We introduce Khovanov homology for ribbon graphs and show that the Khovanov homology of a certain ribbon graph embedded on the Turaev surface of a link is isomorphic to the Khovanov homology of the link (after a grading shift). We also present a spanning quasi-tree model for the Khovanov homology of a ribbon graph.
The all- ribbon graph of a link diagram (defined in Section ) is an oriented ribbon graph embedded on the Turaev surface of the link diagram. Dasbach, Futer, Kalfagianni, Lin, and Stoltzfus [
Champanerkar and Kofman [
The homological and polynomial gradings of a generator in the spanning quasi-tree complex can be expressed via the quasi-tree activities defined by Champanerkar, Kofman, and Stoltzfus [
We provide several applications of the quasi-tree model for the Khovanov homology of ribbon graphs. The first application compares the homological width of and the genus of . Recall that is bigraded with homological grading and polynomial grading , and the summand in the bigrading is denoted . The diagonal grading is defined by , and the summand in diagonal grading is denoted . Let and . The
The genus of is zero precisely if is the all- ribbon graph of some alternating link . Theorem implies that the Khovanov homology of lies on adjacent diagonals, and Theorem implies that is isomorphic to up to a prescribed grading shift. Lee [
Let and denote the set of vertices and edges of respectively. Also, let denote the set of faces of , that is the set of disjoint disks of . If is any finite set, then let denote the number of elements of . A
A ribbon graph where both and its dual (defined in Section ) have no loops is called
This paper is organized as follows. In Section , we review basic definitions for ribbon graphs. In Section , we define the Khovanov homology of ribbon graphs. In Section , we review the construction of Khovanov homology and show if is the all- ribbon graph of a diagram of a link , then up to a grading shift. In Section , we show how to construct the all- ribbon graph of a virtual link diagram and prove a generalization of Theorem for virtual links whose all- ribbon graphs are orientable. In Section , we define Reidemeister moves for ribbon graphs that generalize the Reidemeister moves for both classical and virtual links. We also show that our Khovanov homology of ribbon graphs is invariant under the ribbon graph Reidemeister moves. In Section , we construct the spanning quasi-tree model for Khovanov homology of ribbon graphs. We also show that the gradings in this complex can be express via activity words. In Section , we provide several applications of our spanning quasi-tree model. In Section , we compute the Khovanov homology of an example ribbon graph.
In this section, we provide some basic definitions for ribbon graphs. A
Recall that a ribbon graph can be represented by its two-dimensional regular neighborhood in . There is a natural identification between the faces of and the boundary components of , and we will use the notation to equivalently denote both sets.
If the boundary components of are capped off with disks, then one recovers the surface . In the case where is the all- ribbon graph of a link diagram (defined in Section ), then is known as
Let be the set of spanning ribbon subgraphs of the ribbon graph . Fix a bijection taking to as follows. Suppose that the edges of are and the edges of are . Given a spanning ribbon subgraph of , define to be the spanning ribbon subgraph of whose edge set is The ribbon graphs and can be mutually embedded into (though these embeddings are not necessarily cellular). Let and be two-dimensional regular neighborhoods of and inside of . By taking suitably sized neighborhoods of and one may realize as , which gives a bijection between the boundary components of and the boundary components of . See Figure .
A (possibly non-orientable) ribbon graph can be obtained from an arrow presentation by the following process. Consider each circle of the arrow presentation as the boundary of a disk corresponding to a vertex of the ribbon graph. Glue a band to each pair of marking arrows with the same label such that the orientation of the band agrees with the orientation of the marking arrows, as depicted in Figure .
Once bands are attached to every pair of identically labeled marking arrows, the resulting surface may or may not be orientable. If the resulting surface is orientable, then it is the regular neighborhood of an oriented ribbon graph. We will not consider arrow presentations whose associated ribbon graphs are non-orientable. Figure shows an arrow presentation for the ribbon graph depicted in Figure .
In this section, we introduce Khovanov homology and reduced Khovanov homology for ribbon graphs.
We also prove a result about the Khovanov homology of the dual ribbon graph. The construction closely imitates Khovanov’s original categorification of the Jones polynomial [
In the cube of resolutions complex for the Khovanov homology of links, the vertices in the hypercube correspond to Kauffman states of the link diagram, while in our construction the vertices in the hypercube correspond to subsets of the edge set of the ribbon graph. In both constructions, the -modules associated to the vertices and the maps between those -modules are defined analogously.
A bigraded -module is a -module that has a direct sum decomposition , where each summand is said to have bigrading . Alternatively, one can think of a bigrading on as an assignment of a bigrading to each element in a chosen basis of . If and are bigraded -modules, then both and are bigraded -modules where and . Moreover, if and are integers, then define .
