A Turaev surface approach to Khovanov homology Oliver T. Dasbach Department of MathematicsLouisiana State UniversityBaton Rouge, Louisiana kasten@math.lsu.edu Adam M. Lowrance Department of MathematicsVassar CollegePoughkeepsie, New York adlowrance@vassar.edu

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 first author was partially supported by NSF-DMS 0806539 and NSF-DMS FRG 0456275.The second author was partially supported by NSF-DMS 0602242 (VIGRE)
1 section 1 1 §1 <tag close=". ">1</tag>Introduction 1Introduction

Khovanov [] introduced a categorification of the Jones polynomial now known as Khovanov homology. The Khovanov homology of a link $L$ with diagram $D$, equivalently denoted $⁢ K h ( L )$ or $⁢ K h ( D )$, is a bigraded abelian group with homological grading $i$ and polynomial grading $j$. The reduced version of Khovanov homology, denoted either $⁢ ~ ⁢ K h ( L )$ or $⁢ ~ ⁢ K h ( D )$ is also a bigraded abelian group. The graded Euler characteristic of reduced Khovanov homology is the Jones polynomial $⁢ J L ( q )$ while the graded Euler characteristic of Khovanov homology is $⁢ ( + q q - 1 ) J L ( q )$.

An oriented ribbon graph $G$ is a graph $G$ together with an embedding into an oriented surface $Σ$ such that $∖ Σ G$ is a disjoint union of two-cells. The graph $G$ is called the underlying graph of $G$. Two oriented ribbon graphs $G$ and $G ′$ embedded into surfaces $Σ$ and $Σ ′$ respectively are equivalent if there is an orientation preserving homeomorphism from $Σ$ to $Σ ′$ that restricts to a graph isomorphism from the underlying graph of $G$ to the underlying graph of $G ′$. The embedding into the surface determines a cyclic ordering of the half edges incident to each vertex, and oriented ribbon graphs are often depicted as graphs drawn in the plane with the cyclic ordering around each vertex given by counterclockwise rotation. The genus of $G$, denoted $⁢ g ( G )$, is the genus of the surface $Σ$. All ribbon graphs that we consider are oriented and are referred to just as ribbon graphs.

The all-$A$ 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 [] interpret the Jones polynomial via the all-$A$ ribbon graph of a diagram of the link. We define the Khovanov homology $⁢ K h ( G )$ and reduced Khovanov homology $⁢ ~ ⁢ K h ( G )$ of a ribbon graph $G$. If $G$ is the all-$A$ ribbon graph of a link diagram $L$, then up to a grading shift, the Khovanov homology of the ribbon graph $G$ is isomorphic to the Khovanov homology of the link $L$. If $M$ is a bigraded abelian group and $r$ and $s$ are integers, let $⁢ M [ r ] { s }$ denote the group $M$ but with the homological grading shifted up by $r$ units and the polynomial grading shifted up by $s$ units. The first main theorem of the paper is:

Theorem 1.1 Theorem 1.1 1.1 Theorem 1.1 <tag><text font="bold">Theorem 1.1</text></tag>.

Let $L$ be a link with diagram $D$ and all-$A$ ribbon graph $D$. Suppose that $D$ has $n +$ positive crossings and $n -$ negative crossings. There are grading preserving isomorphisms

$≅ ⁢ K h ( D ) [ - n - ] { - n + ⁢ 2 n - } ⁢ K h ( L ) and$ $⁢ K h ( D ) [ - n - ] { - n + ⁢ 2 n - }$ $≅$ $⁢ K h ( L ) and$ $≅ ⁢ ~ ⁢ K h ( D ) [ - n - ] { - n + ⁢ 2 n - } ⁢ ~ ⁢ K h ( L ) .$ $⁢ ~ ⁢ K h ( D ) [ - n - ] { - n + ⁢ 2 n - }$ $≅$ $⁢ ~ ⁢ K h ( L ) .$

Chmutov [] generalized the notion of the all-$A$ ribbon graph to virtual link diagrams (see Section ). However, the all-$A$ ribbon graph $D$ of a virtual link diagram $D$ is not necessarily orientable. We show that a generalization of Theorem holds whenever $D$ is orientable.

