The diagrammatic Soergel category and sl(2) and sl(3) foams

We define two functors from Elias and Khovanov's diagrammatic Soergel category, one targeting Clark-Morrison-Walker's category of disoriented sl(2) cobordisms and the other the category of (universal) sl(3) foams.


INTRODUCTION
In this paper we define functors between the Elias-Khovanov diagrammatic version of the Soergel category SC defined in [4] and the categories of universal sl(2) and sl(3)-foams defined in [2] and [8].
The Soergel category provides a categorification of the Hecke algebra and was used by Khovanov in [6] to construct a triply graded link homology categorifying the HOMFLYPT polynomial. Elias and Khovanov constructed in [4] a category defined diagrammatically by generators and relations and showed it to be equivalent to SC.
This paper can be seen as a first step towards the construction of a family of functors between SC and the categories of sl(N )-foams for all N ∈ Z + , to be completed in a subsequent paper [9]. The functors F sl(2),n and F sl(3),n are not faithful. In [9] we will extend the construction of these functors to all N . The whole family of functors is faithful in the following sense: if for a morphism f in SC 1 we have F sl(N ),n (f ) = 0 for all N , then f = 0. With these functors one can try to give a graphical interpretation of Rasmussen's [10] spectral sequences from the HOMFLYPT link homology to the sl(N )-link homologies.
The plan of the paper is as follows. In Section 2 we give a brief description of Elias and Khovanov's diagrammatic Soergel category. In Section 3 we describe the category Foam 2 of sl(2)-foams and construct a functor from SC to Foam 2 . Finally in Section 4 we give the analogue of these results for the case of sl(3)-foams.
We have tried to keep this paper reasonably self-contained. Although not mandatory, some acquaintance with [2,4,5,8] is desirable.

THE DIAGRAMMATIC SOERGEL CATEGORY REVISITED
This section is a reminder of the diagrammatics for Soergel categories introduced by Elias and Khovanov in [4]. Actually we give the version which they explained in Section 4.5 of [4] and which can be found in detail in [5].
Fix a positive integer n. The category SC 1 is the category whose objects are finite length sequences of points on the real line, where each point is colored by an integer between 1 and n. We read sequences of points from left to right. Two colors i and j are called adjacent if |i−j| = 1 and distant if |i − j| > 1. The morphisms of SC 1 are given by generators modulo relations. A morphism of SC 1 is a C-linear combination of planar diagrams constructed by horizontal and vertical gluings of the following generators (by convention no label means a generic color j): • Generators involving only one color: It is useful to define the cap and cup as ≡ ≡ • Generators involving two colors: -The 4-valent vertex, with distant colors, read from bottom to top. In this setting a diagram represents a morphism from the bottom boundary to the top. We can add a new colored point to a sequence and this endows SC 1 with a monoidal structure on objects, which is extended to morphisms in the obvious way. Composition of morphisms consists of stacking one diagram on top of the other. We consider our diagrams modulo the following relations. "Isotopy" relations: The relations are presented in terms of diagrams with generic colorings. Because of isotopy invariance, one may draw a diagram with a boundary on the side, and view it as a morphism in SC 1 by either bending the line up or down. By the same reasoning, a horizontal line corresponds to a sequence of cups and caps.
One color relations: Two distant colors: Definition 2.1. The category SC 2 is the category containing all direct sums and grading shifts of objects in SC 1 and whose morphisms are the grading preserving morphisms from SC 1 . Elias and Khovanov's main result in [4] is the following theorem.  (Elias-Khovanov). The category SC is equivalent to the Soergel category in [11]. From Soergel's results from [11] we have the following corollary.

