Narrow Results

Considering extremal properties of one polynomial of virtual knots, we establish estimates for virtual crossing numbers of virtual knots from a given class. This yields minimality of certain diagrams of virtual knots with respect to the virtual crossing number. Infinite series of pairwise distinct minimal virtual knot diagrams are constructed and their properties are discussed.

We construct a new invariant polynomial for long virtual knots. It is a generalization of the Alexander polynomial. We designate it as ζ by meaning an analogy with ζ-polynomial for virtual links. The degree of ζ-polynomial estimates the virtual crossing number. We describe some application of ζ-polynomial for the study of minimal long virtual diagrams with respect to the number of virtual crossings.