# The Diagrammatic Soergel Category andsl(2) and sl(3) Foams

Authors: Pedro Vaz
Journal: International Journal of Mathematics and Mathematical Sciences
 (2010)

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## Abstract

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


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## Body

## 1. Introduction

In this paper we define functors between the Elias-Khovanov diagrammatic version of the Soergel category𝒮𝒞 defined in [1] and the categories of universal sl(2) and sl(3) foams defined in [2, 3].The Soergel category provides a categorification of the Hecke algebra and was used by Khovanov in [4] to construct a triply-graded link homology categorifying the HOMFLYPT polynomial. Elias and Khovanov constructed in [1] a category defined diagrammatically by generators and relations and showed it to be equivalent to 𝒮𝒞.Thesl(2) and sl(3) foams were introduced in [2, 5] and in [3, 6], respectively, to give topological constructions of the sl(2) and sl(3) link homologies.This paper can be seen as a first step towards the construction of a family of functors between𝒮𝒞 and the categories of sl(N)-foams for all N∈ℤ+, to be completed in a subsequent paper [7]. The functors ℱsl(2),n and ℱsl(3),n are not faithful. In [7] 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 𝒮𝒞1 we have ℱsl(N),n(f)=0 for all N, then f=0. With these functors one can try to give a graphical interpretation of Rasmussen's [8] spectral sequences from the HOMFLYPT link homology to the sl(N)-link homologies.The plan of the paper is as follows. In Section2 we give a brief description of Elias and Khovanov's diagrammatic Soergel category. In Section 3 we describe the category Foam2 of sl(2) foams and construct a functor from 𝒮𝒞 to Foam2. 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 [1–3, 9] is desirable.


## 2. The Diagrammatic Soergel Category Revisited

This section is a reminder of the diagrammatics for Soergel categories introduced by Elias and Khovanov in [1]. Actually we give the version which they explained in [1, Section 4.5] and which can be found in detail in [9].Fix a positive integern. The category 𝒮𝒞1 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 𝒮𝒞1 are given by generators modulo relations. A morphism of 𝒮𝒞1 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). (i)
Generators involving only one color are as follows:(2.1)It is useful to define the cap and cup as(2.2)(ii)
Generators involving two colors are as follows:- The 4-valent vertex, with distant colors,(2.3)- and the 6-valent vertex, with adjacent colorsi and j(2.4)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 endows𝒮𝒞1 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.(2.5)(2.6)(2.7)(2.8)(2.9)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 in𝒮𝒞1 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.(2.10)(2.11)(2.12)Two Distant Colors.(2.13)(2.14)(2.15)Two Adjacent Colors.(2.16)(2.17)(2.18)(2.19)Relations Involving Three Colors: (Adjacency is determined by the vertices which appear)(2.20)(2.21)(2.22)Introduce aq-grading on 𝒮𝒞1 declaring that dots have degree 1, trivalent vertices have degree -1 and 4-, and 6-valent vertices have degree 0.Definition 2.1.
The category𝒮𝒞2 is the category containing all direct sums and grading shifts of objects in 𝒮𝒞1 and whose morphisms are the grading-preserving morphisms from 𝒮𝒞1.Definition 2.2.
The category𝒮𝒞 is the Karoubi envelope of the category 𝒮𝒞2.Elias and Khovanov's main result in [1] is the following theorem.Theorem 2.3 (Elias-Khovanov).
The category𝒮𝒞 is equivalent to the Soergel category in [10].From Soergel's results from [10] we have the following corollary.Corollary 2.4.
The Grothendieck algebra of𝒮𝒞 is isomorphic to the Hecke algebra.Notice that𝒮𝒞 is an additive category but not abelian and we are using the (additive) split Grothendieck algebra.In Sections3 and 4 we will define functors from 𝒮𝒞1 to the categories of sl(2) and sl(3) foams. These functors are grading preserving, so they obviously extend uniquely to 𝒮𝒞2. By the universality of the Karoubi envelope, they also extend uniquely to functors between the respective Karoubi envelopes.


