1.1 Motivation from categorical representation theory

It was shown by Khovanov and Thomas [Reference Khovanov and ThomasKT07] that Rouquier complexes give a faithful action of the braid group on the homotopy category of Soergel bimodules. For this reason, the collection of Rouquier complexes is often referred to as a categorification of the braid group, but this is somewhat misleading, as this particular action of the braid group is intricately tied to its Hecke quotient. Soergel [Reference SoergelSoe07] proved that Soergel bimodules over $R_{n}$ categorify the Hecke algebra $\mathbf{H}=\mathbf{H}_{n}$ of the symmetric group $S_{n}$ . Note that $\mathbf{H}$ is linear over the ring $\mathbb{Z}[Q,Q^{-1}]$ , where $Q$ is categorified by the grading shift of an $R$ -bimodule;Footnote ³ for this to work correctly, $R$ is graded so that $\deg x_{i}=2$ for all $1\leqslant i\leqslant n$ . Taking the image of a Rouquier complex in this Grothendieck group yields the familiar quotient map from the braid group (or its group algebra over $\mathbb{Z}[Q,Q^{-1}]$ ) to the Hecke algebra. However, the fact that Rouquier complexes only reflect the ‘Hecke-type’ actions of the braid group is an advantage, not a limitation, as one can lift results from the representation theory of Hecke algebras to study the homotopy category of Soergel bimodules. This paper can be understood and appreciated without a foray into categorical representation theory, but we provide some brief motivation here.

The Hecke algebra admits a sign representation $\operatorname{sgn}_{n}$ , on which each of its standard generators (the images of the overcrossings) acts by $-Q^{-1}$ . The projection from an arbitrary Hecke algebra representation to its isotypic component for the sign representation is an idempotent often known as a (generalized) Jones–Wenzl projector, after the corresponding idempotent in the Temperley–Lieb algebra [Reference JonesJon01, Reference WenzlWen87]. This projection can not be defined in $\mathbf{H}$ itself, requiring certain scalars to be inverted (like $Q+Q^{-1}$ , for example). It can be defined in the base change $\mathbf{H}\otimes _{\mathbb{Z}[Q,Q^{-1}]}\mathbb{Z}((Q))$ .

In [Reference HogancampHog18], the second author constructs an infinite complex of Soergel bimodules $P_{n}$ which categorifies this Jones–Wenzl projector. In this paper we study a finite complex $K_{n}$ which categorifies the ‘renormalized’ Jones–Wenzl projector, a rescaling of the projector which is actually defined within $\mathbf{H}$ before base change. The fact that the Jones–Wenzl idempotent projects to the sign representation is categorified by the fact that the Rouquier complex for an overcrossing, acting by tensor product on $K_{n}$ , will simply act by a homological and a grading shift.

The inductive construction of $K_{n}$ itself also is motivated by the representation theory of the Hecke algebra. When $\operatorname{sgn}_{n}$ is induced from $\mathbf{H}_{n}$ to $\mathbf{H}_{n+1}$ , it splits into two irreducible representations, $\operatorname{sgn}_{n+1}$ and another representation $V$ . This splitting is actually the eigenspace decomposition for the Young–Jucys–Murphy operator $y_{n+1}$ , a certain element of the braid group on $n+1$ strands which commutes with any braid on the first $n$ strands. If $\unicode[STIX]{x1D6FE}$ is the eigenvalue corresponding to $V$ , then $y_{n+1}-\unicode[STIX]{x1D6FE}$ kills $V$ , and thus is equal to the projection to $\operatorname{sgn}_{n+1}$ up to scalar. If $k_{n}\in \mathbf{H}_{n}\subset \mathbf{H}_{n+1}$ denotes the renormalized projection onto the sign representation, then by the previous discussion there is a linear relation

$$\begin{eqnarray}k_{n+1}=k_{n}y_{n+1}-\unicode[STIX]{x1D6FE}k_{n}.\end{eqnarray}$$

On the categorical level, this relation becomes an exact triangle. More precisely, there is a grading shift $\unicode[STIX]{x1D6E4}$ and a chain map $\unicode[STIX]{x1D711}:\unicode[STIX]{x1D6E4}K_{n}\rightarrow K_{n}F(y_{n+1})$ such that $K_{n+1}:=\text{Cone}(\unicode[STIX]{x1D711})$ categorifies the renormalized projection to $\operatorname{sgn}_{n+1}$ . Recall that $F(y_{n+1})$ indicates the Rouquier complex associated to $y_{n+1}$ . It turns out that $\unicode[STIX]{x1D6FE}=Q^{2}$ , and $\unicode[STIX]{x1D6E4}=Q^{2}$ is simply the functor which shifts internal degree up by 2. This chain map is constructed in [Reference HogancampHog18], and we recall the basics in § 2.6.

This is an example of categorical diagonalization, a concept which is developed in forthcoming work.Footnote ⁴ The chain map $\unicode[STIX]{x1D711}$ mentioned above is an eigenmap; in our categorification of various concepts in linear algebra, the cones of eigenmaps are used to categorify the operators $(A-\unicode[STIX]{x1D706}I)$ for an eigenvalue $\unicode[STIX]{x1D706}$ of an operator $A$ . This makes the computation of $\operatorname{HH}^{0}$ of a braid particularly significant, because it describes the space of maps from the (shifted) monoidal identity, which are potential eigenmaps. In fact, the main result of this paper is used as a lemma in [Reference Elias and HogancampEH18] to prove that the full twist in the braid group has enough eigenmaps and therefore is categorically diagonalizable. We use this to construct categorical projections to arbitrary irreducible representations of the Hecke algebra, not just the sign representation.

