Khovanov-Kauffman Homology for embedded Graphs
A discussion given to the question of extending Khovanov homology from links to embedded graphs, by using the Kauffman topological invariant of embedded graphs by associating family of links and knots to a such graph by using some local replacements at each vertex in the graph. This new concept of Khovanov-Kauffman homology of an embedded graph constructed to be the sum of the Khovanov homologies of all the links and knots associated to this graph.
Key words and phrases:Khovanov homology, embedded graphs, Kauffman replacements, graph homology
The idea of categorification the Jones polynomial is known by
Khovanov Homology for links which is a new link invariant introduced by
Khovanov , . For each link in he defined a graded chain complex,
with grading preserving differentials, whose graded Euler characteristic is
equal to the Jones polynomial of the link . The idea of Khovanov Homology
for graphs arises from the same idea of Khovanov homology for links by
the categorifications the chromatic polynomial of graphs.
This was done by L. Helme-Guizon and Y. Rong ,
for each graph G, they defined a graded chain complex whose graded
Euler characteristic is equal to the chromatic polynomial of G.
In our work we try to recall, the Khovanov homology for links.
We discuss the question of extending Khovanov homology from links to embedded graphs. This is based on a result of Kauffman that constructs a topological invariant of embedded graphs in the 3-sphere by associating to such a graph a family of links and knots obtained using some local replacements at each vertex in the graph. He showed that it is a topological invariant by showing that the resulting knot and link types in the family thus constructed are invariant under a set of Reidemeister moves for embedded graphs that determine the ambient isotopy class of the embedded graphs. We build on this idea and simply define the Khovanov homology of an embedded graph to be the sum of the Khovanov homologies of all the links and knots in the Kauffman invariant associated to this graph. Since this family of links and knots is a topologically invariant, so is the Khovanov-Kauffman homology of embedded graphs defined in this manner. We close this paper by giving an example of computation of Khovanov-Kauffman homology for an embedded graph using this definition.
Acknowledgements: The author would like to express his deeply grateful to Prof.Matilde Marcolli for her advices and numerous fruitful discussions. He is also thankful to Louis H. Kauffman and Mikhail Khovanov for their support words and advices. Some parts of this paper done in Max-Planck Institut für Mathematik (MPIM), Bonn, Germany, the author is kindly would like to thank MPIM for their hosting and subsidy during his study there.
2. Khovanov Homology
In the following we recall a homology theory for knots and links embedded in the
3-sphere. We discuss later how to extend it to the case of embedded
2.1. Khovanov Homology for links
In recent years, many papers have appeared that discuss properties of Khovanov Homology theory, which was introduced in . For each link , Khovanov constructed a bi-graded chain complex associated with the diagram for this link and applied homology to get a group , whose Euler characteristic is the normalized Jones polynomial.
He also proved that, given two diagrams and for the same link, the corresponding chain complexes are chain equivalent, hence, their homology groups are isomorphic. Thus, Khovanov homology is a link invariant.
2.2. The Link Cube
Let be a link with crossings. At any small neighborhood of a crossing we can
replace the crossing by a pair of parallel arcs and this operation is called a resolution.
There are two types of these resolutions called -resolution (Horizontal resolution) and
-resolution (Vertical resolution) as illustrated in figure (1).
We can construct a -dimensional cube by applying the and -resolutions times to each crossing to get pictures called smoothings (which are one dimensional manifolds) . Each of these can be indexed by a word of zeros and ones, i.e. . Let be an edge of the cube between two smoothings and , where and are identical smoothings except for a small neighborhood around the crossing that changes from to -resolution. To each edge we can assign a cobordism (orientable surface whose boundary is the union of the circles in the smoothing at either end)
This is a product cobordism except in the neighborhood of the crossing, where it is the obvious saddle cobordism between the and -resolutions. Khovanov constructed a complex by applying a -dimensional TQFT (Topological Quantum Field Theory) which is a monoidal functor, by replacing each vertex by a graded vector space and each edge (cobordism) by a linear map , and we set the group to be the direct sum of the graded vector spaces for all the vertices and the differential on the summand is a sum of the maps for all edges such that Tail() i.e.
where and is chosen such that .
An element of is said to have homological grading and -grading where
for all , is the number of 1’s in , and , represent the number of negative and positive crossings respectively in the diagram .
If is a diagram obtained from by the application of a Reidemeister moves then the complexes and are homotopy equivalent.
For an oriented link with diagram D, the graded Euler characteristic satisfies
where is the normalized Jones Polynomials for a link and
Let and be two links with odd and even number of components then and
For two oriented link diagrams and , the chain complex of the disjoint union is given by
For two oriented links and , the Khovanov homology of the disjoint union satisfies
Let be an oriented link diagram of a link with mirror image diagram of the mirror link . Then the chain complex is isomorphic to the dual of and
3. Khovanov-Kauffman Homology for Embedded Graphs (KKh(G))
3.1. Kauffman’s invariant of Graphs
We give now a survey of the Kauffman theory and show how to associate to an embedded
graph in a family of knots and links. We then use these results to give our definition
of Khovanov homology for embedded graphs.
In  Kauffman introduced a method for producing topological invariants of graphs
embedded in . The idea is to associate a collection of knots and links to a graph so that
this family is an invariant under the expanded Reidemeister moves defined by Kauffman and
reported here in figure (2).
He defined in his work an ambient isotopy for non-rigid (topological) vertices. (Physically, the rigid vertex concept corresponds to a network of rigid disks each with (four) flexible tubes or strings emanating from it.) Kauffman proved that piecewise linear ambient isotopies of embedded graphs in correspond to a sequence of generalized Reidemeister moves for planar diagrams of the embedded graphs.
Let be a graph embedded in . The procedure described by Kauffman of how to associate to a family of knots and links prescribes that we should make a local replacement as in figure (3) to each vertex in . Such a replacement at a vertex connects two edges and isolates all other edges at that vertex, leaving them as free ends. Let denote the link formed by the closed curves formed by this process at a vertex . One retains the link , while eliminating all the remaining unknotted arcs. Define then to be the family of the links for all possible replacement choices,
For example see figure (4).
 Let be any graph embedded in , and presented diagrammatically. Then the family of knots and links , taken up to ambient isotopy, is a topological invariant of .
For example, in the figure (4) the graph is not ambient isotopic to the graph , since contains a non-trivial link.
3.2. Definition of Khovanov-Kauffman Homology for Embedded Graphs
Now we are ready to speak about a new concept of Khovanov-Kauffman homology for embedded graphs by using Khovanov homology for the links (knots) and Kauffman theory of associate a family of links to an embedded graph , as described above.
Let be an embedded graph with the family of links associated to by the Kauffman procedure. Let be the usual Khovanov homology of the link in this family. Then the Khovanov-Kauffman homology for the embedded graph is given by
Its graded Euler characteristic is the sum of the graded Euler characteristics of the Khovanov homology of each link, i.e. the sum of the Jones polynomials,
We show some simple explicit examples.
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