## 3. Thesl(2) Case

### 3.1. Clark-Morrison-Walker's Category of Disorientedsl(2) Foams

In this subsection we review the categoryFoam2 of sl(2) foams following Clark et al. 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.Adisoriented 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 ℝ2 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 Foam2 with a monoidal structure on objects. For example, the diagrams 1n and uj for (1<j<n) are disoriented diagrams: (3.1)Adisoriented 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, (3.2)is a disoriented cobordism from1n to uj.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. Letℂ[t] be the ring of polynomials in t with coefficients in ℂ.Definition 3.1.
The categoryFoam2 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:(i)
the disorientation relations(3.3)(3.4)(3.5)
where i is the imaginary unit,(ii)
and the Bar-Natan (BN) relations(3.6)(3.7)(3.8)
which are only valid away from the disorientations.The universal theory for the original Khovanov homology contains another parameterh, but we have to put h=0 in the Clark-Morrison-Walker's cobordism theory over a field of characteristic zero. Suppose that we have a cylinder with a transversal disoriented circle. Applying (3.8) on one side of the disorientation circle followed by the disoriented relation (3.3) gives a cobordism that is independent of the side chosen to apply (3.8) only if h=0 over a field of characteristic zero.Define aq-grading on ℂ[t] by q(1)=0 and q(t)=4. We introduce a q-grading on Foam2 as follows. Let f be a cobordism with |•| dots and |b| vertical boundary components. The q-grading of f is given by(3.9)q(f)=-χ(f)+2|•|+12|b|,
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 Foam2 is additive and monoidal. More details about Foam2 can be found in [2].


### 3.2. The Functorℱsl(2),n

In this subsection we define a monoidal functorℱsl(2),n between the categories 𝒮𝒞 and Foam2.On Objects.ℱsl(2),n sends the empty sequence to 1n and the one-term sequence (j) to uj with ℱsl(2),n(jk) given by the vertical composite ujuk.On Morphisms
(i)
The empty diagram is sent ton parallel vertical sheets:(3.10)(ii)
The vertical line coloredj is sent to the identity cobordism of uj:
(3.11)
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.(iii)
TheStartDot and EndDot morphisms are sent to saddle cobordisms:(3.12)(iv)
Merge and Split are sent to cup and cap cobordisms:(3.13)(v)
The4-valent vertex with distant colors is given as follows. For j+1<k we have
(3.14)
The case j>k+1 is given by reflection in a horizontal plane.(vi)
The6-valent vertices are sent to zero:(3.15)Notice thatℱsl(2),n respects the gradings of the morphisms. Taking the quotient of 𝒮𝒞 by the 6-valent vertex gives a diagrammatic category 𝒯ℒ categorifying the Temperley-Lieb algebra. According to [11] relations (2.16) and (2.17) can be replaced by a single relation in 𝒯ℒ. The functor ℱsl(2),n descends to a functor between 𝒯ℒ and Foam2.Proposition 3.2.
ℱsl(2),n is a monoidal functor.Proof.
The assignment given byℱsl(2),n clearly respects the monoidal structures of 𝒮𝒞1 and Foam2. So we only need to show that ℱsl(2),n is a functor, that is, it respects the relations (2.5) to (2.22) of Section 2.
”Isotopy Relations”
Relations (2.5) to (2.8) are straightforward to check and correspond to isotopies of their images under ℱsl(2),n which respect the disorientations. Relation (2.9) is automatic since ℱsl(2),n sends all terms to zero. For the sake of completeness we show the first equality in (2.5). We have
(3.16)One Color Relations
For relation (2.10) we have(3.17)
where the first equivalence follows from relations (2.5) and (2.7) and the second from isotopy of the cobordisms involved.
For relation (2.11) we have
(3.18)
Relation (2.12) requires some more work. We have
(3.19)
where the second equality follows from the disoriented relation (3.4) and the third follows from the BN relation (3.8). We also have
(3.20)
and therefore
(3.21)(3.22)
We thus have that
(3.23)Two Distant Colors
Relations (2.13) to (2.15) correspond to isotopies of the cobordisms involved and are straightforward to check.
Adjacent Colors
We prove the case where “blue’’ corresponds toj and ”red“ corresponds to j+1. The relations with colors reversed are proved the same way. To prove relation (2.16) we first notice that(3.24)
which means that
(3.25)
On the other side we have(3.26)
which, using isotopies and the disorientation relation (3.4) twice, can be seen to be equivalent to
(3.27)
which equals
(3.28)
This implies that(3.29)
We now prove relation (2.17). We have isotopy equivalences
(3.30)(3.31)
Therefore we see that
(3.32)
The functorℱsl(2),n sends both sides of relation (2.18) to zero and so there is nothing to prove here. To prove relation (2.19) we start with the equivalence
(3.33)
which is a consequence of the neck-cutting relation (3.8) and the disorientation relations (3.3) and (3.5). We also have
(3.34)
Comparing with (3.21) and (3.22) and using the disoriented relation (3.5), we get
(3.35)Relations Involving Three Colors
Functorℱsl(2),n sends to zero both sides of relations (2.20) and (2.22). Relation (2.21) follows from isotopies of the cobordisms involved.