In this paper, our focus is on computation: the existence of $K_{n}$ is known by other means, and we use the recursive definition of $K_{n}$ to compute link invariants. This strategy is outlined below.

1.2 Our method, decategorified

The Hecke algebra $\mathbf{H}_{n}$ is isomorphic to a quotient of the group algebra $\mathbb{Z}[Q,Q^{-1}][\operatorname{Br}_{n}]$ where we identify

The Jones–Ocneanu trace $\operatorname{Tr}:\mathbf{H}_{n}\rightarrow \mathbb{Z}[Q^{\pm },A^{\pm }]$ is such that $\operatorname{Tr}(\unicode[STIX]{x1D6FD})$ is the Homfly polynomial of the braid closure $\hat{\unicode[STIX]{x1D6FD}}$ . Using the skein relation above, and the formula which defines $\operatorname{Tr}$ , one can in principal compute the Homfly polynomial for any link. In (1.2) we introduce another skein-like relation which is often useful.

There are elements $k_{n}\in \mathbf{H}_{n}$ defined inductively by $k_{1}=1\in \mathbf{H}_{n}$ , and

(1.1)

The element $k_{n}$ is a renormalized projection onto the sign representation: $k_{n}\unicode[STIX]{x1D70E}_{i}^{\pm }=-Q^{\mp }k_{n}=\unicode[STIX]{x1D70E}_{i}^{\pm }k_{n}$ for all $1\leqslant i\leqslant n-1$ , where $\unicode[STIX]{x1D70E}_{i}$ denotes the elementary braid generator. This implies that

(1.2)

The Jones–Ocneanu trace $\operatorname{Tr}(k_{n})$ can easily be computed inductively from this equation using the invariance of $\operatorname{Tr}$ under the Markov moves together with the fact that $k_{n}$ absorbs crossings. We can use $k_{n}$ to compute link invariants as follows.

1. – Assume $\unicode[STIX]{x1D6FD}$ is given. Choose a crossing $x$ in $\unicode[STIX]{x1D6FD}$ . Place $k_{1}$ somewhere in the vicinity of this crossing. Since $k_{1}$ is the identity element of $\mathbf{H}_{1}$ , this does not change the element $\unicode[STIX]{x1D6FD}\in \mathbf{H}_{n}$ .

2. – Apply the relation (1.2); one of the terms will involve the switched crossing $x^{-1}$ , and in the other term $k_{1}$ will have grown to $k_{2}$ , which now has the potential to absorb some crossings.

3. – Repeat. That is, assume that $\unicode[STIX]{x1D6FD}$ is a braid with a $k_{\ell }$ inserted somewhere. After manipulating the diagram, if necessary, arrange the picture so that (1.2) can be applied. In one of the resulting terms, some crossings will be switched, which in good situations will simplify $\unicode[STIX]{x1D6FD}$ . In the other term, $k_{\ell }$ grows in size and can now absorb more crossings, also resulting in a simpler diagram.

If one is lucky, this process can be repeated until the trace $\operatorname{Tr}$ of the resulting terms is trivial to compute. Torus links seem especially well adapted to the application of this trick.

Example 1.1. Let $x=\unicode[STIX]{x1D70E}_{1}\in \mathbf{H}_{2}$ denote the crossing. The trefoil is the $(2,3)$ torus knot, and can be presented as the closure of $x^{3}$ . Equation (1.2) says that $x=-Qk_{2}+Q^{2}x^{-1}$ . Multiplying by $x$ gives

$$\begin{eqnarray}x^{2}=k_{2}+Q^{2}.\end{eqnarray}$$

Multiplying by $x$ again gives

$$\begin{eqnarray}x^{3}=(-Q^{-1})k_{2}+Q^{2}x.\end{eqnarray}$$

The trace of $k_{2}$ is easy to compute, and the trace of $x$ is the Homfly polynomial of the unknot, up to normalization. Thus, the trace of $x^{3}$ is expressed in terms of known quantities. We can continue in this manner, obtaining

$$\begin{eqnarray}x^{2m}=(Q^{2(1-m)}+Q^{2(2-m)}+\cdots +Q^{2(m-1)})k_{2}+Q^{2m}\end{eqnarray}$$

and

$$\begin{eqnarray}x^{2m+1}=(-Q^{-1})(Q^{2(1-m)}+Q^{2(2-m)}+\cdots +Q^{2(m-1)})k_{2}+Q^{2m}x,\end{eqnarray}$$

from which the Homfly polynomials of the $(2,m)$ torus links are readily computed.

1.3 Our method, categorified

In this paper we categorify the method outlined in the previous section. As alluded to earlier in this introduction, the element $k_{n}\in \mathbf{H}_{n}$ gets replaced by a finite complex $K_{n}$ of Soergel bimodules, and the relations (1.1) and (1.2) become exact triangles. More precisely, there is a chain map constructed in [Reference HogancampHog18] from $Q^{2}K_{n}\rightarrow K_{n}F(y_{n+1})$ , and $K_{n+1}$ is defined to be the mapping cone on this map. The fact that $k_{n}$ absorbs crossings becomes the fact that $K_{n}F(\unicode[STIX]{x1D70E}_{i})\simeq TQ^{-1}K_{n}\simeq F(\unicode[STIX]{x1D70E}_{i})K_{n}$ . Here and throughout we use $T$ and $Q$ to denote the functors which increase homological degree and bimodule degree respectively. We have, for instance, an equivalence