Champanerkar and Kofman [] and independently Wehrli [] proved that there exists a complex whose homology is the reduced Khovanov homology of a link $L$ and whose generators are in one-to-one correspondence with the spanning trees of the checkerboard graph of a diagram $D$ of $L$. Such a complex is known as a spanning tree model of Khovanov homology. A ribbon graph $G$ can be represented by a surface $Σ G$ with boundary. The surface $Σ G$ is a two-dimensional regular neighborhood of $G$ in $Σ$, as in Figure . A spanning quasi-tree $T$ of a ribbon graph $G$ is a spanning ribbon subgraph of $G$ such that the surface $Σ T$ has only one boundary component. The spanning quasi-trees of a ribbon graph can be viewed as generalizations of spanning trees of a graph embedded in the plane in the following sense. If the genus of $G$ is zero, then the spanning quasi-trees of $G$ are precisely the spanning trees of the underlying graph of $G$. However, if the genus of $G$ is greater than zero, then $G$ will have spanning quasi-trees of all different genera ranging from zero to the genus of $G$. Champanerkar, Kofman, and Stoltzfus [] show that the spanning trees of the checkerboard graph of a link diagram $D$ are in one-to-one correspondence with the spanning quasi-trees of the all-$A$ ribbon graph of $D$. Therefore, the spanning tree model of Khovanov homology for links may be viewed as a spanning quasi-tree model. We show that a spanning quasi-tree model can be obtained directly from our definition of the Khovanov homology of ribbon graphs.

Theorem 1.2 Theorem 1.2 1.2 Theorem 1.2 <tag><text font="bold">Theorem 1.2</text></tag>.

Let $G$ be a ribbon graph. There exists a complex $⁢ ~ C ( G )$ whose generators are in one-to-one correspondence with the spanning quasi-trees of $G$ and whose homology is $⁢ ~ ⁢ K h ( G )$.

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 []. Corollary gives an expansion of the Jones polynomial as a summation over the spanning quasi-trees where each spanning quasi-tree is assigned a signed monomial in $q$.

We provide several applications of the quasi-tree model for the Khovanov homology of ribbon graphs. The first application compares the homological width of $⁢ ~ ⁢ K h ( G )$ and the genus of $G$. Recall that $⁢ ~ ⁢ K h ( G )$ is bigraded with homological grading $i$ and polynomial grading $j$, and the summand in the $( i , j )$ bigrading is denoted $⁢ ~ ⁢ K h i , j ( G )$. The diagonal grading $δ$ is defined by $= δ - / j 2 i$, and the summand in diagonal grading $δ$ is denoted $⁢ ~ ⁢ K h δ ( G )$. Let $= ⁢ δ max ( G ) max { ≠ | δ ⁢ ~ ⁢ K h δ ( G ) 0 } ,$ and $= δ min min { ≠ | δ ⁢ ~ ⁢ K h δ ( G ) 0 }$. The homological width of $⁢ ~ ⁢ K h ( G )$, denoted $⁢ h w ( ⁢ ~ ⁢ K h ( G ) )$, is defined as $= ⁢ h w ( ⁢ ~ ⁢ K h ( G ) ) + - ⁢ δ max ( G ) ⁢ δ min ( G ) 1$. Our first application is the following theorem.

Theorem 1.3 Theorem 1.3 1.3 Theorem 1.3 <tag><text font="bold">Theorem 1.3</text></tag>.

Let $G$ be a ribbon graph. The genus of $G$ gives an upper bound on the homological width of $⁢ ~ ⁢ K h ( G )$. In particular, we have

$≤ ⁢ h w ( ⁢ ~ ⁢ K h ( G ) ) + ⁢ g ( G ) 1 .$

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 $⁢ K h ( G )$ is isomorphic to $⁢ K h ( L )$ up to a prescribed grading shift. Lee [] proved that the Khovanov homology of an alternating link is supported on adjacent diagonals, and thus Theorem can be viewed as a generalization of Lee’s result.

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 $∖ Σ G$. If $S$ is any finite set, then let $| S |$ denote the number of elements of $S$. A loop of $G$ is an edge that connects a vertex to itself. If a ribbon graph $G$ does not have any loops, then the quasi-tree model implies that the reduced Khovanov homology is isomorphic to $Z$ in its minimum nontrivial polynomial grading. Define $= ⁢ j min ( G ) min { ≠ | j ⊕ ∈ i Z ⁢ K h i , j ( G ) 0 }$.

