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.

Khovanov [

An

Chmutov [

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 $~Kh(G)$ and the genus of $G$. Recall that $~Kh(G)$ is bigraded with homological grading $i$ and polynomial grading $j$, and the summand in the $(i,j)$ bigrading is denoted $~Khi,j(G)$. The diagonal grading $\delta $ is defined by $=\delta -/j2i$, and the summand in diagonal grading $\delta $ is denoted $~Kh\delta (G)$. Let $=\delta max(G)max\{\ne |\delta ~Kh\delta (G)0\},$ and $=\delta minmin\{\ne |\delta ~Kh\delta (G)0\}$. The

The genus of $G$ is zero precisely if $G$ is the all-$A$ ribbon graph of some alternating link $L$. Theorem implies that the Khovanov homology of $G$ lies on adjacent diagonals, and Theorem implies that $Kh(G)$ is isomorphic to $Kh(L)$ up to a prescribed grading shift. Lee [

Let $V(G)$ and $E(G)$ denote the set of vertices and edges of $G$ respectively. Also, let $F(G)$ denote the set of faces of $G$, that is the set of disjoint disks of $\setminus \Sigma G$. If $S$ is any finite set, then let $|S|$ denote the number of elements of $S$. A

A ribbon graph where both $G$ and its dual $G*$ (defined in Section ) have no loops is called

Corollary implies that the homological width of an adequate ribbon graph is determined by its genus (see Corollary ).

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 $G$ is the all-$A$ ribbon graph of a diagram of a link $L$, then $\cong Kh(G)Kh(L)$ up to a grading shift. In Section , we show how to construct the all-$A$ ribbon graph of a virtual link diagram and prove a generalization of Theorem for virtual links whose all-$A$ 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 $G$ can be represented by its two-dimensional regular neighborhood $\Sigma G$ in $\Sigma $. There is a natural identification between the faces of $G$ and the boundary components of $\Sigma G$, and we will use the notation $F(G)$ to equivalently denote both sets.

If the boundary components of $\Sigma G$ are capped off with disks, then one recovers the surface $\Sigma $. In the case where $G$ is the all-$A$ ribbon graph of a link diagram $D$ (defined in Section ), then $\Sigma $ is known as

Let $S(G)$ be the set of spanning ribbon subgraphs of the ribbon graph $G$. Fix a bijection $\to S(G)S(G*)$ taking $H$ to $^H$ as follows. Suppose that the edges of $G$ are $e1,\dots ,en$ and the edges of $G$ are $e1*,\dots ,en*$. Given a spanning ribbon subgraph $H$ of $G$, define $^H$ to be the spanning ribbon subgraph of $G*$ whose edge set is $=E(^H)\{\in ei*E(G*)|\notin eiE(H)\}.$ The ribbon graphs $H$ and $^H$ can be mutually embedded into $\Sigma $ (though these embeddings are not necessarily cellular). Let $\Sigma H$ and $\Sigma ^H$ be two-dimensional regular neighborhoods of $H$ and $^H$ inside of $\Sigma $. By taking suitably sized neighborhoods of $H$ and $^H$ one may realize $\Sigma $ as $\cup \Sigma H\Sigma ^H$, which gives a bijection $\Phi $ between the boundary components $F(H)$ of $H$ and the boundary components $F(^H)$ of $^H$. See Figure .

Chmutov [

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 $Z$-modules associated to the vertices and the maps between those $Z$-modules are defined analogously.

A bigraded $Z$-module $M$ is a $Z$-module that has a direct sum decomposition $=M\oplus \in i,jZMi,j$, where each summand $Mi,j$ is said to have bigrading $(i,j)$. Alternatively, one can think of a bigrading on $M$ as an assignment of a bigrading $(i,j)$ to each element in a chosen basis of $M$. If $=M\oplus i,jMi,j$ and $=N\oplus k,lNk,l$ are bigraded $Z$-modules, then both $\oplus MN$ and $\otimes MN$ are bigraded $Z$-modules where $=(\oplus MN)m,n\oplus Mm,nNm,n$ and $=(\otimes MN)m,n\oplus =+ikm,=+jln\otimes Mi,jNk,l$. Moreover, if $r$ and $s$ are integers, then define $=(M[r]\{s\})i,jM-ir,-js$.

