}\right]\right.$$ $$\left. \left[\begin{array}{ccc} 1 & 1 & 0\\ 0 & 1 & 0\\ 1 & 1 & 1\end{array}\right], \left[\begin{array}{ccc} 1 & 0 & 1\\ 0 & 1 & 0\\ 1 & 0 & 0\end{array}\right], \left[\begin{array}{ccc} 1 & 1 & 1\\ 0 & 1 & 0\\ 1 & 1 & 0\end{array}\right], \left[\begin{array}{ccc} 1 & 0 & 0\\ 0 & 1 & 1\\ 1 & 0 & 1\end{array}\right] \right\}$$
No surprise—since $q>1$, there are more flags than permutation matrices: $$\begin{array}{ccc}|SL_3(\mathbb{F}_2)|/|B(\mathbb{F}_2)|&=&(2^2+2+1)(2+1)=21\\ |S_3| &=& 3\cdot 2 = 6\end{array}$$
Definition. For $B',B'' \in G/B$, write $B' \xrightarrow{s_i} B''$ if $B',B''$ differ in their $i$th subspace. Extends by reduced words to $B' \xrightarrow{w} B''$ for $w \in W$, called relative position.
Since permutation matrices are in relative position $s_i$ exactly for weak order covers, the flag variety is a sort of thickening of the weak order. (Permutations thicken to Schubert cells.)
"Claim". The Weyl group $W$ is the flag variety $G(\mathbb{F}_q)/B(\mathbb{F}_q)$ at $q = 1$.



Braid Varieties

We want to encode positive links as walks in the flag variety $G(\mathbb{F}_q)/B(\mathbb{F}_q)$.
  • Alexander: every oriented link arises as a braid closure $\widehat{\alpha}$
    (called positive if $\alpha\in B_n^+$):
  • Markov: any two braid closures representing the same oriented link are connected by moves of the form:
          
Example. $T_{3,4}$ is the braid closure of $(s_1s_2)^4$ but also of $(s_1s_2s_3)^3$.

Unipotent Braid Varieties

Let $U$ be the unipotent radical of $B$.
Definition. (Trinh, W.) The unipotent braid variety for a positive link $\beta$ is the set of walks in $G/B$ that start and end in the same Schubert cell (possibly at different flags), together with an element of $u \in U$ recording this difference: $$Z_\beta:=\{ (u, B_0\xrightarrow{\beta_1} B_1 \xrightarrow{\beta_2} \cdots \xrightarrow{\beta_\ell} B_\ell) : u \in U, B_0 = u B_\ell \}.$$
Example. Fix $G=\mathrm{SL}_3(\mathbb{F}_q)$.
Starting from any flag, there are (trivially) $q^8$ walks of the form $\color{red}s\color{blue}t\color{red}s\color{blue}t\color{red}s\color{blue}t\color{red}s\color{blue}t$.
But there are (nontrivially) a $q$-parking number
that begin and end in the same Schubert cell: $$|Z_{\color{red}s\color{blue}t\color{red}s\color{blue}t\color{red}s\color{blue}t\color{red}s\color{blue}t}(\mathbb{F}_q)|=(q-1)^2(1+q+q^2+q^3)^2.$$

Unipotent Braid Varieties

Definition. (Trinh, W.) The unipotent braid variety for a positive link $\beta$ is the set of walks in $G/B$ that start and end in the same Schubert cell (possibly at different flags), together with an element of $u \in U$ recording this difference: $$Z_\beta:=\{ (u, B_0\xrightarrow{\beta_1} B_1 \xrightarrow{\beta_2} \cdots \xrightarrow{\beta_\ell} B_\ell) : u \in U, B_0 = u B_\ell \}.$$
This definition is a new braid variety variant, different from previous variants introduced by Trinh and subsequently studied by Bezrukavnikov, Boixeda-Alvarez, McBreen, Yun including:
  • Steinberg braid varieties (with a $G$-action, rather than a $B$-action); and
  • braid varieties $\mathcal{U}(\beta)$ built using the full unipotent variety of $G$ (similar to the Springer resolution).
Definition. (Trinh, W.) The unipotent braid variety for a positive link $\beta$ is the set of walks in $G/B$ that start and end in the same Schubert cell (possibly at different flags), together with an element of $u \in U$ recording this difference: $$Z_\beta:=\{ (u, B_0\xrightarrow{\beta_1} B_1 \xrightarrow{\beta_2} \cdots \xrightarrow{\beta_\ell} B_\ell) : u \in U, B_0 = u B_\ell \}.$$
It vastly improves on a cheap trick (i.e., unpalatable to algebraic geometers)
we used to count in our work with Galashin and Lam: $$ \sum_{w \in W} \underbrace{\sum_{\chi \in \mathrm{Irr}(W)} \cdots}_{\text{a tricky trace over }H_W} = \sum_{\chi \in \mathrm{Irr}(W)} \underbrace{\sum_{w \in W} \cdots .}_{\text{becomes an average over }W}$$
  • The data of $u \in U$ also explains an artificial factor of $q^{-\ell(w)}$ in our computations.
  • Counting for $\beta=c^p$ still relies on the not-so-cheap exotic Fourier transform.
  • By having one unified variety, we are able to use results of Borho-MacPherson to connect the point counts to the Hecke algebra.
  • Following Jones, we can compute walks in $Z_\beta(\mathbb{F}_q)$ as a trace in $H_W$.

