The decategorification of bordered Khovanov homology
In ,  the author showed how to decompose the Khovanov homology of a link into the algebraic pairing of a type D structure and a type A structure (as defined in bordered Floer homology), whenever a diagram for is decomposed into the union of two tangles. Since Khovanov homology is the categorification of a version of the Jones polynomial, it is natural to ask what the type A and type D structures categorify, and how their pairing is encoded in the decategorifications. In this paper, the author constructs the decategorifications of these two structures, in a manner similar to Ina Petkova’s decategorification of bordered Floer homology, , and shows how they recover the Jones polynomial. We also give a new proof of the mutation invariance of the Jones polynomial which uses these decomposition techniques.
1. Background and motivation
In , M. Khovanov describes a homology theory whose “Euler characteristic” is a reparametrization of the Jones polynomial of an oriented link in . In particular, he uses a link diagram for and defines a bigraded, free Abelian group whose generators are decorated resolutions of . The group admits a differential for which the homology is an invariant of the link .
Since the differential is we can decompose as a direct sum of chain complexes where is the subgroup of whose second grading, called the quantum grading, is equal to . The “Euler characteristic” of is taken to be the polynomial
where is the usual Euler characteristic of a finitely generated chain complex with free chain groups. This polynomial has the form , where is the number of positive/negative crossings in and can be computed from the unoriented link diagram. If we let be an unknot, and be a link with an unlinked and unknotted component , then satisfies
Thus is a version of the unnormalized Jones polynomial.
Due to this relationship, made more precise below, Khovanov homology is called a categorification of the Jones polynomial, and the polynomial is called the decategorification of Khovanov homology. These terms are used since the polynomial arises from a process more general than just taking Euler characteristics. Despite both the homology and its decategorification giving link invariants, the homology is known to be stronger than the Jones polynomial, and has many additional properties related to the cobordism on links.
In ,  the author, inspired by bordered Floer homology, describes a similar construction for tangles111There are several other constructions of Khovanov homology for tangles, , , , , . The formal algebraic structure of these tangle invariants mimics the formal structure of Ozsváth, Lipshitz, and Thurston’s description of bordered Floer homology, , which also provided a road map for constructing a gluing theory for Khovanov homology.
The construction in  takes a tangle diagram in a disc , with a marked point :
and associates to it a bigraded differential module over a differential bigraded algebra , where has endpoints on . Each differential is in the respective bigradings, and the action preserves the bigradings on its source and target. The homotopy type of as an -module is invariant under the three Reidemeister moves applied to swatches in the diagram , and thus defines a tangle invariant.
The main goal in defining these algebraic structures is to obtain a complete gluing theory for Khovanov homology. For instance, let be a link diagram in an oriented sphere , and , where each closed disk inherits its orientation from and is a circle in . We will orient as the boundary of . Suppose is transverse to away from its crossings. Then and are tangles in oriented disks. If we choose , then each disk also has a marked point in its boundary. We illustrate this situation in the diagram below, where we have taken to be the point at infinity. is then the left half plane, and is the right half plane.
To assign the bigraded differential module described above. To the construction in  assigns a bigraded Abelian group and a map , which satisfies the structure relation for a type D structure, .
There is an algebraic construction which pairs differential graded modules and type structures to obtain a normal chain complex. This can be adapted to the bigraded setting and applied to compute . The main result of  is that is chain homotopy equivalent to , the original link diagram. Furthermore, preserves chain homotopy equivalence when we alter either factor by a homotopy equivalence (in the respective categories of -modules and type structures), so this effects a gluing of the homotopy types of the invariants assigned to the tangles. As it is the homotopy types that are invariants of the tangles and links, and not the raw algebraic object, this provides a complete gluing theory for Khovanov homology.
Having defined and it is natural to ask if there is any analog of these objects, and their pairing, in the simpler world of the Jones polynomial. In this paper we provide the answer to this questions by describing the decategorifications of and . The next section elaborates on the notion of decategorification we will use. In section 3 the decategorification is described in the abstract. The section 4 we provide more detail about the algebras involved, which allows us to give a concrete description of the decategorification in section 5. This will all be described for the type A structure. In section 6 we show how to interpret these results for type D structures. Then we can show how to recover the Jones polynomial in section 7 and reprove its invariance under mutation in section 8.
The concrete description can be used to generalize this paper to other settings, and exhibit a kind of planar algebra structure, a topic the author pursues in a sequel, .
Convention: If is a bigraded module, then is the bigraded module with .