Let be a ribbon graph with edges , and let denote the -dimensional hypercube. Denote the vertices and edges of by and respectively. A vertex in the hypercube is an -tuple of ’s and ’s. There is a directed edge from a vertex to a vertex if there exists a with such that , , and if , then . Define the height of a vertex by . The set of spanning ribbon subgraphs of is in one-to-one correspondence with the vertices of the hypercube . Each vertex is associated to the spanning ribbon subgraph of whose edge set is .
There are -modules associated to each vertex in and morphisms associated to each edge in . Let be the free -module with basis elements and , and suppose that has bigrading and has bigrading . Associate the -module to each . One should view this as associating one tensor factor of to each boundary component of . Define to be the direct sum . The -module is bigraded, and we write . The summand is said to have
Suppose that there is a directed edge from a vertex to a vertex . The spanning ribbon subgraph can be obtained from the spanning ribbon subgraph by adding a single edge.
Adding the edge to either merges two boundary components of into one boundary component of or splits one boundary component of into two boundary components of . Define to be the identity on the tensor factors corresponding to boundary components that do not change when adding the edge . If adding the edge to merges two boundary components, then define to be the map on the tensor factors corresponding to merging boundary components, and if adding the edge to splits one boundary component into two, then define to be the map on the tensor factor corresponding to the splitting boundary component.
Suppose that and are vertices in that agree in all but two coordinates and , and whose and coordinates are given by their subscripts. Let , and be the edges in the hypercube from to , from to , from to , and from to respectively. The edge maps around this square commute, that is
In order to ensure that , it is necessary that the edge maps around any square anti-commute. An
Proposition below states the choice of edge assignment does not change the isomorphism type of the chain complex. If we wish to highlight the choice of the edge assignment, we denote the complex by ; however we will often hide this choice and denote the complex by only or just .
Suppose the edges of are and fix an ordering on the edges where if and only if . An edge assignment on can be constructed as follows. Suppose that is a directed edge from vertex to vertex , where and differ only at the th coordinate. Suppose that , and define where .
The differential is defined by taking the signed sum of the edge maps . Define . Observe that preserves the polynomial grading, and thus where . Since the signed edge maps around any square of the hypercube anticommute, it follows that . The Khovanov homology of the ribbon graph is defined to be
The construction of the chain complex depends on an edge assignment . However, using a proof adapted from Ozsváth, Rasmussen, and Szabó [
The hypercube is a simplicial complex. We consider the edge assignments and as -cochains in where is the space of -chains and is the field of two elements. Since both edge assignments assign a to an odd number of edges around each square, it follows that is a -cocycle. Because the hypercube is contractible, the product of the edge assignments is the coboundary of a -cochain, that is there exists such that if is an edge between vertices and .
Let be the map which when restricted to is multiplication by . Then is an isomorphism from . ∎
In the construction of , one associates a tensor factor of to each boundary component of . Suppose that there is a marked point on the boundary of a vertex of that misses the bands attached for each edge. Let denote the set of boundary components of without marked points. Note that . For each vertex , one can consider as , where the and are the two summands of associated to the boundary component of that contains the marked point. Define where as before, the corresponds to the boundary component of that contains the marked point.
Let , and define . Since the range of is a subset of , it follows that forms a chain complex. The homology of this chain complex is called the reduced Khovanov homology of .
Throughout this subsection, let be a ribbon graph and let be the dual ribbon graph. In what follows we show that the Khovanov complex of is isomorphic to the dual complex of the Khovanov complex of .
If is a -module, then define the dual of by , and if is a -module homomorphism, then the dual homomorphism is defined by . Let denote the complex
The dual complex is the complex where and is the dual of . When there is a polynomial grading on that preserves (as is the case with the Khovanov homology defined above), define to have the opposite polynomial grading, i.e. .
We prove the proposition for ; the result for is proved similarly. Let be an -dimensional hypercube with vertex set and edge set . The one-skeleton of the hypercube is the same underlying graph as the one-skeleton of except that the edges in are in the opposite direction as the edges in . If is a vertex in , define its dual vertex in to be the vertex where for . The complexes and will use the hypercube , while the complex will use the dual hypercube .
First, we show that for each vertex , we have a grading preserving isomorphism
Next we show that if is an edge in from to and is the dual edge in from to , then the edge maps and commute with and . Finally, we note that an edge assignment for the hypercube induces an edge assignment for the dual hypercube , giving us the desired isomorphism of complexes.
Define a basis of by
Fix an isomorphism where and , and define an isomorphism by . The map sends summands in the -bigrading of to the -bigrading of . As noted in Section , there is a canonical bijection from the boundary components of to the boundary components of given by the gluing map in . The bijection induces an isomorphism that sends the summand in the -bigrading of to the -bigrading of . The composition is the desired isomorphism.
If and are the dual maps of and respectively, then
Let be the edge maps in the dual complex defined using and . Since and , it follows that .
An edge assignment gives an edge assignment by . Therefore, up to the prescribed grading shift, the complexes and are isomorphic. Proposition states that the choice of edge assignment does not change the isomorphism type of the complex, and the result follows. ∎