Theorem 1.4 Theorem 1.4 1.4 Theorem 1.4 <tag><text font="bold">Theorem 1.4</text></tag>.

Suppose that $G$ is a ribbon graph with no loops. Then $= ⁢ j min ( G ) - 1 | ⁢ V ( G ) |$ and

$⊕ ∈ i Z ⁢ ~ ⁢ K h i , - 1 | ⁢ V ( G ) | ( G ) ≅ ⁢ ~ ⁢ K h 0 , - 1 | ⁢ V ( G ) | ( G ) ≅ Z .$

A ribbon graph where both $G$ and its dual $G *$ (defined in Section ) have no loops is called adequate. If a ribbon graph is adequate, then a corollary to Theorem states that the Khovanov homology in the maximum polynomial grading is also isomorphic to $Z$. This corollary is the generalization to ribbon graphs of results by Khovanov [] and Abe []. Define $= ⁢ j max ( G ) max { ≠ | j ⊕ ∈ i Z ⁢ ~ ⁢ K h i , j ( G ) 0 }$.

Corollary 1.5 1.5 1.5 Corollary 1.5 <tag><text font="bold">Corollary 1.5</text></tag>.

Let $G$ be an adequate ribbon graph. Then $= ⁢ j min ( G ) - 1 | ⁢ V ( G ) |$, $= ⁢ j max ( G ) - + | ⁢ E ( G ) | | ⁢ F ( G ) | 1$, and

$⊕ ∈ i Z ⁢ ~ ⁢ K h i , ⁢ j min ( G ) ( G ) ≅ ⁢ ~ ⁢ K h 0 , - 1 | ⁢ V ( G ) | ( G ) ≅ ⁢ Z and ⊕ ∈ i Z ⁢ ~ ⁢ K h i , ⁢ j max ( G ) ( G ) ≅ ⁢ ~ ⁢ K h | ⁢ E ( G ) | , - + | ⁢ E ( G ) | | ⁢ F ( G ) | 1 ( G ) ≅ Z .$

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 $≅ ⁢ K h ( G ) ⁢ K h ( L )$ 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.

Acknowledgment: The first author thanks Christian Blanchet for fruitful discussions on Khovanov homology during a visit to Paris. We also thank the referee, whose suggestion to add Sections and greatly improved the paper.

2 section 2 2 §2 <tag close=". ">2</tag>Ribbon graphs 2Ribbon graphs

In this section, we provide some basic definitions for ribbon graphs. A ribbon subgraph $H$ of a ribbon graph $G$ is a subgraph $H$ of the underlying graph $G$ of $G$ such that the cyclic order of the edges in $H$ around each vertex is inherited from the cyclic order of the edges in $G$. A ribbon subgraph $H$ of $G$ is spanning if $= ⁢ V ( H ) ⁢ V ( G )$.

Recall that a ribbon graph $G$ can be represented by its two-dimensional regular neighborhood $Σ G$ in $Σ$. There is a natural identification between the faces of $G$ and the boundary components of $Σ G$, and we will use the notation $⁢ F ( G )$ to equivalently denote both sets.

If the boundary components of $Σ G$ are capped off with disks, then one recovers the surface $Σ$. In the case where $G$ is the all-$A$ ribbon graph of a link diagram $D$ (defined in Section ), then $Σ$ is known as the Turaev surface of $D$. The dual ribbon graph $G *$ is constructed as follows. The vertices of $G *$ are the centers of the faces of $G$. There is a one-to-one correspondence between the edges of $G$ and $G *$. Locally, each edge $e$ in $G$ has two (not necessarily distinct) disks attached to it to form $Σ$. For each edge $e$ in $G$, there is a dual edge $e *$ in $G *$ such that the endpoints of $e *$ correspond to the two (not necessarily distinct) disks attached along $e$, and such that $e$ and $e *$ transversely intersect exactly once in $Σ$. The cyclic order of the edges around each vertex in $G *$ is given by its embedding into $Σ$. Chmutov [] and Moffatt [] provide other notions of duality in ribbon graphs. Compare also with [].