The Hecke algebra and walks in $G(\mathbb{F}_q)/B(\mathbb{F}_q)$

The Hecke algebra $H_W$ is generated by linear operators $T_s$ on $\mathbb{Z}[G(\mathbb{F}_q)/B(\mathbb{F}_q)]$ defined by $$T_s \cdot B' = \sum_{B' \xrightarrow{s} B''} B''.$$
These satisfy the usual commutations and braid relations, and (check!) $$T_s^2 = (q-1)T_s+q,$$ so that by multiplying by $q^{-1/2} T_s^{-1}$ and rearranging we get a suspiciously familiar $$q^{-1/2} T_s-q^{1/2} T_s^{-1}=(q^{1/2}-q^{-1/2})$$

The Hecke algebra and walks in $G(\mathbb{F}_q)/B(\mathbb{F}_q)$

$$T_{st}=\left(\begin{array}{r|rr|rr|rrrr|rrrr|rrrrrrrr} 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline 0 & 0 & 0 & 1 & 1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 1 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 \\ \hline 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 \\ \hline 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 \\ \hline 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \end{array}\right) \mapsto \left(\begin{array}{rrrrrr} 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & q & q - 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & q & q - 1 \\ q^{2} & 0 & q^{2} - q & 0 & 0 & q - 1 \\ 0 & q^{2} & 0 & q^{2} - q & q^{2} - q & q^{2} - 2 q + 1 \end{array}\right)$$ $$|G/B|\times |G/B| \text{-sized matrix} \mapsto |W| \times |W|\text{-sized matrix} $$

Parabolic Unipotent Braid Varieties

Our parabolic analogue coarsens the start and end in same cell condition.

For $J \subseteq S$, write $P_J = B W_J B$ and let $U_J$ be the unipotent radical of $P_J$.
Definition. (Trinh, W.) The parabolic unipotent braid variety
for a positive link $\beta$ is: $$Z^J_\beta:=\{ (u, B_0\xrightarrow{\beta_1} B_1 \xrightarrow{\beta_2} \cdots \xrightarrow{\beta_\ell} B_\ell) : u \in U_J, B_0 = u B_\ell \}.$$ (For $J=\emptyset$, $P_\emptyset = B$ and $Z_\beta=Z^\emptyset_\beta$.)
We show that $Z^J_\beta$ is stratified into disjoint $P_J$-stable subvarieties $Z^{[w]}_\beta$: $$Z^J_\beta = \bigsqcup_{w \in W/W_J} Z^{[w]}_\beta$$ defined by the condition $P_J B_\ell= P_J w^{-1} B$ for some $w \in W/W_J$.

Parabolic Unipotent Braid Varieties

Parabolic Unipotent Braid Varieties

Deodhar Decompositions

Further refining $Z^{[w]}_\beta$ according to which Schubert cell contains $B_i$ for each $i$, we obtain Deodhar-type cells indexed by combinatorial objects coming from subwords of $\beta$ obeying Mellit's mantra:

if you can go up, you must go up.

Theorem. (Trinh, W.) Each $Z^{[w]}_\beta$ has a Deodhar-type cell decomposition into cells of the form $\mathbb{A}^{d} \times \mathbb{G}_m^e$, so that $$|Z^J_\beta(\mathbb{F}_q)| \propto \sum_{\substack{w \in W/W_J \\ w \text{ maximal rep.}}} \sum_{\substack{\omega \text{ distinguished}\\ w \text{ subword of }\beta}} q^{d_\omega} (q-1)^{e_\omega}.$$

Deodhar Decompositions

        



Results

Catalan Combinatorics

For $\beta=c^p$ with $p$ coprime to $h$, we obtain rational parabolic $q$-Catalan numbers that interpolate between $W$-parking and $W$-Catalan numbers.
Theorem (Trinh, W.). $$|Z^J_{c^p}(\mathbb{F}_q)| \propto \prod_{i=1}^r \frac{[p-e_i(W_J)]_q}{[d_i(W_J)]_q},$$ with $|Z^\emptyset_{c^p}(\mathbb{F}_q)|\propto [p]_q^r$ and $|Z^S_{c^p}(\mathbb{F}_q)|\propto \prod_{i=1}^r\frac{[p-e_i]_q}{[d_i]_q}.$
        

HOMFLY Coefficients

Theorem. (Trinh, W.) Each $a$-degree of the HOMFLY polynomial $\mathcal{P}(\widehat{\beta})$ is a sum of Deodhar-cell point counts: $$[a^{2k}]\mathcal{P}(\widehat{\beta}) \propto \sum_{\substack{w \in S_n\\ \mathrm{des}(w)=[n-1-k]}} |Z^{[w]}_\beta(\mathbb{F}_q)|.$$

$$\mathcal{P}(T_{4,5})\Big|_{q=1} \propto \color{blue}{14}\color{black} - 21a^2+9a^4-a^6.$$


Thank you!