The Cycle Space of the Graph of Circles for a Resolution of a Generalized Cross Graph

Proving the mutation invariance of (odd) Khovanov homology a new construction of odd Khovanov homology for link diagrams was given. This construction uses the cycle space of the graph of circles for a resolution of a link diagram. The cycle space in this case was described. This construction was generalized for defining the odd Khovanov homology of graph-links. But it was not shown that the given relations indeed generate the cycle space of the graph of circles in the case of virtual link diagrams. We are eliminating this deficiency and describe the cycle space of the graph of circles for a resolution of any cross graph. Then, we generalize this construction for the case of generalized cross graphs and show that non-realizable graphs can have both all realizable cycle spaces of graphs of circles and a non-realizable one.