Khovano Homology and Embedded Graphs

TL;DR

This paper introduces a cobordism group for embedded graphs using fusion and fission operations, and extends Khovanov homology from links to embedded graphs via local replacements and associated link families.

Contribution

It constructs a new cobordism group for embedded graphs and proposes an extension of Khovanov homology to these graphs through local link replacements.

Findings

Defined a cobordism group for embedded graphs.

Extended Khovanov homology to embedded graphs.

Sum of Khovanov homologies of associated links.

Abstract

We construct a cobordism group for embedded graphs in two different ways, first by using sequences of two basic operations, called "fusion" and "fission", which in terms of cobordisms correspond to the basic cobordisms obtained by attaching or removing a 1-handle, and the other one by using the concept of a 2-complex surface with boundary is the union of these knots. 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 homology of an embedded graph constructed to be the sum of the Khovanov homologies of all the links and knots.