## 3.1. Clark-Morrison-Walker's Category of Disorientedsl(2) Foams

In this subsection we review the categoryFoam2 of sl(2) foams following Clark et al. 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.Adisoriented 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 ℝ2 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 Foam2 with a monoidal structure on objects. For example, the diagrams 1n and uj for (1<j<n) are disoriented diagrams: (3.1)Adisoriented 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, (3.2)is a disoriented cobordism from1n to uj.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. Letℂ[t] be the ring of polynomials in t with coefficients in ℂ.Definition 3.1.
The categoryFoam2 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:(i)
the disorientation relations(3.3)(3.4)(3.5)
where i is the imaginary unit,(ii)
and the Bar-Natan (BN) relations(3.6)(3.7)(3.8)
which are only valid away from the disorientations.The universal theory for the original Khovanov homology contains another parameterh, but we have to put h=0 in the Clark-Morrison-Walker's cobordism theory over a field of characteristic zero. Suppose that we have a cylinder with a transversal disoriented circle. Applying (3.8) on one side of the disorientation circle followed by the disoriented relation (3.3) gives a cobordism that is independent of the side chosen to apply (3.8) only if h=0 over a field of characteristic zero.Define aq-grading on ℂ[t] by q(1)=0 and q(t)=4. We introduce a q-grading on Foam2 as follows. Let f be a cobordism with |•| dots and |b| vertical boundary components. The q-grading of f is given by(3.9)q(f)=-χ(f)+2|•|+12|b|,
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 Foam2 is additive and monoidal. More details about Foam2 can be found in [2].