Corollary 2.4. The Grothendieck algebra of SC is isomorphic to the Hecke algebra.
Notice that SC is an additive category but not abelian and we are using the (additive) split Grothendieck algebra.
In Sections 3 and 4 we will define functors from SC 1 to the categories of sl(2) and sl(3) foams. These functors are grading preserving, so they obviously extend uniquely to SC 2 . By the universality of the Karoubi envelope, they also extend uniquely to functors between the respective Karoubi envelopes.
3. THE sl(2) CASE 3.1. Clark-Morrison-Walker's category of disoriented sl(2) foams. In this subsection we review the category Foam 2 of sl(2) foams following Clark, Morrison and Walker's construction in [2]. This category was introduced in [2] to modify Khovanov's link homology theory making it properly functorial with respect to link cobordisms. Actually we will use the version with dots of Clark-Morrison-Walker's original construction in [2]. Recall that we obtain one from the other by replacing each dot by 1/2 times a handle.
A disoriented arc is an arc composed by oriented segments with oppositely oriented segments separated by a mark pointing to one of these segments. A disoriented diagram consists of a collection D of disoriented arcs in the strip in R 2 bounded by the lines y = 0, 1 containing the boundary points of D. We allow diagrams containing oriented and disoriented circles. Disoriented diagrams can be composed vertically which endows Foam 2 with a monoidal structure on objects. For example, the diagrams 1 n and u j for (1 < j < n) are disoriented diagrams: A disoriented cobordism between disoriented diagrams is a 2d cobordism which can be decorated with dots and with seams separating differently oriented regions and such that the vertical boundary of each cobordism is a set (possibly empty) of vertical lines. Disorientation seams can have one out of two possible orientations which we identify with a fringe. We read cobordisms from bottom to top. For example, · · · · · · 1 j n is a disoriented cobordism from 1 n to u j .
Cobordism composition consists of placing one cobordism on top of the other and the monoidal structure is given by vertical composition which corresponds to placing one cobordism behind the other in our pictures. Let C[t] be the ring of polynomials in t with coefficients in C.
Definition 3.1. The category Foam 2 is the category whose objects are disoriented diagrams, and whose morphisms are C[t]-linear combinations of isotopy classes of disoriented cobordisms, modulo some relations, where i is the imaginary unit, • and the Bar-Natan (BN) relations, which are only valid away from the disorientations.
The universal theory for the original Khovanov homology contains another parameter h, but we have to put h = 0 in the Clark-Morrison-Walker's cobordism theory over a field of characteristic zero. For suppose we have a cylinder with a transversal disoriented circle. Applying (24) on one side of the disorientation circle followed by the disoriented relation (19) gives a cobordism that is independent of the side chosen to apply (24) only if h = 0 over a field of characteristic zero.
Define a q-grading on C[t] by q(1) = 0 and q(t) = 4. We introduce a q-grading on Foam 2 as follows. Let f be a cobordism with |•| dots and |b| vertical boundary components. The q-grading of f is given by where χ is the Euler characteristic. For example the degree of a saddle is 1 while the degree of a cap or a cup is −1. The category Foam 2 is additive and monoidal. More details about Foam 2 can be found in [2].
3.2. The functor F sl(2),n . In this subsection we define a monoidal functor F sl(2),n between the categories SC and Foam 2 .
On objects: F sl(2),n sends the empty sequence to 1 n and the one-term sequence (j) to u j with F sl(2),n (jk) given by the vertical composite u j u k .
On morphisms: • The empty diagram is sent to n parallel vertical sheets: • The vertical line colored j is sent to the identity cobordism of u j : The remaining n − 2 vertical parallel sheets on the r.h.s. are not shown for simplicity, a convention that we will follow from now on. • The StartDot and EndDot morphisms are sent to saddle cobordisms: The case j > k + 1 is given by reflection in a horizontal plane.
• The 6-valent vertices are sent to zero: Notice that F sl(2),n respects the gradings of the morphisms. Taking the quotient of SC by the 6-valent vertex gives a diagrammatic category T L categorifying the Temperley-Lieb algebra. According to [3] relations (12) and (13) can be replaced by a single relation in T L. The functor F sl (2),n descends to a functor between T L and Foam 2 . (2),n is a monoidal functor.
Proof. The assignment given by F sl (2),n clearly respects the monoidal structures of SC 1 and Foam 2 . So we only need to show that F sl(2),n is a functor, i.e. it respects the relations (1) to (18) of Section 2.
"Isotopy relations": Relations (1) to (4) are straightforward to check and correspond to isotopies of their images under F sl(2),n which respect the disorientations. Relation (5) is automatic since F sl (2),n send all terms to zero. For the sake of completeness we show the first equality in (1). We have F sl(2),n j j j j + 1 j j + 1 = ∼ = = F sl(2),n One color relations: For relation (6) we have where the first equivalence follows from relations (1) and (3) and the second from isotopy of the cobordisms involved. For relation (7) we have Relation (8) requires some more work. We have F sl(2),n j j j j + 1 j j + 1 where the second equality follows from the disoriented relation (20) and the third from the BN relation (24). We also have We thus have that F sl(2),n + F sl(2),n = 2 F sl(2),n .
Two distant colors: Relations (9) to (11) correspond to isotopies of the cobordisms involved and are straightforward to check.

Adjacent colors:
We prove the case where "blue" corresponds to j and "red" corresponds to j + 1. The relations with colors reversed are proved the same way. To prove relation (12) we first notice that This implies that 0 = F sl(2),n = F sl(2),n + F sl(2),n .
We now prove relation (13). We have isotopy equivalences Therefore we see that The functor F sl(2),n sends both sides of relation (14) to zero and so there is nothing to prove here. To prove relation (15) we start with the equivalence F sl(2),n j j + 1 j + 2 which is a consequence of the neck cutting relation (24) and the disorientation relations (19) and (21). We also have F sl(2),n j j + 1 j + 2 j j + 1 j + 2 Comparing with Equations (26) and (27) and using the disoriented relation (21) we get Relations involving three colors: Functor F sl (2),n sends to zero both sides of relations (16) and (18). Relation (17) follows from isotopies of the cobordisms involved.