Let $G$ be a ribbon graph with edges $e1,\dots ,en$, and let $\{0,1\}n$ denote the $n$-dimensional hypercube. Denote the vertices and edges of $\{0,1\}n$ by $V(n)$ and $E(n)$ respectively. A vertex $=I(m1,\dots ,mn)$ in the hypercube is an $n$-tuple of $0$’s and $1$’s. There is a directed edge $\in \xi E(n)$ from a vertex $=I(m1,\dots ,mn)$ to a vertex $=J(m\prime 1,\dots ,m\prime n)$ if there exists a $k$ with $1\le k\le n$ such that $=mk0$, $=m\prime k1$, and if $\ne ik$, then $=mim\prime i$. Define the height $h(I)$ of a vertex $=I(m1,\dots ,mn)$ by $=h(I)\sum =i1nmi$. The set $S(G)$ of spanning ribbon subgraphs of $G$ is in one-to-one correspondence with the vertices of the hypercube $V(G)$. Each vertex $I=(m1,\dots ,mn)\in V(G)$ is associated to the spanning ribbon subgraph $G(I)$ of $G$ whose edge set is $=E(G(I))\{ei|=mi1\}$.

There are $Z$-modules associated to each vertex in $V(n)$ and morphisms associated to each edge in $E(n)$. Let $V$ be the free $Z$-module with basis elements $v+$ and $v-$, and suppose that $v+$ has bigrading $(0,1)$ and $v-$ has bigrading $(0,-1)$. Associate the $Z$-module $=V(G(I))V\otimes F(G(I))[h(I)]\{h(I)\}$ to each $\in IV(n)$. One should view this as associating one tensor factor of $V$ to each boundary component of $\Sigma H$. Define $CKh(G)$ to be the direct sum $\oplus \in IV(n)V(G(I))$. The $Z$-module $CKh(G)$ is bigraded, and we write $=CKh(G)\oplus i,jCKhi,j(G)$. The summand $CKhi,j(G)$ is said to have

Suppose that there is a directed edge $\in \xi E(n)$ from a vertex $I=(m1,\dots ,mn)\in V(n)$ to a vertex $J=(m\prime 1,\dots ,m\prime n)\in V(n)$. The spanning ribbon subgraph $G(J)$ can be obtained from the spanning ribbon subgraph $G(I)$ by adding a single edge.
The

Adding the edge $e$ to $G(I)$ either merges two boundary components of $\Sigma G(I)$ into one boundary component of $\Sigma G(J)$ or splits one boundary component of $\Sigma G(I)$ into two boundary components of $\Sigma G(J)$. Define $:d\xi \to V(G(I))V(G(J))$ to be the identity on the tensor factors corresponding to boundary components that do not change when adding the edge $e$. If adding the edge $e$ to $G(I)$ merges two boundary components, then define $d\xi $ to be the map $:m\to V\otimes VV$ on the tensor factors corresponding to merging boundary components, and if adding the edge $e$ to $G(I)$ splits one boundary component into two, then define $d\xi $ to be the map $:\Delta \to V\otimes VV$ on the tensor factor corresponding to the splitting boundary component.

Suppose that $I00,I10,I01,$ and $I11$ are vertices in $V(n)$ that agree in all but two coordinates $k$ and $l$, and whose $k$ and $l$ coordinates are given by their subscripts. Let $\xi *0,\xi 0*,\xi 1*$, and $\xi *1$ be the edges in the hypercube from $I00$ to $I10$, from $I00$ to $I01$, from $I10$ to $I11$, and from $I01$ to $I11$ respectively. The edge maps around this square commute, that is

In order to ensure that $=\circ dd0$, it is necessary that the edge maps around any square anti-commute. An

Suppose the edges of $G$ are $e1,\dots ,en$ and fix an ordering on the edges where $$ if and only if $$. An edge assignment on $E(n)$ can be constructed as follows. Suppose that $\xi $ is a directed edge from vertex $I$ to vertex $J$, where $I$ and $J$ differ only at the $k$th coordinate. Suppose that $=I(m1,\dots ,mn)$, and define $=\u03f5(\xi )(-1)l$ where $$.