## 3.2. The Functorℱsl(2),n

In this subsection we define a monoidal functorℱsl(2),n between the categories 𝒮𝒞 and Foam2.On Objects.ℱsl(2),n sends the empty sequence to 1n and the one-term sequence (j) to uj with ℱsl(2),n(jk) given by the vertical composite ujuk.On Morphisms
(i)
The empty diagram is sent ton parallel vertical sheets:(3.10)(ii)
The vertical line coloredj is sent to the identity cobordism of uj:
(3.11)
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.(iii)
TheStartDot and EndDot morphisms are sent to saddle cobordisms:(3.12)(iv)
Merge and Split are sent to cup and cap cobordisms:(3.13)(v)
The4-valent vertex with distant colors is given as follows. For j+1<k we have
(3.14)
The case j>k+1 is given by reflection in a horizontal plane.(vi)
The6-valent vertices are sent to zero:(3.15)Notice thatℱsl(2),n respects the gradings of the morphisms. Taking the quotient of 𝒮𝒞 by the 6-valent vertex gives a diagrammatic category 𝒯ℒ categorifying the Temperley-Lieb algebra. According to [11] relations (2.16) and (2.17) can be replaced by a single relation in 𝒯ℒ. The functor ℱsl(2),n descends to a functor between 𝒯ℒ and Foam2.Proposition 3.2.
ℱsl(2),n is a monoidal functor.Proof.
The assignment given byℱsl(2),n clearly respects the monoidal structures of 𝒮𝒞1 and Foam2. So we only need to show that ℱsl(2),n is a functor, that is, it respects the relations (2.5) to (2.22) of Section 2.
”Isotopy Relations”
Relations (2.5) to (2.8) are straightforward to check and correspond to isotopies of their images under ℱsl(2),n which respect the disorientations. Relation (2.9) is automatic since ℱsl(2),n sends all terms to zero. For the sake of completeness we show the first equality in (2.5). We have
(3.16)One Color Relations
For relation (2.10) we have(3.17)
where the first equivalence follows from relations (2.5) and (2.7) and the second from isotopy of the cobordisms involved.
For relation (2.11) we have
(3.18)
Relation (2.12) requires some more work. We have
(3.19)
where the second equality follows from the disoriented relation (3.4) and the third follows from the BN relation (3.8). We also have
(3.20)
and therefore
(3.21)(3.22)
We thus have that
(3.23)Two Distant Colors
Relations (2.13) to (2.15) correspond to isotopies of the cobordisms involved and are straightforward to check.
Adjacent Colors
We prove the case where “blue’’ corresponds toj and ”red“ corresponds to j+1. The relations with colors reversed are proved the same way. To prove relation (2.16) we first notice that(3.24)
which means that
(3.25)
On the other side we have(3.26)
which, using isotopies and the disorientation relation (3.4) twice, can be seen to be equivalent to
(3.27)
which equals
(3.28)
This implies that(3.29)
We now prove relation (2.17). We have isotopy equivalences
(3.30)(3.31)
Therefore we see that
(3.32)
The functorℱsl(2),n sends both sides of relation (2.18) to zero and so there is nothing to prove here. To prove relation (2.19) we start with the equivalence
(3.33)
which is a consequence of the neck-cutting relation (3.8) and the disorientation relations (3.3) and (3.5). We also have
(3.34)
Comparing with (3.21) and (3.22) and using the disoriented relation (3.5), we get
(3.35)Relations Involving Three Colors
Functorℱsl(2),n sends to zero both sides of relations (2.20) and (2.22). Relation (2.21) follows from isotopies of the cobordisms involved.


## 4. Thesl(3) Case

### 4.1. The Category Foam3 of sl(3) Foams

In this subsection we review the categoryFoam3 of sl(3) foams introduced by the author and Mackaay in [3]. This category was introduced to universally deform Khovanov's construction in [6] leading to the sl(3)-link homology theory.We follow the conventions and notation of [3]. 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 prefoam f is the set of points of f which has a neighborhood homeomorphic to the letter Y times an interval. Singular arcs are disjoint. Interpreted as morphisms, we read prefoams from bottom to top by convention; foam composition consists of placing one prefoam on top of the other. The orientation of the singular arcs is by convention as in the zip and the unzip: (4.1)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. Afoam 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 relationsℓ=(3D,CN,S,Θ) on foams, as well as the closure relation, which are explained below. (3D)(CN)(S)(Θ)Theclosure relation says that any ℂ[a,b,c]-linear combination of foams, all of which having 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 relationsℓ, one can prove the identities below (for detailed proofs see [3]).(Bamboo)(RD)(Bubble)(DR)(SqR)(Dot Migration)In this paper we will work with open webs and open foams.Definition 4.1.
Foam3 is the category whose objects are webs Γ inside a horizontal strip in ℝ2 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 diagrams1n and vj are objects of Foam3: (4.2)The categoryFoam3 is additive and monoidal, with the monoidal structure given as in Foam2. The category Foam3 is also additive and graded. The q-grading in ℂ[a,b,c] is defined as (4.3)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 (4.4)q(f)=-2χ(f)+χ(∂f)+2|•|+|b|,
where χ denotes the Euler characteristic and ∂f is the boundary of f.