Let $⁢ S ( G )$ be the set of spanning ribbon subgraphs of the ribbon graph $G$. Fix a bijection $→ ⁢ S ( G ) ⁢ S ( G * )$ taking $H$ to $^ H$ as follows. Suppose that the edges of $G$ are $e 1 , … , e n$ and the edges of $G$ are $e 1 * , … , e n *$. Given a spanning ribbon subgraph $H$ of $G$, define $^ H$ to be the spanning ribbon subgraph of $G *$ whose edge set is $= ⁢ E ( ^ H ) { ∈ e i * ⁢ E ( G * ) | ∉ e i ⁢ E ( H ) } .$ The ribbon graphs $H$ and $^ H$ can be mutually embedded into $Σ$ (though these embeddings are not necessarily cellular). Let $Σ H$ and $Σ ^ H$ be two-dimensional regular neighborhoods of $H$ and $^ H$ inside of $Σ$. By taking suitably sized neighborhoods of $H$ and $^ H$ one may realize $Σ$ as $∪ Σ H Σ ^ H$, which gives a bijection $Φ$ between the boundary components $⁢ F ( H )$ of $H$ and the boundary components $⁢ F ( ^ H )$ of $^ H$. See Figure .

Chmutov [] introduced a representation of a ribbon graph called an arrow presentation. An arrow presentation is a collection of non-nested circles in the plane together with a collection of oriented, labeled arcs lying on the circles, called marking arrows, such that each label appears on exactly two marking arrows. Two arrow presentations are equivalent if one can be obtained from the other by reversing all arrows on a circle and reversing the cyclic order of the arrows along it, by reversing the orientation of all marking arrows that belong to some subset of the labels, or by changing the labeling set.

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 .

3 section 3 3 §3 <tag close=". ">3</tag>Khovanov homology of ribbon graphs 3Khovanov homology of ribbon graphs

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 [] (especially as interpreted by Bar-Natan [] and Viro []).

3.1 subsection 3.1 3.1 §3.1 <tag close=". ">3.1</tag>Khovanov homology for ribbon graphs 3.1Khovanov homology for ribbon graphs

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 $= M ⊕ ∈ i , j Z M i , j$, where each summand $M i , j$ 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 $= M ⊕ i , j M i , j$ and $= N ⊕ k , l N k , l$ are bigraded $Z$-modules, then both $⊕ M N$ and $⊗ M N$ are bigraded $Z$-modules where $= ( ⊕ M N ) m , n ⊕ M m , n N m , n$ and $= ( ⊗ M N ) m , n ⊕ = + i k m , = + j l n ⊗ M i , j N k , l$. Moreover, if $r$ and $s$ are integers, then define $= ( ⁢ M [ r ] { s } ) i , j M - i r , - j s$.

Let $G$ be a ribbon graph with edges $e 1 , … , e n$, 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 $= I ( m 1 , … , m n )$ in the hypercube is an $n$-tuple of $0$’s and $1$’s. There is a directed edge $∈ ξ ⁢ E ( n )$ from a vertex $= I ( m 1 , … , m n )$ to a vertex $= J ( m ′ 1 , … , m ′ n )$ if there exists a $k$ with $1 ≤ k ≤ n$ such that $= m k 0$, $= m ′ k 1$, and if $≠ i k$, then $= m i m ′ i$. Define the height $⁢ h ( I )$ of a vertex $= I ( m 1 , … , m n )$ by $= ⁢ h ( I ) ∑ = i 1 n m i$. 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 $I = ( m 1 , … , m n ) ∈ ⁢ V ( G )$ is associated to the spanning ribbon subgraph $⁢ G ( I )$ of $G$ whose edge set is $= ⁢ E ( ⁢ G ( I ) ) { e i | = m i 1 }$.

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 $= ⁢ V ( ⁢ G ( I ) ) ⁢ V ⊗ ⁢ F ( ⁢ G ( I ) ) [ ⁢ h ( I ) ] { ⁢ h ( I ) }$ to each $∈ I ⁢ V ( n )$. One should view this as associating one tensor factor of $V$ to each boundary component of $Σ H$. Define $⁢ C K h ( G )$ to be the direct sum $⊕ ∈ I ⁢ V ( n ) ⁢ V ( ⁢ G ( I ) )$. The $Z$-module $⁢ C K h ( G )$ is bigraded, and we write $= ⁢ C K h ( G ) ⊕ i , j ⁢ C K h i , j ( G )$. The summand $⁢ C K h i , j ( G )$ is said to have homological grading $i$ and polynomial grading $j$. It will sometimes be useful to consider all summands of $⁢ C K h ( G )$ in a particular homological grading without specifying the polynomial grading; therefore, we let $= ⁢ C K h i , * ( G ) ⊕ ∈ j Z ⁢ C K h i , j ( G )$.