The differential $:di\to CKhi,*(G)CKh+i1,*(G)$ is defined by taking the signed sum of the edge maps $d\xi $. Define $:=di\sum =|\xi |i\u03f5(\xi )d\xi $. Observe that $di$ preserves the polynomial grading, and thus $=di\sum \in jZdi,j$ where $=di,jdi|CKhi,j(G)$. Since the signed edge maps around any square of the hypercube anticommute, it follows that $=\circ dd0$. The Khovanov homology of the ribbon graph $G$ is defined to be

where $=Khi,j(G)ker/di,jimd-i1,j$.

The construction of the chain complex $CKh(G)$ depends on an edge assignment $:\u03f5\to E(n)\{\pm 1\}$. However, using a proof adapted from Ozsváth, Rasmussen, and Szabó [

The hypercube $\{0,1\}n$ is a simplicial complex. We consider the edge assignments $\u03f5$ and $\u03f5\prime $ as $1$-cochains in $Hom(C1,F2)$ where $C1$ is the space of $1$-chains and $F2$ is the field of two elements. Since both edge assignments assign a $-1$ to an odd number of edges around each square, it follows that $\cdot \u03f5\u03f5\prime $ is a $1$-cocycle. Because the hypercube is contractible, the product of the edge assignments $\cdot \u03f5\u03f5\prime $ is the coboundary of a $0$-cochain, that is there exists $:\eta \to V(n)\{\pm 1\}$ such that $=\eta (I)\eta (J)\u03f5(\xi )\u03f5\prime (\xi )$ if $\xi $ is an edge between vertices $I$ and $J$.

Let $:\psi \to (CKh(G),d\u03f5)(CKh(G),d\u03f5\prime )$ be the map which when restricted to $V(G(I))$ is multiplication by $\eta (I)$. Then $\psi $ is an isomorphism from $(CKh(G),d\u03f5)\to (CKh(G,d\u03f5\prime )$. ∎

In the construction of $CKh(G)$, one associates a tensor factor of $V$ to each boundary component of $\Sigma G(I)$. Suppose that there is a marked point on the boundary of a vertex of $\Sigma G$ that misses the bands attached for each edge. Let $~F(G(I))$ denote the set of boundary components of $\Sigma G(I)$ without marked points. Note that $=|~F(G(I))|-|F(G(I))|1$. For each vertex $\in IV(n)$, one can consider $V(G(I))$ as $=\otimes VV\otimes ~F(G(I))\oplus (\otimes Zv+V\otimes ~F(G(I)))(\otimes Zv-V\otimes ~F(G(I)))$, where the $Zv+$ and $Zv-$ are the two summands of $V$ associated to the boundary component of $\Sigma G(I)$ that contains the marked point. Define $=~V(G(I))(\otimes Zv-V\otimes ~F(G(I)))\{1\}$ where as before, the $Zv-$ corresponds to the boundary component of $\Sigma G(I)$ that contains the marked point.

Let $=~CKh(G)\oplus \in IV(n)~V(G(I))$, and define $:=~dd|~CKh(G)$. Since the range of $~d$ is a subset of $~CKh(G)$, it follows that $(~CKh(G),~d)$ forms a chain complex. The homology of this chain complex $~Kh(G)$ is called the reduced Khovanov homology of $G$.

Throughout this subsection, let $G$ be a ribbon graph and let $G*$ be the dual ribbon graph. In what follows we show that the Khovanov complex of $G*$ is isomorphic to the dual complex of the Khovanov complex of $G$.

If $M$ is a $Z$-module, then define the dual of $M$ by $=M*Hom(M,Z)$, and if $:f\to MN$ is a $Z$-module homomorphism, then the dual homomorphism $:f*\to M*N*$ is defined by $=f*(\varphi )\circ \varphi f$. Let $(C,\partial )$ denote the complex

The dual complex $(C*,\partial *)$ is the complex where $=(C*)i(C-i)*$ and $(\partial *)i$ is the dual of $\partial --i1$. When there is a polynomial grading on $C$ that $\partial $ preserves (as is the case with the Khovanov homology defined above), define $C*$ to have the opposite polynomial grading, i.e. $=(C*)i,j(C-i,-j)*$.

