  We construct graph-valued analogues of the Kuperberg sl(3) and G2 invariants
for virtual knots. The restriction of the sl(3) or G2 invariants for classical
knots coincides with the usual Homflypt sl(3) invariant and G2 invariants. For
virtual knots and graphs these invariants provide new graphical information
that allows one to prove minimality theorems and to construct new invariants
for free knots (unoriented and unlabeled Gauss codes taken up to abstract
Reidemeister moves). 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. The paper
generalizes these structures to virtual braids and discusses the relationship
with the original Penrose bracket for graph colorings.