(1.3)

where the notation $A\simeq (B\rightarrow C)$ means that there is an exact triangle

$$\begin{eqnarray}C\rightarrow A\rightarrow B\rightarrow T^{-1}C.\end{eqnarray}$$

The distinguished triangle (1.3) will be essentially the only weapon we need to attack our computations. Suppose $F(\unicode[STIX]{x1D6FD})$ is a Rouquier complex that we would like to study. Iterated application of the above exact triangle results in a certain kind of filtered complex (a convolution of a twisted complex; see below) which is homotopy equivalent to $F(\unicode[STIX]{x1D6FD})$ , and whose subquotients are tensor products of Rouquier complexes and some $K_{\ell }$ . In favorable situations these have Hochschild cohomologies which are easy to compute. There then arises the problem of recovering the homology of the total complex from the homology of its constituents. For certain computations, we will see that this very serious complication is nullified by an equally serious miracle: the miracle of parity.

We first explain what sorts of filtered complexes we will use. Suppose $A_{i}$ ( $i\in I$ ) is a family of complexes, indexed by a finite partially ordered set $I$ . Suppose $d_{ij}:A_{j}\rightarrow A_{i}$ are a collection of linear maps such that:

1. – $d_{ij}$ increases homological degree by 1;

2. – $d_{ii}$ is the given differential on $A_{i}$ ;

3. – $d_{ij}=0$ unless $i\geqslant j$ ;

4. – the total differential $d_{\text{tot}}:=\sum _{i\geqslant j}d_{ij}$ satisfies $d_{\text{tot}}^{2}=0$ .

Then $C:=(\bigoplus _{i\in I}A_{i},d_{\text{tot}})$ is a chain complex, which we call a convolution of the $A_{i}$ . More precisely, this is the convolution of a one-sided twisted complex; see [Reference Bondal and KapranovBK90] for more details on this construction in homological algebra. We will also say that $C=\bigoplus _{i}A_{i}$ with twisted differential, to indicate that the differential is not merely the sum of the differentials on the $A_{i}$ . The differential on this complex may be quite complicated, possibly sending terms in homological degree $m$ inside $A_{j}$ to terms in homological degree $m+1$ inside many different $A_{i}$ , but it does respect the order on $I$ . Thus, convolutions can also be thought of as certain kinds of filtered complexes, whose subquotients are the $A_{i}$ .

Note that any exact triangle

$$\begin{eqnarray}A_{2}\rightarrow C\rightarrow A_{1}\rightarrow T^{-1}A_{2}\end{eqnarray}$$

gives rise to an equivalence $C\simeq (A_{1}\oplus A_{2})$ with twisted differential. Iterated mapping cones can be regarded as convolutions in a similar way.

Our main application is to the Rouquier complex $\text{FT}_{n}$ associated to the full twist braids. In § 3 we iterate the equivalence (1.3), obtaining a convolution description of $F(y_{n})$ , the Rouquier complex associated to the Young–Jucys–Murphy braid. Using the relation $\text{FT}_{n}=\text{FT}_{n-1}F(y_{n})$ , we then prove the following.

In the antilexicographic order we regard $(\ast ,1)$ as larger than $(\ast ,0)$ , where $\ast$ denotes any sequence of zeroes and ones. For each sequence $v\in \{0,1\}^{n}$ , which we will call a shuffle, there is a complex which we call $D_{v}$ . For example, here is $D_{10101101}$ , which occurs (up to shift) in the expression of $\text{FT}_{8}$ :

Inside $v$ , the zeroes indicate which strands are connected to the full twist $\text{FT}_{k}$ , and the ones indicate which are connected to $K_{\ell }$ , for $k+\ell =n$ .

Then, of course, one wants to compute the Hochschild homology of the complexes $D_{v}$ . Let us be precise. Given a complex $F$ of Soergel bimodules, let $\operatorname{HH}^{i}(C)$ denote the complex obtained by applying the functor $\operatorname{HH}^{i}$ to each bimodule, and let $\operatorname{HH}(C)=\oplus \operatorname{HH}^{i}(C)$ . Let $\operatorname{HHH}(C)$ denote the cohomology of the complex $\operatorname{HH}(C)$ .

Because Hochschild cohomology of a complex $C$ is unchanged by conjugation $C\mapsto FCF^{-1}$ for any invertible complex $F$ , we can move part of $D_{v}$ from the bottom to the top, yielding the complex $C_{v}^{\prime }$ :

Note that $\text{FT}_{n}=C_{00\cdots 0}^{\prime }$ .

For purely combinatorial reasons, we work instead with a similar complex $C_{v}$ , which is defined by the same expression as $C_{v}^{\prime }$ , but with $K_{\ell }$ replaced by its reduced version $\hat{K}_{\ell }$ . Reduced complexes are discussed in § 4.3. The effect this has on Poincaré polynomials is multiplication by a factor of $(1-Q^{2})$ .

Let $v\cdot w$ denote the concatenation of two shuffles (sequences of zeroes and ones). For any shuffle $v$ , we can use our distinguished triangle for $K_{n}$ to prove the following.