Suppose that there is a directed edge $∈ ξ ⁢ E ( n )$ from a vertex $I = ( m 1 , … , m n ) ∈ ⁢ V ( n )$ to a vertex $J = ( m ′ 1 , … , m ′ n ) ∈ ⁢ V ( n )$. The spanning ribbon subgraph $⁢ G ( J )$ can be obtained from the spanning ribbon subgraph $⁢ G ( I )$ by adding a single edge. The height of the edge $ξ$ is defined as $= | ξ | ⁢ E ( ⁢ G ( I ) )$, which is the height of the vertex from which the edge originates. Each edge in the hypercube has an associated map $: d ξ → ⁢ V ( ⁢ G ( I ) ) ⁢ V ( ⁢ G ( J ) )$. Define $Z$-linear maps $: m → V ⊗ V V$ and $: Δ → ⊗ V V V$ by

$: m → V ⊗ V V : m { ↦ ⊗ v + v - v - ↦ ⊗ v + v + v + ↦ ⊗ v - v + v - ↦ ⊗ v - v - 0 : Δ → ⊗ V V V : Δ { ↦ v + + ⊗ v + v - ⊗ v - v + ↦ v - ⊗ v - v - .$

Adding the edge $e$ to $⁢ G ( I )$ either merges two boundary components of $Σ ⁢ G ( I )$ into one boundary component of $Σ ⁢ G ( J )$ or splits one boundary component of $Σ ⁢ G ( I )$ into two boundary components of $Σ ⁢ G ( J )$. Define $: d ξ → ⁢ V ( ⁢ G ( I ) ) ⁢ V ( ⁢ G ( J ) )$ 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 $d ξ$ to be the map $: m → V ⊗ V V$ 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 $d ξ$ to be the map $: Δ → V ⊗ V V$ on the tensor factor corresponding to the splitting boundary component.

Suppose that $I 00 , I 10 , I 01 ,$ and $I 11$ 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 $ξ * 0 , ξ 0 ⁣ * , ξ 1 ⁣ *$, and $ξ * 1$ be the edges in the hypercube from $I 00$ to $I 10$, from $I 00$ to $I 01$, from $I 10$ to $I 11$, and from $I 01$ to $I 11$ respectively. The edge maps around this square commute, that is

$= ∘ d ξ 1 ⁣ * d ξ * 0 ∘ d ξ * 1 d ξ 0 ⁣ * .$

In order to ensure that $= ∘ d d 0$, it is necessary that the edge maps around any square anti-commute. An edge assignment on $⁢ E ( n )$ is a map $: ϵ → ⁢ E ( n ) { ± 1 }$ such that each square in the hypercube has an odd number of edges $ξ$ for which $= ⁢ ϵ ( ξ ) - 1$. Given such an edge assignment, we have

$= ⁢ ∘ ⁢ ϵ ( ξ 1 ⁣ * ) d ξ 1 ⁣ * ϵ ( ξ * 0 ) d ξ * 0 - ⁢ ∘ ⁢ ϵ ( ξ * 1 ) d ξ * 1 ϵ ( ξ 0 ⁣ * ) d ξ 0 ⁣ * .$

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 $( ⁢ C K h ( G ) , d ϵ )$; however we will often hide this choice and denote the complex by only $( ⁢ C K h ( G ) , d )$ or just $⁢ C K h ( G )$.

Suppose the edges of $G$ are $e 1 , … , e n$ 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 $ξ$ is a directed edge from vertex $I$ to vertex $J$, where $I$ and $J$ differ only at the $k$th coordinate. Suppose that $= I ( m 1 , … , m n )$, and define $= ⁢ ϵ ( ξ ) ( - 1 ) l$ where .