We prove the proposition for $CKh(G*)$; the result for $~CKh(G*)$ is proved similarly. Let $\{^0,1\}n$ be an $n$-dimensional hypercube with vertex set $^V(n)$ and edge set $^E(n)$. The one-skeleton of the hypercube $\{^0,1\}n$ is the same underlying graph as the one-skeleton of $\{0,1\}n$ except that the edges in $^E(n)$ are in the opposite direction as the edges in $E(n)$. If $=I(m1,\dots ,mn)$ is a vertex in $V(n)$, define its dual vertex $=^I(^m1,\dots ,^mn)$ in $^V(n)$ to be the vertex where $\equiv +mi^mimod12$ for $1\le i\le n$. The complexes $CKh(G)$ and $CKh(G*)$ will use the hypercube $\{0,1\}n$, while the complex $CKh(G)*$ will use the dual hypercube $\{^0,1\}n$.

First, we show that for each vertex $\in IV(n)$, we have a grading preserving isomorphism

Next we show that if $\xi $ is an edge in $E(n)$ from $I$ to $J$ and $^\xi $ is the dual edge in $^E(n)$ from $^I$ to $^J$, then the edge maps $:d\xi \to V(G*(I))V(G*(J))$ and $:d\xi *\to V(G(^I))*V(G(^J))*$ commute with $\circ \Psi \Phi *$ and $(\circ \Psi \Phi *)-1$. Finally, we note that an edge assignment for the hypercube $\{0,1\}n$ induces an edge assignment for the dual hypercube $\{^0,1\}n$, giving us the desired isomorphism of complexes.

Define a basis $\{v-*,v+*\}$ of $V*$ by

Fix an isomorphism $:\psi \to V*V$ where $=\psi (v-*)v-$ and $=\psi (v+*)v+$, and define an isomorphism $:\Psi \to V(G(I))*V(G(I))$ by $=\Psi \otimes \psi \cdots \psi $. The map $\Psi $ sends summands in the $(i,j)$-bigrading of $CKh(G)*$ to the $(-i,-j)$-bigrading of $CKh(G)$. As noted in Section , there is a canonical bijection $\Phi $ from the boundary components of $\Sigma G*(I)$ to the boundary components of $\Sigma G(^I)$ given by the gluing map in $=\Sigma \cup \Sigma G*(I)\Sigma G(^I)$. The bijection induces an isomorphism $:\Phi *\to V(G*(I))V(G(^I))$ that sends the summand in the $(i,j)$-bigrading of $CKh(G)$ to the $(-ni,-nj)$-bigrading of $CKh(G*)$. The composition $:\circ \Psi \Phi *\cong \to V(G*(I))V(G(^I))*[n]\{n\}$ is the desired isomorphism.

If $m*$ and $\Delta *$ are the dual maps of $m$ and $\Delta $ respectively, then

Let $d\xi *$ be the edge maps in the dual complex defined using $m*$ and $\Delta *$. Since $=m*(\otimes v*\pm v*\pm )(m(\otimes v\pm v\pm ))*$ and $=\Delta *(v*\pm )(\Delta (v\pm ))*$, it follows that $=\circ \Psi \Phi *d\xi \circ d\xi *\Psi \Phi *$.

An edge assignment $:\u03f5\to E(n)\{\pm 1\}$ gives an edge assignment $:^\u03f5\to ^E(n)\{\pm 1\}$ by $:=^\u03f5(^\xi )\u03f5(\xi )$. Therefore, up to the prescribed grading shift, the complexes $(CKh(G*),d\u03f5)$ and $(CKh(G)*,d*^\u03f5)$ are isomorphic. Proposition states that the choice of edge assignment does not change the isomorphism type of the complex, and the result follows. ∎

The following corollary follows from Proposition and the relationship between the homology of a complex and the homology of its dual.