Next, we can use some relatively easy arguments involving the complex $K_{n}$ to prove that $\operatorname{HH}(C_{v\cdot 1})$ is just a direct sum of shifted copies of $\operatorname{HH}(C_{v})$ . For readers familiar with knot theory, this last statement should be thought of as analogous to the Markov move; it allows us to reduce the number of strands by $1$ . Finally, a simple observation (pertaining to reduced complexes) allows one to replace the computation of $\operatorname{HH}(C_{000\cdots 0})$ with $\operatorname{HH}(C_{100\cdots 0})$ . Combining these three operations, we obtain a recursive convolution description of any $\operatorname{HH}(C_{v})$ . This is the main result of § 4.5.

Let us return to the computation of the Hochschild cohomology of the Rouquier complex for the full twist $\text{FT}_{n}$ on $n$ strands.

We are interested in the cohomology $\operatorname{HHH}(C_{v})$ of the complexes $\operatorname{HH}(C_{v})$ . However, in general, the cohomology of a convolution of complexes is not the direct sum of the cohomology of the individual complexes; instead, there is a spectral sequence relating the two. Our final argument comes from observing a parity miracle! We prove inductively that $\operatorname{HHH}(C_{v})$ is concentrated in even homological degrees. This forces every spectral sequence in sight to degenerate at the $E_{1}$ page, and implies that our convolution description of $\operatorname{HH}(C_{v})$ gives a direct sum description of $\operatorname{HHH}(C_{v})$ . See Theorem 4.22 and its proof for further discussion of this parity argument.

Thus, we have a recursive formula for the triply graded cohomologies $\operatorname{HHH}(C_{v})$ , and as a special case, a formula for $\operatorname{HHH}(\text{FT}_{n})$ . We discuss this formula in the next section.

Let us pause to point out one of the subtleties we have ignored above. One can conjugate a complex by a braid and obtain a non-isomorphic complex with the same Hochschild cohomology. We begin to apply this operation freely in § 4. Above, we have stated that $\operatorname{HH}(C_{v\cdot 0})$ is a convolution of $\operatorname{HH}(C_{0\cdot v})$ and $\operatorname{HH}(C_{1\cdot v})$ , but the same statement does not hold for the original complexes. Instead, some conjugate of $C_{v\cdot 0}$ is a convolution of a conjugate of $C_{0\cdot v}$ and a conjugate of $C_{1\cdot v}$ (conjugating by different braids for each term). For purposes of Hochschild cohomology, this imprecision is harmless. However, were one to try to actually construct a chain map from $R$ to $C_{v}$ using this computation of $\operatorname{HHH}^{0}(C_{v})$ , then one would need to keep track of conjugation more carefully, which would be rather difficult.

On the other hand, the work done in § 3 describes full twist as a genuine convolution of complexes $D_{v}$ , not complexes up to conjugation. This result is not actually needed or used in the recursive computation of $\operatorname{HHH}(C_{v})$ which is our main result. We include this auxiliary result because it can be used to construct chain maps from $R$ to $\text{FT}_{n}$ . Our main theorem implies that the complexes $D_{v}$ satisfy a parity condition, and therefore our convolution description of $\text{FT}_{n}$ induces a direct sum decompositions on Hochschild cohomology. In particular, any chain map (up to homotopy) from $R$ to the complex $D_{v}$ (an element of $\operatorname{HHH}^{0}(D_{v})$ ) can be extended uniquely (up to homotopy) to a chain map from $R$ to the entire complex $\text{FT}_{n}$ . We use this fact to construct eigenmaps to the full twist in [Reference Elias and HogancampEH18].

In addition, the convolution description involving $D_{v}$ from § 3 can be adapted to other torus links, whereas the results § 4 are fundamentally tied to the case of $(n,n)$ -torus links.

1.4 The recursive formula

We will find it convenient to use a non-standard choice of variables for our Poincaré series. We let $t=T^{2}Q^{-2}$ , $q=Q^{2}$ , and $a=Q^{-2}A$ , where $T$ denotes the usual homological degree, $Q$ the bimodule degree (also called internal degree, or quantum degree), and $A$ the Hochschild degree. For instance, the Poincaré series of the polynomial ring $R=\mathbb{Q}[x_{1},\ldots ,x_{n}]$ is written $1/(1-q)^{n}$ , and the Poincaré series of its Hochschild cohomology is $(1-q)^{-n}(1+a)^{n}$ . In general, all our Poincaré series will be power series in the variables $q$ , $a$ , and $t^{1/2}$ .

Proposition 1.5. There is a unique family of polynomials $f_{v}(q,a,t)$ , indexed by integers $n\geqslant 0$ and binary sequences $v\in \{0,1\}^{n}$ , satisfying $f_{\emptyset }=1$ together with

(1.4a) $$\begin{eqnarray}\displaystyle & \displaystyle f_{v\cdot 1}(q,a,t)=(t^{|v|}+a)f_{v}, & \displaystyle\end{eqnarray}$$

(1.4b) $$\begin{eqnarray}\displaystyle & \displaystyle f_{v\cdot 0}(q,a,t)=qf_{0\cdot v}+f_{1\cdot v}. & \displaystyle\end{eqnarray}$$

Here $|v|:=v_{1}+\cdots +v_{n}$ is the number of ones of $v=(v_{1},\ldots ,v_{n})\in \{0,1\}^{n}$ .

The following theorem, together with the fact that the $f_{v}$ are rational functions in $q,a,t$ rather than $q,a,t^{1/2}$ implies that the parity miracle holds.

These results are restated and proved in § 4.6. The proof of Theorem 1.6 comes from the convolution description of $\operatorname{HH}(C_{v})$ discussed in the previous section.