The differential $: d i → ⁢ C K h i , * ( G ) ⁢ C K h + i 1 , * ( G )$ is defined by taking the signed sum of the edge maps $d ξ$. Define $:= d i ∑ = | ξ | i ⁢ ϵ ( ξ ) d ξ$. Observe that $d i$ preserves the polynomial grading, and thus $= d i ∑ ∈ j Z d i , j$ where $= d i , j d i | ⁢ C K h i , j ( G )$. Since the signed edge maps around any square of the hypercube anticommute, it follows that $= ∘ d d 0$. The Khovanov homology of the ribbon graph $G$ is defined to be

$= ⁢ K h ( G ) ⊕ ∈ i , j Z ⁢ K h i , j ( G ) ,$

where $= ⁢ K h i , j ( G ) ker ⁢ / d i , j im d - i 1 , j$.

The construction of the chain complex $⁢ C K h ( G )$ depends on an edge assignment $: ϵ → ⁢ E ( n ) { ± 1 }$. However, using a proof adapted from Ozsváth, Rasmussen, and Szabó [], one can show that complexes with different edge assignments are isomorphic.

Proposition 3.1 3.1 3.1 Proposition 3.1 <tag><text font="bold">Proposition 3.1</text></tag>.

Let $ϵ$ and $ϵ ′$ be edge assignments on $⁢ E ( n )$. Then $≅ ( ⁢ C K h ( G ) , d ϵ ) ( ⁢ C K h ( G ) , d ϵ ′ )$.

Proof.

The hypercube ${ 0 , 1 } n$ is a simplicial complex. We consider the edge assignments $ϵ$ and $ϵ ′$ as $1$-cochains in $Hom ( C 1 , F 2 )$ where $C 1$ is the space of $1$-chains and $F 2$ 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 $⋅ ϵ ϵ ′$ is a $1$-cocycle. Because the hypercube is contractible, the product of the edge assignments $⋅ ϵ ϵ ′$ is the coboundary of a $0$-cochain, that is there exists $: η → ⁢ V ( n ) { ± 1 }$ such that $= ⁢ η ( I ) η ( J ) ⁢ ϵ ( ξ ) ϵ ′ ( ξ )$ if $ξ$ is an edge between vertices $I$ and $J$.

Let $: ψ → ( ⁢ C K h ( G ) , d ϵ ) ( ⁢ C K h ( G ) , d ϵ ′ )$ be the map which when restricted to $⁢ V ( ⁢ G ( I ) )$ is multiplication by $⁢ η ( I )$. Then $ψ$ is an isomorphism from $( C K h ( G ) , d ϵ ) → ( C K h ( G , d ϵ ′ )$. ∎

3.2 subsection 3.2 3.2 §3.2 <tag close=". ">3.2</tag>Reduced homology 3.2Reduced homology

In the construction of $⁢ C K h ( G )$, one associates a tensor factor of $V$ to each boundary component of $Σ ⁢ G ( I )$. Suppose that there is a marked point on the boundary of a vertex of $Σ G$ that misses the bands attached for each edge. Let $⁢ ~ F ( ⁢ G ( I ) )$ denote the set of boundary components of $Σ ⁢ G ( I )$ without marked points. Note that $= | ⁢ ~ F ( ⁢ G ( I ) ) | - | ⁢ F ( ⁢ G ( I ) ) | 1$. For each vertex $∈ I ⁢ V ( n )$, one can consider $⁢ V ( ⁢ G ( I ) )$ as $= ⊗ V V ⊗ ⁢ ~ F ( ⁢ G ( I ) ) ⊕ ( ⊗ ⁢ Z v + V ⊗ ⁢ ~ F ( ⁢ G ( I ) ) ) ( ⊗ ⁢ Z v - V ⊗ ⁢ ~ F ( ⁢ G ( I ) ) )$, where the $⁢ Z v +$ and $⁢ Z v -$ are the two summands of $V$ associated to the boundary component of $Σ ⁢ G ( I )$ that contains the marked point. Define $= ⁢ ~ V ( ⁢ G ( I ) ) ⁢ ( ⊗ ⁢ Z v - V ⊗ ⁢ ~ F ( ⁢ G ( I ) ) ) { 1 }$ where as before, the $⁢ Z v -$ corresponds to the boundary component of $Σ ⁢ G ( I )$ that contains the marked point.

