Graphical constructions for the sl(3), C₂ and G₂ invariants for virtual knots, virtual braids and free knots

Abstract

We construct graph-valued analogues of the Kuperberg sl(3) and G2 invariants for virtual knots. The restriction of the sl(3) and 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.

The generalized Yamada polynomials of virtual spatial graphs

“…In [19], degree 4 vertices are considered as special crossings, i.e., in a virtual graph, there are three types of transverse double points. However, Kauffman and Manturov in [12] use the term "virtual graph" to mean a 4-regular graph with virtual crossings only. We therefore note that from our point of view an acceptable usage is "virtual graph" for arbitrary cyclic graph (See Definition 2.8) with virtual crossings immersed in the plane.…”