For the reader’s edification, here are the complete power series for $\operatorname{HHH}(\text{FT}_{n})$ for $n=1,2,3$ :

$$\begin{eqnarray}\displaystyle & \displaystyle f_{0}(q,a,t)=\frac{1+a}{1-q}, & \displaystyle \nonumber\\ \displaystyle & \displaystyle f_{00}(q,a,t)=\frac{1+a}{(1-q)^{2}}(q+t-qt+a), & \displaystyle \nonumber\end{eqnarray}$$

$$\begin{eqnarray}\displaystyle f_{000}(q,a,t) & = & \displaystyle \frac{1+a}{(1-q)^{3}} (\,(t^{3}q^{2}+q^{3}t^{2}-2t^{2}q^{2}-2tq^{3}-2qt^{3}+t^{3}+q^{3}+tq^{2}+qt^{2}+tq)\nonumber\\ \displaystyle & & \displaystyle +\,(t^{2}q^{2}-2tq^{2}-2qt^{2}+t^{2}+q^{2}+tq+t+q)a+a^{2} ).\nonumber\end{eqnarray}$$

The recursion can be unraveled into the equivalent recursion below, which is more complicated but faster to implement.

Definition 1.7. For each integer $n\geqslant 0$ , we let $[n]:=\{1,\ldots ,n\}$ . We identify subsets $v\subset [n]$ with binary sequences $v\in \{0,1\}^{n}$ . For each such $v\subset [n]$ , we define a rational function $f_{v}(q,a,t)$ by $f_{\emptyset }=1$ , together with the following rules:

(1.5a) $$\begin{eqnarray}\displaystyle & \displaystyle f_{000\cdots 0}(q,a,t)=\frac{1}{1-q}f_{100\cdots 0}(q,a,t), & \displaystyle\end{eqnarray}$$

(1.5b) $$\begin{eqnarray}\displaystyle & \displaystyle f_{11\cdots 1}(q,a,t)=\mathop{\prod }_{i=1}^{n}(t^{i-1}+a)\quad (n~\text{indices}), & \displaystyle\end{eqnarray}$$

(1.5c) $$\begin{eqnarray}\displaystyle & \displaystyle f_{v}(q,a,t)=\mathop{\sum }_{w\subset [k]}{\mathcal{P}}_{v,w}(a,t)q^{k-|w|}f_{w}(q,a,t),\quad v\neq (0\cdots 0)\text{ and }v\neq (1\cdots 1). & \displaystyle\end{eqnarray}$$

Here $k=n-|v|$ is the number of zeroes in $v$ . Definition 5.3 contains the description of ${\mathcal{P}}_{v,w}(a,t)$ , which is a product of $n-k$ factors each of the form $(t^{\ell +m}+a)$ for various numbers $\ell$ and $m$ depending on the sequences $v$ and $w$ .

The equivalence between these recursions is proven in § 5, which contains various such numerological considerations. For example, one can show that both rule (1.5a) and (1.5b) are actually just consequences of rule (1.5c) when applied verbatim, although this is not obvious.

Note that the contribution to higher Hochschild gradings comes only from a factor in rule (1.5b) and the factor ${\mathcal{P}}_{v,w}(a,t)$ in rule (1.5c); both of these become explicit monomials in $t$ upon setting $a=0$ . Thus, to understand the zeroth Hochschild degree coefficient, i.e. the polynomial $f_{v}(q,0,t)$ , one use a simplified recursion relation. Using this, we prove the following closed formula for the power series $f_{00\cdots 0}(q,0,t)$ , also known as the Poincaré series of the zeroth Hochschild cohomology $\operatorname{HHH}^{0}(\text{FT})$ .

Theorem 1.9. The Hochschild degree zero part of the unreduced triply graded homology of  $(n,n)$ torus links has Poincaré series equal to

$$\begin{eqnarray}F_{n}(q,t)=\mathop{\sum }_{\unicode[STIX]{x1D70E}}t^{a(\unicode[STIX]{x1D70E})+b(\unicode[STIX]{x1D70E})}q^{c(\unicode[STIX]{x1D70E})},\end{eqnarray}$$

where the sum is over functions $\unicode[STIX]{x1D70E}:\{1,\ldots ,n\}\rightarrow \mathbb{Z}_{{\geqslant}0}$ , and the integers $a(\unicode[STIX]{x1D70E})$ , $b(\unicode[STIX]{x1D70E})$ , $c(\unicode[STIX]{x1D70E})$ are defined by:

1. (i) $a(\unicode[STIX]{x1D70E})=\sum _{k\geqslant 0}\binom{|\unicode[STIX]{x1D70E}^{-1}(k)|}{2}$ ;

2. (ii) $b(\unicode[STIX]{x1D70E})$ is the number of pairs $(i,j)\in \{1,\ldots ,n\}$ such that $i<j$ and $\unicode[STIX]{x1D70E}(j)=\unicode[STIX]{x1D70E}(i)+1$ ;

3. (iii) $c(\unicode[STIX]{x1D70E})=\sum _{i=1}^{n}\unicode[STIX]{x1D70E}(i)$ .

Example 1.11. In case $n=2$ , $F_{2}(q,t)$ is the sum of monomials appearing in the following diagram.

After rearranging, this becomes $F_{2}(q,t)=t/(1-q)+q/(1-q)^{2}$ , agreeing with $f_{00}(q,0,t)$ which was computed above.

The proof of this closed formula from Theorem 1.6 is a simple combinatorial argument, and is found in § 5. Unfortunately, the polynomials ${\mathcal{P}}_{v,w}(q,a,t)$ are sufficiently complicated so that we have been unable to produce a closed formula for the higher Hochschild degrees along these lines.

