From Colorings to Virtual Links and Back - A Sequel

We review the formulation of graph colorings in terms of colored circuits (formations) and how it applies to a proof of the Penrose formula. We generalize the Penrose formula - once to obtain coloring counts for all cubic graphs - then to make a new perfect matching polynomial. The new perfect matching polynomial is mapped by a correspondence to virtual link theory to meet a generalized bracket polynomial. The appropriate virtual knot theory for this construction has two virtual crossings that detour over each other and over the classical crossings. A Reidemeister Two configuration with the two distinctvirtual crossings does not cancel. This form of Doubled Virtual Knot Theory is the correct image for the expanded Penrose evaluation and leads to new virtual invariants and new examples of virtualization phenomena between classical and virtual knot theory as well as new relationships between coloring theory and virtual knot theory.