Let $= ⁢ ~ ⁢ C K h ( G ) ⊕ ∈ I ⁢ V ( n ) ⁢ ~ V ( ⁢ G ( I ) )$, and define $:= ~ d d | ⁢ ~ ⁢ C K h ( G )$. Since the range of $~ d$ is a subset of $⁢ ~ ⁢ C K h ( G )$, it follows that $( ⁢ ~ ⁢ C K h ( G ) , ~ d )$ forms a chain complex. The homology of this chain complex $⁢ ~ ⁢ K h ( G )$ is called the reduced Khovanov homology of $G$.

3.3 subsection 3.3 3.3 §3.3 <tag close=". ">3.3</tag>Homology of the dual ribbon graph 3.3Homology of the dual ribbon graph

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 $: f → M N$ is a $Z$-module homomorphism, then the dual homomorphism $: f * → M * N *$ is defined by $= ⁢ f * ( ϕ ) ∘ ϕ f$. Let $( C , ∂ )$ denote the complex

$⁢ ⋯ C i ∂ i → C + i 1 → ⋯ .$

The dual complex $( C * , ∂ * )$ is the complex where $= ( C * ) i ( C - i ) *$ and $( ∂ * ) i$ is the dual of $∂ - - i 1$. When there is a polynomial grading on $C$ that $∂$ preserves (as is the case with the Khovanov homology defined above), define $C *$ to have the opposite polynomial grading, i.e. $= ( C * ) i , j ( C - i , - j ) *$.

Proposition 3.2 3.2 3.2 Proposition 3.2 <tag><text font="bold">Proposition 3.2</text></tag>.

Let $G$ be a ribbon graph with $n$ edges, and let $G *$ be the dual ribbon graph. The Khovanov complex of $G *$ is isomorphic to the dual of the Khovanov complex of $G$, that is

$≅ ⁢ C K h ( G * ) ⁢ C K h ( G ) * [ n ] { n } and$ $⁢ C K h ( G * )$ $≅$ $⁢ C K h ( G ) * [ n ] { n } and$ $≅ ⁢ ~ ⁢ C K h ( G * ) ⁢ ~ ⁢ C K h ( G ) * [ n ] { n } .$ $⁢ ~ ⁢ C K h ( G * )$ $≅$ $⁢ ~ ⁢ C K h ( G ) * [ n ] { n } .$
Proof.

We prove the proposition for $⁢ C K h ( G * )$; the result for $⁢ ~ ⁢ C K h ( G * )$ 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 $= I ( m 1 , … , m n )$ is a vertex in $⁢ V ( n )$, define its dual vertex $= ^ I ( ^ m 1 , … , ^ m n )$ in $⁢ ^ V ( n )$ to be the vertex where $≡ + m i ^ m i mod 1 2$ for $1 ≤ i ≤ n$. The complexes $⁢ C K h ( G )$ and $⁢ C K h ( G * )$ will use the hypercube ${ 0 , 1 } n$, while the complex $⁢ C K h ( G ) *$ will use the dual hypercube ${ ^ 0 , 1 } n$.

First, we show that for each vertex $∈ I ⁢ V ( n )$, we have a grading preserving isomorphism

$: ∘ Ψ Φ * ≅ → ⁢ V ( ⁢ G * ( I ) ) ⁢ V ( ⁢ G ( ^ I ) ) * [ n ] { n } .$

Next we show that if $ξ$ is an edge in $⁢ E ( n )$ from $I$ to $J$ and $^ ξ$ is the dual edge in $⁢ ^ E ( n )$ from $^ I$ to $^ J$, then the edge maps $: d ξ → ⁢ V ( ⁢ G * ( I ) ) ⁢ V ( ⁢ G * ( J ) )$ and $: d ξ * → ⁢ V ( ⁢ G ( ^ I ) ) * ⁢ V ( ⁢ G ( ^ J ) ) *$ commute with $∘ Ψ Φ *$ and $( ∘ Ψ Φ * ) - 1$. 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

$= ⁢ v - * ( v - ) 0 , = ⁢ v - * ( v + ) 1 , = ⁢ v + * ( v - ) 1 , = ⁢ v + * ( v + ) 0 .$