We conclude with a recursion for the normalized polynomials $\widetilde{f}_{v}(q,a,t):=(1-q)^{k}f_{v}(q,a,t)$ , where $k$ is the number of zeroes in $v$ . The recursion of Proposition 1.5 immediately gives rise to

(1.6a) $$\begin{eqnarray}\displaystyle & \displaystyle \widetilde{f}_{v\cdot 1}(q,a,t)=(t^{|v|}+a)\widetilde{f}_{v}, & \displaystyle\end{eqnarray}$$

(1.6b) $$\begin{eqnarray}\displaystyle & \displaystyle \widetilde{f}_{v\cdot 0}(q,a,t)=q\widetilde{f}_{0\cdot v}+(1-q)\widetilde{f}_{1\cdot v}. & \displaystyle\end{eqnarray}$$

Computer experiments suggest the following.

This symmetry would follow from a formula of Gorsky, Negut, and Rasmussen, which we discuss now.

1.5 Flag Hilbert schemes and a magic formula

According to the remarkable work of Gorsky, Negut, and Rasmussen [Reference Gorsky and NeguțGN15, Reference Gorsky, Neguț and RasmussenGNR16], triply graded link homology can be extracted from flag Hilbert schemes.Footnote ⁵ Roughly, the picture looks like this: there is a space $\operatorname{FHilb}_{n}(\mathbb{C}^{2})$ , which parametrizes flags of ideals $I_{1}\subset \cdots \subset I_{n}\in \mathbb{C}[x,y]$ such that $I_{i}/I_{i-1}$ is 1-dimensional. Associated to each $n$ -strand braid, Gorsky, Negut, and Rasmussen conjecture that there exists a line bundle (or sheaf, or complex of sheaves) on $\operatorname{FHilb}_{n}(\mathbb{C}^{2})$ whose space of global sections recovers $\operatorname{HHH}^{0}(F(\unicode[STIX]{x1D6FD}))$ . The sheaves are meant to be equivariant with respect to an obvious action of $\mathbb{C}^{\ast }\times \mathbb{C}^{\ast }$ , and the variables $q,t$ correspond to weights with respect to this action. The variable $a$ can also be accounted for with more work. A combination of Atiyah–Bott localization and careful analysis of the flag Hilbert scheme near its torus fixed points yields a remarkably simple combinatorial formula which, given their conjecture, will describe the knot homology of positive torus links. Now we state the formulas, and we will make no further mention of the geometric foundations which motivate them.

Let $\unicode[STIX]{x1D706}$ be a Young diagram, drawn in the ‘English style’ as in the following.

Suppose a box $c$ is in the $i$ th column and $j$ th row. Here columns and rows are counted left-to-right and top-to-bottom, starting at zero. To such a box we associated the monomial $z_{c}=t^{i}q^{j}$ .Footnote ⁶ To a Young diagram, we let $z_{\unicode[STIX]{x1D706}}=z_{\unicode[STIX]{x1D706}}(q,t)$ be the product of $z_{c}$ as $c$ ranges over all the boxes of  $\unicode[STIX]{x1D706}$ . Note that $z_{\unicode[STIX]{x1D706}}(q,t)=z_{\unicode[STIX]{x1D706}^{t}}(t,q)$ , where $\unicode[STIX]{x1D706}^{t}$ is the transposed partition.

A box in $\unicode[STIX]{x1D706}$ is removable if $\unicode[STIX]{x1D706}\smallsetminus c$ is a Young diagram. Now we discuss boxes, i.e. coordinates $(i,j)$ , which need not be in the given Young diagram $\unicode[STIX]{x1D706}$ . We call $c\notin \unicode[STIX]{x1D706}$ an outer corner of $\unicode[STIX]{x1D706}$ if the top left corner of $c$ coincides with the bottom right corner of a removable box in $\unicode[STIX]{x1D706}$ . We call a box $c\notin \unicode[STIX]{x1D706}$ an inner corner of $\unicode[STIX]{x1D706}$ if $\unicode[STIX]{x1D706}\cup c$ is a Young diagram. Pictorially, we have the following.

The inner corners are darkly shaded, and the outer corners are lightly shaded. Let $\operatorname{In}(\unicode[STIX]{x1D706})$ and $\operatorname{Out}(\unicode[STIX]{x1D706})$ denote the sets of inner and outer corners of $\unicode[STIX]{x1D706}$ . If $c\in \operatorname{In}(\unicode[STIX]{x1D706})$ , then we define

$$\begin{eqnarray}f_{\unicode[STIX]{x1D706},c}(q,t):=\frac{\mathop{\prod }_{d\in \operatorname{Out}(\unicode[STIX]{x1D706})}(z_{c}-z_{d})}{\mathop{\prod }_{e\in \operatorname{In}(\unicode[STIX]{x1D706})\smallsetminus \{c\}}(z_{c}-z_{e})}.\end{eqnarray}$$

It is easy to observe that $f_{\unicode[STIX]{x1D706},c}(q,t)=f_{\unicode[STIX]{x1D706}^{t},c^{t}}(t,q)$ , where $c^{t}$ is the corresponding transposed inner corner of $\unicode[STIX]{x1D706}^{t}$ .