### 4.2. The Functorℱsl(3),n

In this subsection we define a monoidal functorℱsl(3),n between the categories 𝒮𝒞 and Foam3.On Objects
ℱsl(3),n sends the empty sequence to 1n and the one-term sequence (j) to vj with ℱsl(3),n(jk) given by the vertical composite vjvk.On Morphisms
(i)
As before the empty diagram is sent ton parallel vertical sheets:(4.5)(ii)
The vertical line coloredj is sent to the identity foam of vj:(4.6)(iii)
TheStartDot and EndDot morphisms are sent to the zip and the unzip, respectively:(4.7)(iv)
Merge and Split are sent to the digon annihilation and creation, respectively:(4.8)(v)
The4-valent vertex with distant colors is showen as follows. For j+1<k we have.(4.9)
The case j>k+1 is given by reflection around a horizontal plane.(vi)
For the6-valent vertex we have(4.10)The case with the colors switched is given by reflection in a vertical plane.Notice thatℱsl(3),n respects the gradings of the morphisms.Proposition 4.2.
ℱsl(3),n is a monoidal functor.Proof.
The assignment given byℱsl(3),n clearly respects the monoidal structures of 𝒮𝒞1 and Foam3. To prove that it is a monoidal functor we need only to show that it is actually a functor, that is, it respects relations (2.5) to (2.22) of Section 2.Isotopy Relations
Relations (2.5) to (2.9) correspond to isotopies of their images under ℱsl(3),n, and we leave its check to the reader.One-Color Relations
Relation (2.10) is straightforward and left to the reader. For relation (2.11) we have(4.11)
the last equality following from the (Bubble) relation.
For relation (2.12) we have(4.12)
where the second equality follows from the (DR) relation. We also have
(4.13)
which is given by (RD). Using (Dot Migration) one obtains
(4.14)(4.15)
and therefore, we have that
(4.16)Two Distant Colors
Relations (2.13) to (2.15) correspond to isotopies of the foams involved and are straightforward to check.Adjacent Colors
We prove the case where ”blue“ corresponds toj and ”red“ corresponds to j+1. The relations with colors reversed are proved the same way. To prove relation (2.16) we first notice that(4.17)(4.18)
We also have an isotopy equivalence
(4.19)
which in turn is isotopy equivalent to the foam obtained by putting
(4.20)
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 ℱsl(3),n and the other isotopic to ℱsl(3),n.
We now prove relation (2.17). We have(4.21)
Applying (SqR) to the middle square we obtain two terms. One is isotopic to -ℱsl(3),n and the other gives ℱsl(3),n after using the (Bamboo) relation.
We now prove relation (2.18) in the form(4.22)The images of the l.h.s. and r.h.s. under ℱsl(3),n are isotopic to
(4.23)
respectively, and both give the same foam after applying the (Bamboo) relation.
Relation (2.19) follows from a straightforward computation and is left to the reader.
Relations Involving Three Colors
Relations (2.20) and (2.21) follow from isotopies of the cobordisms involved.
We prove relation (2.22) in the form(4.24)
We claim that ℱsl(3),n sends both sides to zero. Since the images of both sides of (4.24) 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 (4.24) 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 Section 4.1. The r.h.s. corresponds to
(4.25)
followed by
(4.26)
The foamf2 is isotopic to(4.27)
Using this, we can also see that the foams corresponding with(4.28)(4.29)
are isotopic. We see that the foam we have contains
(4.30)
which corresponds to a foam containing , which is zero by the (Bubble) relation.


## 4.1. The Category Foam3 of sl(3) Foams

In this subsection we review the categoryFoam3 of sl(3) foams introduced by the author and Mackaay in [3]. This category was introduced to universally deform Khovanov's construction in [6] leading to the sl(3)-link homology theory.We follow the conventions and notation of [3]. 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 prefoam f is the set of points of f which has a neighborhood homeomorphic to the letter Y times an interval. Singular arcs are disjoint. Interpreted as morphisms, we read prefoams from bottom to top by convention; foam composition consists of placing one prefoam on top of the other. The orientation of the singular arcs is by convention as in the zip and the unzip: (4.1)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. Afoam 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 relationsℓ=(3D,CN,S,Θ) on foams, as well as the closure relation, which are explained below. (3D)(CN)(S)(Θ)Theclosure relation says that any ℂ[a,b,c]-linear combination of foams, all of which having 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 relationsℓ, one can prove the identities below (for detailed proofs see [3]).(Bamboo)(RD)(Bubble)(DR)(SqR)(Dot Migration)In this paper we will work with open webs and open foams.Definition 4.1.
Foam3 is the category whose objects are webs Γ inside a horizontal strip in ℝ2 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 diagrams1n and vj are objects of Foam3: (4.2)The categoryFoam3 is additive and monoidal, with the monoidal structure given as in Foam2. The category Foam3 is also additive and graded. The q-grading in ℂ[a,b,c] is defined as (4.3)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 (4.4)q(f)=-2χ(f)+χ(∂f)+2|•|+|b|,
where χ denotes the Euler characteristic and ∂f is the boundary of f.