Fix an isomorphism $: ψ → V * V$ where $= ⁢ ψ ( v - * ) v -$ and $= ⁢ ψ ( v + * ) v +$, and define an isomorphism $: Ψ → ⁢ V ( ⁢ G ( I ) ) * ⁢ V ( ⁢ G ( I ) )$ by $= Ψ ⊗ ψ ⋯ ψ$. The map $Ψ$ sends summands in the $( i , j )$-bigrading of $⁢ C K h ( G ) *$ to the $( - i , - j )$-bigrading of $⁢ C K h ( G )$. As noted in Section , there is a canonical bijection $Φ$ from the boundary components of $Σ ⁢ G * ( I )$ to the boundary components of $Σ ⁢ G ( ^ I )$ given by the gluing map in $= Σ ∪ Σ ⁢ G * ( I ) Σ ⁢ G ( ^ I )$. The bijection induces an isomorphism $: Φ * → ⁢ V ( ⁢ G * ( I ) ) ⁢ V ( ⁢ G ( ^ I ) )$ that sends the summand in the $( i , j )$-bigrading of $⁢ C K h ( G )$ to the $( - n i , - n j )$-bigrading of $⁢ C K h ( G * )$. The composition $: ∘ Ψ Φ * ≅ → ⁢ V ( ⁢ G * ( I ) ) ⁢ V ( ⁢ G ( ^ I ) ) * [ n ] { n }$ is the desired isomorphism.

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

$: m * → V * ⊗ V * V * : m * { ↦ v + * + ⊗ v + * v - * ⊗ v - * v + * ↦ v - * ⊗ v - * v - * : Δ * → ⊗ V * V * V * : Δ * { ↦ ⊗ v + * v - * v - * ↦ ⊗ v + * v + * v + * ↦ ⊗ v - * v + * v - * ↦ ⊗ v - * v - * 0 .$

Let $d ξ *$ be the edge maps in the dual complex defined using $m *$ and $Δ *$. Since $= ⁢ m * ( ⊗ v * ± v * ± ) ( ⁢ m ( ⊗ v ± v ± ) ) *$ and $= ⁢ Δ * ( v * ± ) ( ⁢ Δ ( v ± ) ) *$, it follows that $= ∘ Ψ Φ * d ξ ∘ d ξ * Ψ Φ *$.

An edge assignment $: ϵ → ⁢ E ( n ) { ± 1 }$ gives an edge assignment $: ^ ϵ → ⁢ ^ E ( n ) { ± 1 }$ by $:= ⁢ ^ ϵ ( ^ ξ ) ⁢ ϵ ( ξ )$. Therefore, up to the prescribed grading shift, the complexes $( ⁢ C K h ( G * ) , d ϵ )$ and $( ⁢ C K h ( G ) * , d * ^ ϵ )$ 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.

Corollary 3.3 3.3 3.3 Corollary 3.3 <tag><text font="bold">Corollary 3.3</text></tag>.

Let $G$ be a ribbon graph with $n$ edges, and let $G *$ be the dual ribbon graph. There are isomorphisms

$≅ ⊗ ⁢ K h i , j ( G * ) Q ⊗ ⁢ K h - n i , - n j ( G ) Q$ $⊗ ⁢ K h i , j ( G * ) Q$ $≅$ $⊗ ⁢ K h - n i , - n j ( G ) Q$ $≅ ⁢ Tor ( ⁢ K h i , j ( G * ) ) ⁢ Tor ( ⁢ K h + - n i 1 , + - n j 1 ( G ) )$ $⁢ Tor ( ⁢ K h i , j ( G * ) )$ $≅$ $⁢ Tor ( ⁢ K h + - n i 1 , + - n j 1 ( G ) )$ $≅ ⊗ ⁢ ~ ⁢ K h i , j ( G * ) Q ⊗ ⁢ ~ ⁢ K h - n i , - n j ( G ) Q$ $⊗ ⁢ ~ ⁢ K h i , j ( G * ) Q$ $≅$ $⊗ ⁢ ~ ⁢ K h - n i , - n j ( G ) Q$ $≅ ⁢ Tor ( ⁢ ~ ⁢ K h i , j ( G * ) ) ⁢ Tor ( ⁢ ~ ⁢ K h + - n i 1 , + - n j 1 ( G ) ) .$ $⁢ Tor ( ⁢ ~ ⁢ K h$