A graphical construction of the sl(3) invariant for virtual knots

Abstract

We construct a graph-valued analogue of the Homypt sl(3) invariant for virtual knots. The restriction of this invariant for classical knots coincides with the usual Homypt sl(3) invariant, and for virtual knots and graphs it provides new information that allows one to prove minimality theorems and to construct new invariants for free knots. A novel feature of this approach is that some knots are of sufficient complexity that they evaluate themselves in the sense that the invariant is the knot itself seen as a combinatorial structure.