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

Abstract

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

1. Introduction

In this paper we define functors between the Elias-Khovanov diagrammatic version of the Soergel category defined in [1] and the categories of universal and 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 .

The and foams were introduced in [2, 5] and in [3, 6], respectively, to give topological constructions of the and link homologies.

This paper can be seen as a first step towards the construction of a family of functors between and the categories of -foams for all , to be completed in a subsequent paper [7]. The functors and are not faithful. In [7] we will extend the construction of these functors to all . The whole family of functors is faithful in the following sense: if for a morphism in we have for all , then . With these functors one can try to give a graphical interpretation of Rasmussen’s [8] spectral sequences from the HOMFLYPT link homology to the -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 of foams and construct a functor from to . Finally in Section 4 we give the analogue of these results for the case of 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 [4, Section ] and which can be found in detail in [9].

Fix a positive integer . The category is the category whose objects are finite length sequences of points on the real line, where each point is colored by an integer between and . We read sequences of points from left to right. Two colors and are called adjacent if and distant if . The morphisms of are given by generators modulo relations. A morphism of is a -linear combination of planar diagrams constructed by horizontal and vertical gluings of the following generators (by convention no label means a generic color ). (i) Generators involving only one color are as follows:   It is useful to define the cap and cup as (ii) Generators involving two colors are as follows: - The 4-valent vertex, with distant colors, - and the 6-valent vertex, with adjacent colors and 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 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 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.

Two Adjacent Colors.

Relations Involving Three Colors: (Adjacency is determined by the vertices which appear)

Introduce a -grading on declaring that dots have degree , trivalent vertices have degree and -, and -valent vertices have degree .

Definition 2.1. The category is the category containing all direct sums and grading shifts of objects in and whose morphisms are the grading-preserving morphisms from .

Definition 2.2. The category is the Karoubi envelope of the category .

Elias and Khovanov’s main result in [1] is the following theorem.

Theorem 2.3 (Elias-Khovanov). The category 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 of is isomorphic to the Hecke algebra.

Notice that 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 to the categories of and foams. These functors are grading preserving, so they obviously extend uniquely to . By the universality of the Karoubi envelope, they also extend uniquely to functors between the respective Karoubi envelopes.

3. The Case

3.1. Clark-Morrison-Walker’s Category of Disoriented Foams

In this subsection we review the category of 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 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 of disoriented arcs in the strip in bounded by the lines containing the boundary points of . We allow diagrams containing oriented and disoriented circles. Disoriented diagrams can be composed vertically, which endows with a monoidal structure on objects. For example, the diagrams and for () are disoriented diagrams:

A disoriented cobordism between disoriented diagrams is a 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, is a disoriented cobordism from to .

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 be the ring of polynomials in with coefficients in .

Definition 3.1. The category is the category whose objects are disoriented diagrams, and whose morphisms are -linear combinations of isotopy classes of disoriented cobordisms, modulo some relations: (i) the disorientation relations where is the imaginary unit, (ii) 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 , but we have to put 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 on one side of the disorientation circle followed by the disoriented relation gives a cobordism that is independent of the side chosen to apply only if over a field of characteristic zero.

Define a -grading on by and . We introduce a -grading on as follows. Let be a cobordism with dots and vertical boundary components. The -grading of is given by where is the Euler characteristic. For example, the degree of a saddle is while the degree of a cap or a cup is . The category is additive and monoidal. More details about can be found in [2].

3.2. The Functor

In this subsection we define a monoidal functor between the categories and .

On Objects. sends the empty sequence to and the one-term sequence to with given by the vertical composite .

On Morphisms (i) The empty diagram is sent to parallel vertical sheets: (ii) The vertical line colored is sent to the identity cobordism of :   The remaining vertical parallel sheets on the r.h.s. are not shown for simplicity, a convention that we will follow from now on. (iii) The StartDot and EndDot morphisms are sent to saddle cobordisms: (iv) Merge and Split are sent to cup and cap cobordisms: (v) The 4-valent vertex with distant colors is given as follows. For we have   The case is given by reflection in a horizontal plane. (vi) The 6-valent vertices are sent to zero:

Notice that respects the gradings of the morphisms. Taking the quotient of by the 6-valent vertex gives a diagrammatic category categorifying the Temperley-Lieb algebra. According to [11] relations and can be replaced by a single relation in . The functor descends to a functor between and .

Proposition 3.2. is a monoidal functor.

Proof. The assignment given by clearly respects the monoidal structures of and . So we only need to show that is a functor, that is, it respects the relations to of Section 2.

“Isotopy Relations”

Relations to are straightforward to check and correspond to isotopies of their images under which respect the disorientations. Relation is automatic since sends all terms to zero. For the sake of completeness we show the first equality in . We have

One Color Relations

For relation we have where the first equivalence follows from relations and and the second from isotopy of the cobordisms involved.

