Introduction to Relations between Trivalent Graphs and Virtual Knot Theory

This talk will begin by discussing the Penrose formula for counting the number of proper edge three colorings of a trivalent (planar) graph and its relationships with both perfect matchings of trivalent graphs and with structures in virtual knot theory. We then build a natural functor from the category of virtual trivalent graphs and the category of virtual knot and link diagrams, showing how virtual link invariants inform the structure of the graphs and how the structure of the graphs informs the virtual knot theory. This is joint work with Scott Baldridge, Ben McCarty and William Rushworth.