2. Background on decategorification
First, we provide a little more detail concerning the method. Above, we referred to the polynomial as a “Euler characteristic.” A better, and more sophisticated version of this statement can be obtained through the use of Grothendieck groups. For this paper, we will need a generalization of the following version of the Grothendieck construction, taken from :
Let be a bigraded associative algebra. The Grothendieck group is the -module generated by the elements where is a finitely generated, bigraded, projective -module, and subject to the relations that when there is a short exact sequence , and .
When is (in bigrading ) the relations above confirm that . Now is in gradings for . Thus, . Since is free, so is so the image of (in bigrading ) in the Grothendieck group is just the rank of . Thus,
from which it readily follows that .
This can be extended to the case where is a finitely generated, bigraded projective -module with a -differential. Following the pattern, the decategorifications of and should be the elements in a Grothendieck group for the differential bigraded algebra where is the number of unclosed components in .
Let be a bigraded differential graded algebra with differential. We will consider right differential graded modules over , since that is the structure of over . Following , we aim to define the Grothendieck group . We need some additional definitions.
is the triangulated category found from the category of (right) bigraded differential modules with differential by quotienting out by the morphisms homotopic to zero. is the derived category found by localizing at its quasi-isomorphisms.
Here it is understood that the homotopies, chain maps, etc. do not change the quantum (second) grading. The distinguished triangles in the triangulated structure are those diagrams triangle isomorphic to a standard triangle
where is a morphism of differential graded modules, and is the mapping cone of : the module equipped with the differential
A differential graded module is said to be projective if, given any (right) differential graded module with trivial homology, the complex , defined in , has trivial homology.
is the full subcategory of whose objects are the projective differential graded modules.
In Part II, section 10 of  a bar construction is described which takes any differential graded module and finds a quasi-isomorphic projective differential graded module . Thus, is equivalent to the derived category .
An object of is compact if the natural inclusion
is an isomorphism for any collection of objects .
is the full subcategory of whose objects are compact, projective modules over
We are now in a position to define .
Let be a bigraded differential graded algebra with differential. is the Grothendieck group of the category . More specifically, is the Abelian group with a generator for each compact, projective differential bigraded (right) module over , with differential, subject to the relations for each distinguished triangle
It follows from the definition that (from the distinguished triangle coming from the mapping cone of the identity on ), and that is a -module, where the action is . In particular, .
3. The Grothendieck group of
We apply this construction to . More details about this algebra will be given below. For this computation all that is required are the following properties, :
is the quotient of a quiver algebra defined by an acyclic directed graph ,
The relations defining this quotient consist of identities involving paths of length ,
The differential is non-trivial only on paths of length .
Let be the algebra of (orthogonal) idempotents in . We will now prove that
is isomorphic to the module spanned by the idempotents corresponding to vertices in .
Note that this is in keeping with the computation of Grothendieck groups for acyclic quiver algebras with relations, . We have switched to since the quantum grading on is half-integral, but no other changes are necessary to the above construction.
Proof: Let be a right bigraded differential module over . Since is projective and compact in we may use a homotopy equivalent representative of which is finitely generated. Since is acyclic and directed, there is a vertex which has no out edges. Let be the corresponding idempotent. now consider . This is a submodule of since the action of any element of arises as the image of the action of path elements in . As no path starts at , the only element which acts non-trivially on is . Furthermore, is a subcomplex of with its differential since
for any . Thus the image of any element in under will be in . Thus there is a distinguished triangle
in . Therefore, , so . Now is still a module over the differential graded algebra , but only the elements of will act non-trivially, where is the directed graph . Since is acyclic, it also has a vertex with no outward edges. Due to the orthogonality of the idempotents . Applying this reasoning repeatedly, and using that is finitely generated, we arrive at
Since the action of is essentially trivial on , we can consider to be a bigraded complex over with differential, just as above. Thus where is the module with a in bigrading , trivial differential, and . Thus we see that the module spanned by the idempotents of maps surjectively to .
Before continuing we note that this is precisely what happens for bound quiver algebras, , as they too have Jordan-Hölder sequences of this type.
We show that this map is an isomorphism by constructing a module for which for any collection of polynomials with integer coefficients. For each term we have a copy of if the sign is , and if the sign is . We take the direct sum over all such terms. The differential is taken to be trivial. We do this for all , and then define the action of to be trivial for every element that is the image of a path of of length , and let act nontrivially only on the copies of . It is clear that this cannot be simplified further, and has image in . Thus there are no relations among the elements . .
This provides a strategy for computing the image , which we will describe presently. First, we spend some times on the idempotent sub-algebra .