The category Foam 3 of sl(3) foams.
In this subsection we review the category Foam 3 of sl(3)-foams introduced by the author and Mackaay in [8]. This category was introduced to universally deform Khovanov's construction in [7] leading to the sl(3)-link homology theory. We follow the conventions and notation of [8]. Recall that a web is a trivalent planar graph, where near each vertex all edges are oriented away from it or all edges are oriented towards it. We also allow webs without vertices, which are oriented loops. A pre-foam is a cobordism with singular arcs between two webs. A singular arc in a pre-foam f is the set of points of f which have a neighborhood homeomorphic to the letter Y times an interval. Singular arcs are disjoint. Interpreted as morphisms, we read pre-foams from bottom to top by convention, foam composition consists of placing one pre-foam on top of the other. The orientation of the singular arcs is by convention as in the zip and the unzip, and , respectively. Pre-foams can have dots which can move freely on the facet to which they belong but are not allowed to cross singular arcs. A foam is an isotopy class of pre-foams. Let C[a, b, c] be the ring of polynomials in a, b, c with coefficients in C.
We impose the set of relations ℓ = (3D, CN, S, Θ) on foams, as well as the closure relation, which are explained below. 1, 2, 0) or a cyclic permutation −1 (α, β, δ) = (2, 1, 0) or a cyclic permutation 0 else The closure relation says that any C[a, b, c]-linear combination of foams, all of which have the same boundary, is equal to zero if and only if any common way of closing these foams yields a C[a, b, c]-linear combination of closed foams whose evaluation is zero.
Using the relations ℓ one can prove the identities below (for detailed proofs see [8]).
In this paper we will work with open webs and open foams.
Definition 4.1. Foam 3 is the category whose objects are webs Γ inside a horizontal strip in R 2 bounded by the lines y = 0, 1 containing the boundary points of Γ and whose morphisms are C[a, b, c]-linear combinations of foams inside that strip times the unit interval such that the vertical boundary of each foam is a set (possibly empty) of vertical lines.
For example, the diagrams 1 n and v j are objects of Foam 3 : The category Foam 3 is additive and monoidal, with the monoidal structure given as in Foam 2 . The category Foam 3 is also additive and graded. The q-grading in C[a, b, c] is defined as q(1) = 0, q(a) = 2, q(b) = 4, q(c) = 6 and the degree of a foam f with | • | dots and |b| vertical boundary components is given by where χ denotes the Euler characteristic and ∂f is the boundary of f .

4.2.
The functor F sl(3),n . In this subsection we define a monoidal functor F sl(3),n between the categories SC and Foam 3 .
On objects: F sl(3),n sends the empty sequence to 1 n and the one-term sequence (j) to v j with F sl(3),n (jk) given by the vertical composite v j v k .

On morphisms:
• As before the empty diagram is sent to n parallel vertical sheets: The case j > k + 1 is given by reflection around a horizontal plane.
• For the 6-valent vertex we have: The case with the colors switched is given by reflection in a vertical plane. Notice that F sl(3),n respects the gradings of the morphisms. Proof. The assignement given by F sl (3),n clearly respects the monoidal structures of SC 1 and Foam 3 . To prove it is a monoidal functor we need only to show that it is actually a functor, that is, it respects the relations (1) to (18) of Section 2.
"Isotopy relations": Relations (1) to (5) correspond to isotopies of their images under F sl(3),n and we leave its check to the reader.
One color relations: Relation (6) is straightforward and left to the reader. For relation (7) we have the last equality following from the (Bubble) relation. For relation (8) we have where the second equality follows from the (DR) relation. We also have which is given by (RD). Using (Dot Migration) one obtain F sl(3),n j j j j + 1 j j + 1 j j + 1 and therefore, we have that Two distant colors: Relations (9) to (11) correspond to isotopies of the foams involved and are straightforward to check.
Adjacent colors: We prove the case where "blue" corresponds to j and "red" corresponds to j + 1. The relations with colors reversed are proved the same way. To prove relation (12) we first notice that We also have an isotopy equivalence The common boundary of these two foams contains two squares. Putting (SqR) on the square on the right glued with the identity foam everywhere else gives two terms, one isotopic to F sl(3),n and the other isotopic to F sl(3),n . We now prove relation (13). We have Applying (SqR) to the middle square we obtain two terms. One is isotopic to −F sl (3),n and the other gives F sl (3),n after using the (Bamboo) relation. We now prove relation (14) in the form = .
The images of the l.h.s. and r.h.s. under F sl(3),n are isotopic to j j + 1 j + 2 and j j + 1 j + 2 respectively, and both give the same foam after applying the (Bamboo) relation. Relation (15) follows from a straightforward computation and is left to the reader. (16) and (17) follow from isotopies of the cobordisms involved. We prove relation (18) in the form (30) = .

Relations involving three colors: Relations
We claim that F sl(3),n sends both sides to zero. Since the images of both sides of (30) can be obtained from each other using a symmetry relative to a vertical plane placed between the sheets labelled j + 1 and j + 2 it suffices to show that one side of (30) is sent to zero. The foams involved are rather complicated and hard to visualize. To make the computations easier we use movies (two dimensional diagrams) for the whole foam and implicitly translate some bits to three dimensional foams to apply isotopy equivalences or relations from Subsection 4.1. The r.h.s. corresponds to The foam f 2 is isotopic to .
Using this we can also see that the foams corresponding with and are isotopic. We see that the foam we have contains , which corresponds to a foam containing , which is zero by the (Bubble) relation.