Remark 1.14. Let $\{x_{1},\ldots ,x_{n}\}$ and $\{y_{1},\ldots ,y_{n+1}\}$ be two families of abstract variables. It is not a hard exercise to show that

$$\begin{eqnarray}\mathop{\sum }_{i=1}^{n+1}\frac{\mathop{\prod }_{j}(y_{i}-x_{j})}{\mathop{\prod }_{k\neq i}(y_{i}-y_{k})}=1.\end{eqnarray}$$

Applying this general formula to the definition above, one obtains

(1.7) $$\begin{eqnarray}\mathop{\sum }_{c\in \operatorname{In}(\unicode[STIX]{x1D706})}f_{\unicode[STIX]{x1D706},c}=1,\end{eqnarray}$$

a fact which has nothing to do with the combinatorics of partitions.

A standard tableau $T$ can be thought of as a sequence of Young diagrams $T=(\unicode[STIX]{x1D706}_{1},\ldots ,\unicode[STIX]{x1D706}_{n})$ such that $\unicode[STIX]{x1D706}_{1}=\Box$ and $\unicode[STIX]{x1D706}_{i+1}\smallsetminus \unicode[STIX]{x1D706}_{i}=\Box$ . Let $\operatorname{Sh}(T)=\unicode[STIX]{x1D706}_{n}$ denote the shape of $T$ . To each tableau we set

$$\begin{eqnarray}f_{T}(q,t):=\mathop{\prod }_{i=1}^{n-1}f_{\unicode[STIX]{x1D706}_{i},c_{i}}(q,t),\end{eqnarray}$$

where $c_{i}\in \operatorname{In}(\unicode[STIX]{x1D706}_{i})$ is the box such that $\unicode[STIX]{x1D706}_{i+1}=\unicode[STIX]{x1D706}_{i}\cup c_{i}$ . Once more, $f_{T}(q,t)=f_{T^{t}}(t,q)$ , where $T^{t}$ is the transposed tableau.

Finally, associated to a Young diagram $\unicode[STIX]{x1D706}$ , let $g_{\unicode[STIX]{x1D706}}(q,a,t)$ denote the product over all boxes $c\in \unicode[STIX]{x1D706}$ of $(1+az_{c}^{-1})$ .

Conjecture 1.15 (Magic formula).

Let $F_{n,r}(q,a,t)$ denote the Poincaré series of $\operatorname{HHH}^{0}(\text{FT}_{n}^{\otimes r})$ . Then

(1.8) $$\begin{eqnarray}(1-q)^{n}F_{n,r}(q,a,t)=\mathop{\sum }_{T}z_{\operatorname{Sh}(T)}^{r}g_{\operatorname{ Sh}(T)}f_{T},\end{eqnarray}$$

a sum over all tableaux with $n$ boxes. In particular, the right-hand side is symmetric under replacing $q$ with $t$ , and thus so is the left-hand side. We remind the reader that $z_{\operatorname{Sh}(T)}$ and $f_{T}$ are functions of $q$ and $t$ , while $g_{\operatorname{Sh}(T)}$ is a function of $q$ , $a$ , and $t$ .

When $r=0$ , the formula yields $1=\sum _{T}g_{\operatorname{Sh}(\unicode[STIX]{x1D706})}(q,a,t)f_{T}(q,t)$ , whose $a$ -degree zero part follows from (1.7). We have verified the magic formula for $r=1$ and for $1\leqslant n\leqslant 4$ , using a Mathematica notebook which is available upon request.

According to the magic formula, the $a$ -degree $n$ part of the Poincaré series of the $(n,n)$ torus link is supposed to be $(1-q)^{-n}$ times

$$\begin{eqnarray}\mathop{\sum }_{T}z_{\operatorname{Sh}(T)}z_{\operatorname{Sh}(T)}^{-1}f_{T}(q,t)=1.\end{eqnarray}$$

The factor $z_{\operatorname{Sh}(T)}^{-1}$ comes from taking the $a$ -degree $n$ part of $g_{\operatorname{Sh}(T)}(q,a,t)$ . This instance of the magic formula can be proved directly from our recursive description of this series (see Remark 1.12).

Another consequence of the magic formula concerns the sub-maximal part of the Poincaré series of full twists.

Conjecture 1.16. The $a$ -degree $n-1$ part of the Poincaré series of $\operatorname{HHH}(\text{FT}_{n})$ is a geometric progression

$$\begin{eqnarray}\frac{1}{(1-q)^{n}}\frac{1-(q+t-qt)^{n}}{(1-q)(1-t)}=\frac{1}{(1-q)^{n}}(1+(q+t-qt)+\cdots +(q+t-qt)^{n-1}).\end{eqnarray}$$

We expect that this is not difficult to prove, but we do not do so here. We have verified this conjecture up to $n=7$ using computer calculations.

In the algorithm to compute $f_{00\cdots 0}(q,a,t)$ using (1.5c), one travels from the zero sequence $(00\cdots 0)\in \{0,1\}^{n}$ to the sequence $\emptyset \in \{0,1\}^{0}$ by repeatedly choosing subsets (i.e. smaller sequences in $\{0,1\}^{k}$ ) of the previous set of zeroes. Our instinct indicates that such a sequence of sequences can be thought of as encoding the entries in a Robinson–Shensted row-bumping algorithm, and can thus be assigned a tableau. The contribution to $f_{00\cdots 0}(q,a,t)$ coming from this sequence of sequences and the contribution to $\sum _{T}z_{\operatorname{Sh}(T)}g_{\operatorname{Sh}(T)}f_{T}$ coming from the tableau $T$ have many superficial similarities, but no direct relation has yet been found.