## 4.2. The Functorℱsl(3),n

In this subsection we define a monoidal functorℱsl(3),n between the categories 𝒮𝒞 and Foam3.On Objects
ℱsl(3),n sends the empty sequence to 1n and the one-term sequence (j) to vj with ℱsl(3),n(jk) given by the vertical composite vjvk.On Morphisms
(i)
As before the empty diagram is sent ton parallel vertical sheets:(4.5)(ii)
The vertical line coloredj is sent to the identity foam of vj:(4.6)(iii)
TheStartDot and EndDot morphisms are sent to the zip and the unzip, respectively:(4.7)(iv)
Merge and Split are sent to the digon annihilation and creation, respectively:(4.8)(v)
The4-valent vertex with distant colors is showen as follows. For j+1<k we have.(4.9)
The case j>k+1 is given by reflection around a horizontal plane.(vi)
For the6-valent vertex we have(4.10)The case with the colors switched is given by reflection in a vertical plane.Notice thatℱsl(3),n respects the gradings of the morphisms.Proposition 4.2.
ℱsl(3),n is a monoidal functor.Proof.
The assignment given byℱsl(3),n clearly respects the monoidal structures of 𝒮𝒞1 and Foam3. To prove that it is a monoidal functor we need only to show that it is actually a functor, that is, it respects relations (2.5) to (2.22) of Section 2.Isotopy Relations
Relations (2.5) to (2.9) correspond to isotopies of their images under ℱsl(3),n, and we leave its check to the reader.One-Color Relations
Relation (2.10) is straightforward and left to the reader. For relation (2.11) we have(4.11)
the last equality following from the (Bubble) relation.
For relation (2.12) we have(4.12)
where the second equality follows from the (DR) relation. We also have
(4.13)
which is given by (RD). Using (Dot Migration) one obtains
(4.14)(4.15)
and therefore, we have that
(4.16)Two Distant Colors
Relations (2.13) to (2.15) correspond to isotopies of the foams involved and are straightforward to check.Adjacent Colors
We prove the case where ”blue“ corresponds toj and ”red“ corresponds to j+1. The relations with colors reversed are proved the same way. To prove relation (2.16) we first notice that(4.17)(4.18)
We also have an isotopy equivalence
(4.19)
which in turn is isotopy equivalent to the foam obtained by putting
(4.20)
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 ℱsl(3),n and the other isotopic to ℱsl(3),n.
We now prove relation (2.17). We have(4.21)
Applying (SqR) to the middle square we obtain two terms. One is isotopic to -ℱsl(3),n and the other gives ℱsl(3),n after using the (Bamboo) relation.
We now prove relation (2.18) in the form(4.22)The images of the l.h.s. and r.h.s. under ℱsl(3),n are isotopic to
(4.23)
respectively, and both give the same foam after applying the (Bamboo) relation.
Relation (2.19) follows from a straightforward computation and is left to the reader.
Relations Involving Three Colors
Relations (2.20) and (2.21) follow from isotopies of the cobordisms involved.
We prove relation (2.22) in the form(4.24)
We claim that ℱsl(3),n sends both sides to zero. Since the images of both sides of (4.24) 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 (4.24) 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 Section 4.1. The r.h.s. corresponds to
(4.25)
followed by
(4.26)
The foamf2 is isotopic to(4.27)
Using this, we can also see that the foams corresponding with(4.28)(4.29)
are isotopic. We see that the foam we have contains
(4.30)
which corresponds to a foam containing , which is zero by the (Bubble) relation.



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*Source: 612360-2010-06-10.xml*