4. The Idempotent sub-algebra of
The vertices of the quiver defining correspond to certain planar configurations of circles and decorations, called cleaved links in  and . More specifically, let be an oriented two-dimensional sphere.
A cleaved link in consists of the following data
a smoothly embedded circle , called the equator for ,
a marked point
an identification of the closures of the two open discs as and , called the inside and outside discs, respectively, where each is oriented from ,
the orientation of induced as the boundary of , and
a (possibly empty) set of simple closed curves which each non-trivially and transversely intersect away from
For a cleaved link in an oriented sphere , is the ordered set whose elements are points of intersection between and equipped with the ordering inherited from the orientation of . The cardinality of is .
Note that will be ordered opposite the orientation of .
A cleaved link in a sphere is equivalent to another cleaved link in if there is an orientation preserving diffeomorphism which preserves each of the structures in the definition. In particular,
, and thus
maps each diffeomorphically to a circle
It follows that induces an order preserving bijection .
A decorated, cleaved link is a cleaved link and a map . The map is called the decoration.
Two decorated, cleaved links are equivalent if there is an equivalence of the undecorated cleaved links with .
The set of equivalence classes of decorated, cleaved links with will be denoted .
In there is an idempotent for each equivalence class of decorated, cleaved links. By theorem 8, we know that we should be interested in the modules spanned by these equivalence classes.
For each , is the free -module generated by the elements of . The generator corresponding to will be denoted .
Examples: When , . It is possible to include this in the framework above, by allowing to be empty. Then has a generator corresponding to the equivalence class for , oriented as the boundary of the unit ball, with , and . We take to be the upper hemisphere, since that endows with the same orientation it inherits from being the boundary of in . However, we do not obtain a different cleaved link by taking to be the lower hemisphere since is orientation preserving when restricted to , takes and to themselves, and carries the upper hemisphere to the lower hemisphere.
Convention for describing generators: Before giving more examples we describe how the choice of , and the ordering of , allows each generator to be identified by combinatorial data. Each generator is determined by two planar matchings of : a planar matching embedded in and the other embedded in . A planar matching on enumerated points is uniquely determined by a permutation of the even numbers by the rule that the even number in the permutation is the endpoint of the arc starting at . We will describe both and by these permutations, as specified by . To finish encoding we need to specify the decoration on each circle in . We do this by first ordering the circles by the order in which we first meet the circles if we start at and walk around according to its orientation. Thus the circle containing will always come first in our ordering. This is equivalent to the rule if, and only if, the smallest subscript of any occurring in is less than the smallest subscript of any occurring in . With this ordering, a list of elements from corresponds to a choice of decorations on the circles of : the entry in the list is the decoration on the circle in the ordering. For example, in Figure 1 the generator is specified by and . The decoration occurs on the circle through the point , which is the larger circle in the picture, while the smaller, and second circle, is decorated with a .
The generators of are cleaved links whose circles intersect its equator exactly twice. For each circle that intersects another circle there must be at least two intersections. Consequently . can be decorated with either a or a . Thus corresponding to this choice of decoration. Then has two corresponding generators, which we will write and , so .
For , there are twelve generators, depicted in Figure 1. The inside disc is the shaded disk, while the outside disk is the complement in the sphere. Thus, in terms of the combinatorial data the type generators have while .
has generators, which will not be listed here.
5. Computing the decategorification of
Let be an oriented tangle diagram in an oriented two-dimensional disk with a marked point , and the boundary circle oriented as the boundary of . Let be the set of crossings in . Let be the number of positive and negative crossings in . In this section we explain how to compute when has endpoints. From the computation in section 3 we know that
We will describe the generators of , and the action of the idempotents, presently. We will also describe the bigradings for each generator. However, There are a few preliminaries before these descriptions.
First, we will consider as being embedded in a sphere , oriented compatibly with the orientation of . Let . Along with the marked point , this effects a decomposition of into inside and outside discs.
An APS-resolution of is a map . For each APS-resolution, , there is a planar diagram in , called a APS-resolution diagram, where is the diagram in obtained by locally replacing (disjoint) neighborhoods of the crossings of using the following rule for each crossing :
A resolution of is a pair where is an APS-resolution of , and is a planar matching of embedded in . The resolution diagram for is the diagram in found by gluing to along . The set of resolutions will be denoted .
For each the homological grading is
A decorated resolution for is a resolution and a map
The circles in are of two types: 1) free circles – those which do not intersect , and 2) cut circles – those which do. The set of free circles will be denoted while the cut circles will be denoted .