1.6 Organization of the paper

In § 2 we provide some background. We describe various elements of the braid group, including full twists, Young–Jucys–Murphy elements, shuffle braids, and shuffle twists. In § 2.4 we briefly recall Soergel’s categorification of the Hecke algebra and Rouquier’s categorification of the braid group. In § 2.5 we define convolutions of complexes, and give the crucial argument involving the degeneration of a spectral sequence thanks to parity considerations. In § 2.6 we recall the main result of [Reference HogancampHog18], a complex which categorifies a renormalized Jones–Wenzl projector, and state its properties. The specifics of the Soergel–Rouquier construction need not concern the reader, as all we will use in this paper are facts about the braid group and the results of [Reference HogancampHog18].

In § 3 we find a convolution description of the full twist $\text{FT}_{n}$ in terms of certain complexes $D_{v}$ associated to $v\in \{0,1\}^{n}$ .

In § 4 we switch to a Hochschild frame of mind. Since Hochschild cohomology of a complex is invariant under conjugation by Rouquier complexes of braids, we will allow ourselves to freely conjugate complexes. In § 4.2 we discuss another result of [Reference HogancampHog18] which is an analog of the Markov move on braid closures: a relationship between the Hochschild homologies of the Jones–Wenzl projector on $n$ strands and the projector on $n-1$ strands. We also discuss reduced complexes in § 4.3, finally describing complexes $C_{v}$ in § 4.4 which may be thought of as reduced versions of conjugates of $D_{v}$ . Finally, in §§ 4.5 and 4.6 we state and prove the main result, which is a convolution description of the Hochschild cohomology of $C_{v}$ in terms of the Hochschild cohomologies of smaller $C_{w}$ , which is an analogue of the recursion of Proposition 1.5.

In § 5 we prove some combinatorial results which justify Theorem 1.9, our closed form solution for $\operatorname{HHH}^{0}(\text{FT})$ , and show that the two recursive formulas agree.

In the appendix, we include without proof some computations for other $(n,m)$ torus links. These were obtained by techniques entirely analogous to the computation for $(n,n)$ torus links, and many of them have not appeared in the literature before.

1.7 Notation

We collect here some of our notational conventions, for the reader’s convenience. Unfamiliar concepts will be explained in due course. Soergel bimodules are graded. We denote by $(1)$ the grading shift, so that $M(1)^{i}=M^{i+1}$ . We let $Q=(-1)$ denote the functor which increases the degree of each element. Complexes of Soergel bimodules are bigraded. The differentials always preserve the bimodule degree, and increase homological degree by 1. The shift in homological degree is denoted by $\langle 1\rangle$ , so that $C(a)\langle b\rangle ^{i,j}=C^{i+a,j+b}$ . We denote by ${\mathcal{K}}^{b}({\mathcal{A}})$ the homotopy category of finite complexes over an additive category ${\mathcal{A}}$ . Isomorphism in ${\mathcal{K}}^{b}({\mathcal{A}})$ , that is, chain homotopy equivalence, is denoted by $\simeq$ . The existence of a distinguished triangle

$$\begin{eqnarray}A\rightarrow B\rightarrow C\overset{\unicode[STIX]{x1D6FF}}{\rightarrow }A\langle 1\rangle\end{eqnarray}$$

will be indicated by writing $B\simeq (C\overset{\unicode[STIX]{x1D6FF}}{\rightarrow }A)$ . We also let $T=\langle -1\rangle$ denote the functor which increases homological degree by 1. If $\unicode[STIX]{x1D6FD}$ is a braid, we denote the braid exponent by $e(\unicode[STIX]{x1D6FD})$ ; this is the signed number of crossings in a diagram representing $\unicode[STIX]{x1D6FD}$ . The Rouquier complex $F(\unicode[STIX]{x1D6FD})$ is normalized so that if $\unicode[STIX]{x1D6FD}$ is a positive braid, then there is a chain map $(TQ^{-1})^{e(\unicode[STIX]{x1D6FD})}R\rightarrow F(\unicode[STIX]{x1D6FD})$ which is the inclusion of the degree $e(\unicode[STIX]{x1D6FD})$ chain bimodule, whereas if $\unicode[STIX]{x1D6FD}$ is a negative braid, there is a chain map $F(\unicode[STIX]{x1D6FD})\rightarrow (TQ^{-1})^{e(\unicode[STIX]{x1D6FD})}R$ which is the projection onto the degree $e(\unicode[STIX]{x1D6FD})$ bimodule. Note, in [Reference Abel and HogancampAH15] and [Reference HogancampHog18], the shifts $(k)$ and $\langle \ell \rangle$ would have been denoted $(-k)$ and $\langle -\ell \rangle$ , respectively.

Hochschild cohomology gives rise to a functor $\operatorname{HH}$ whose input is a graded bimodule, and whose output is a bigraded vector space. The additional grading is called the Hochschild grading, and shifts in the Hochschild grading are denoted by $A$ . Extending to complexes gives a functor from complexes of graded bimodules to complexes of bigraded vector spaces. These are triply graded objects, so all together we have the shift functors $Q,A,T$ . If $C$ is a complex of bimodules, then the homology of $\operatorname{HH}(C)$ is denoted by $\operatorname{HHH}(C)$ .

We also find it convenient to introduce $t=T^{2}Q^{-2}$ , $q=Q^{2}$ , and $a=AQ^{-2}$ . One might call these the geometric variables, since they appear most naturally in the connection with Hilbert schemes. When convenient, we express our degree shifts and Poincaré series in terms of these variables.