For relation we have

Relation requires some more work. We have where the second equality follows from the disoriented relation and the third follows from the BN relation . We also have and therefore We thus have that

Two Distant Colors

Relations to correspond to isotopies of the cobordisms involved and are straightforward to check.

Adjacent Colors

We prove the case where “blue” corresponds to and “red” corresponds to . The relations with colors reversed are proved the same way. To prove relation we first notice that which means that

On the other side we have which, using isotopies and the disorientation relation twice, can be seen to be equivalent to which equals

This implies that

We now prove relation . We have isotopy equivalences Therefore we see that

The functor sends both sides of relation to zero and so there is nothing to prove here. To prove relation we start with the equivalence which is a consequence of the neck-cutting relation and the disorientation relations and . We also have

Comparing with and and using the disoriented relation , we get

Relations Involving Three Colors

Functor sends to zero both sides of relations and . Relation follows from isotopies of the cobordisms involved.

4. The Case

4.1. The Category Foam₃ of Foams

In this subsection we review the category of foams introduced by the author and Mackaay in [3]. This category was introduced to universally deform Khovanov’s construction in [6] leading to the -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 is the set of points of 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: 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 be the ring of polynomials in with coefficients in .

We impose the set of relations on foams, as well as the closure relation, which are explained below.

The closure relation says that any -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 -linear combination of closed foams whose evaluation is zero.

Using the relations , one can prove the identities below (for detailed proofs see [3]).

In this paper we will work with open webs and open foams.

Definition 4.1. is the category whose objects are webs inside a horizontal strip in bounded by the lines containing the boundary points of and whose morphisms are -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 and are objects of :

The category is additive and monoidal, with the monoidal structure given as in . The category is also additive and graded. The -grading in is defined as and the degree of a foam with dots and vertical boundary components is given by where denotes the Euler characteristic and is the boundary of .

4.2. The Functor

In this subsection we define a monoidal functor between the categories and .

On Objects

sends the empty sequence to and the one-term sequence to with given by the vertical composite .

On Morphisms (i) As before the empty diagram is sent to parallel vertical sheets: (ii) The vertical line colored is sent to the identity foam of : (iii) The StartDot and EndDot morphisms are sent to the zip and the unzip, respectively: (iv) Merge and Split are sent to the digon annihilation and creation, respectively: (v) The 4-valent vertex with distant colors is showen as follows. For we have.   The case is given by reflection around a horizontal plane. (vi) For the 6-valent vertex we have   The case with the colors switched is given by reflection in a vertical plane.   Notice that respects the gradings of the morphisms.

Proposition 4.2. is a monoidal functor.

Proof. The assignment given by clearly respects the monoidal structures of and . To prove that it is a monoidal functor we need only to show that it is actually a functor, that is, it respects relations to of Section 2.
Isotopy Relations
Relations to correspond to isotopies of their images under , and we leave its check to the reader.
One-Color Relations
Relation is straightforward and left to the reader. For relation we have the last equality following from the relation.
For relation we have where the second equality follows from the relation. We also have which is given by . Using one obtains and therefore, we have that Two Distant Colors
Relations to correspond to isotopies of the foams involved and are straightforward to check.
Adjacent Colors
We prove the case where “blue” corresponds to and “red” corresponds to . The relations with colors reversed are proved the same way. To prove relation we first notice that We also have an isotopy equivalence which in turn is isotopy equivalent to the foam obtained by putting The common boundary of these two foams contains two squares. Putting on the square on the right glued with the identity foam everywhere else gives two terms, one isotopic to and the other isotopic to .
We now prove relation . We have Applying to the middle square we obtain two terms. One is isotopic to and the other gives after using the relation.
We now prove relation in the form The images of the l.h.s. and r.h.s. under are isotopic to respectively, and both give the same foam after applying the relation.
Relation follows from a straightforward computation and is left to the reader.
Relations Involving Three Colors
Relations and follow from isotopies of the cobordisms involved.
We prove relation in the form We claim that sends both sides to zero. Since the images of both sides of can be obtained from each other using a symmetry relative to a vertical plane placed between the sheets labelled and , it suffices to show that one side of 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 followed by The foam is isotopic to Using this, we can also see that the foams corresponding with are isotopic. We see that the foam we have contains which corresponds to a foam containing , which is zero by the relation.

Acknowledgments

The author would like to thank Mikhail Khovanov and Ben Elias for valuable comments on a previous version of this paper. The author also thanks Ben Elias for sharing reference [11]. The author was financially supported by FCT (Portugal), postdoc Grant no. SFRH/BPD/46299/2008.