The quantum grading of a decorated resolution is
The bigraded -module obtained from by forgetting the differential, and the module structure over , is generated by the decorated resolutions, shifted by the homological and quantum gradings:
To describe the action of an idempotent first notice that there is a map taking a decorated resolution to its boundary cleaved circle: is the cleaved circle where is the restriction of to the cut circles. This map is specified by the choice of marked point ; different choices will result in different maps.
Then the action of on is the linear extension of
Consequently, is spanned by the generators whose boundary cleaved circle is exactly .
Since is a bigraded chain complex over , we know how to compute . Let be the simple module over with in bigrading and such that acts by the identity, while the action of the rest of is trivial. Each generator gives a copy of in bigrading . Thus,
Under the isomorphism with , this implies that, in ,
We illustrate with a few example computations.
Example 1: Suppose is a tangle in with no boundary, . Then is the regular Khovanov homology, and is just .
Example 2: Suppose consists of two points, oriented as described in the section on cleaved links. Then the Grothendieck group is , which is two dimensional over with basis and . There is only one right matching which can be used, and resolutions come in pairs depending upon whether the single cut circle is adorned with a or . Now and there is a number such that while . Consequently, there is a polynomial for which
Example 3: We consider the tangle in Figure 2 which has two arcs and a single positive crossing, with the choice of marked point as illustrated. The resolution consists of two vertical arcs, which can be glued to either outside matching or . For there is a single circle and thus two possibilities for . We then have , since , and . Furthermore, in the notation of Figure 1. Thus, these generators contribute terms . For the matching there are cut circles . Once again, the homological grading is , and the quantum grading is , so we have terms . For the resolution, we have two circles for matching which result in generators . Now the homological grading is and the quantum grading is . Therefore, we obtain terms . Finally, for the matching we have and , so we have terms . In total,
Since the homotopy type of is an invariant of the tangle , the decategorifications in are also invariants of the tangles. This can be proven directly, and the constructions above generalized, a process which we will return to in the sequel.
6. The decategorification of type D structures
Above we mentioned that there are two type of invariants associated to a tangle. More specifically, suppose we have an an oriented tangle diagram in a oriented disc with a marked point . Let be the boundary of with the opposite orientation. Let , ordered according to the orientation on . Once again we think of as embedded in an oriented sphere , but now is the inside disc for this decomposition of .
To this configuration is associated a different type of algebraic invariant: a type structure . This is a map which satisfies a certain structural identity that will not play a role in this paper. Type D structures also have Grothendieck groups, and, following Ina Petkova in  and the results of Lipshitz, Oszváth, and Thurston in , the computations are essentially the same as above222The various bimodule constructions of  showing the equivalence of categories can be adapted to the bordered Khovanov setting. Thus, the previous section more or less tells us how to compute the decategorification of the type structure.
However, the point of introducing the two structures was to obtain a gluing , so we will adapt the definition of the decategorification of the type structure to reflect this pairing in the decategorifications.
We will think of the decategorification of as the map defined by , where . Accordingly, we will think of the decategorification of as an element in the dual , and thus as an element in .
We will describe this map somewhat tersely: the generators of are the triples where is an APS-resolution of the diagram , is a planar matching of in (an inside matching) and . The homological and quantum gradings of this generator are computed identically to those in , described above. Likewise, each generator has a boundary obtained by erasing all the free circles. The action of the idempotent on the left of is trivial on generators whose boundary is different from and the identity on those generators for which is the boundary.
The map is the linear extension of the following map on generators of :
Example: We consider the tangle in Figure 2 which is an outside disc containing a single positive crossing. To specify the map we need to specify the image on each of the generators in Figure 1. Note, however, that the arcs coming from resolving the crossing will be those outside the shaded discs in Figure 1. Thus a -resolution of the crossing gives as the outside matching. If we choose for the inside matching, then we obtain the generators . For there is one positive circle, so and . Thus . However, if we choose as the inside matching, then we obtain the generators . Since has and we can compute . To obtain we need to choose as both the inside and outside matching. Thus we need the resolution of the tangle diagram. Then and . Similar computations for each of the generators produce the following map:
Note that the map is in principal determined by the image of the cleaved links with all decorations, since changing a to a does not change the APS-resolutions and matching for those states. It does multiply by due to the change in quantum grading. Thus, once we know that we know that , for example.
7. Recovering the Jones polynomial
Suppose is an oriented sphere decomposed as , where and are two oriented discs who intersect only on their common boundary, and that boundary is oriented as the boundary of . Let . Suppose further that is an oriented link diagram in which intersects