Khovanov-Kauffman Homology for embedded Graphs

TL;DR

This paper proposes a new homology theory for embedded graphs by extending Khovanov homology through associating links and knots to the graphs using local replacements, summing their homologies.

Contribution

It introduces the Khovanov-Kauffman homology for embedded graphs, combining link invariants with graph topology for the first time.

Findings

Defines the Khovanov-Kauffman homology for embedded graphs

Shows how to associate links and knots to graphs via local replacements

Constructs the homology as a sum of link and knot homologies

Abstract

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.