Narrow Results

The first aim of this paper is to give an example of a virtual knot which vanishes all the known invariants and to show that it is a non-classical (in particular, non-trivial) knot. The second is to study some properties of the Z-polynomial of a virtual knot with virtual crossing number one and to show that there are infinitely many virtual knots with virtual crossing number two.

In this paper we prove a Markov theorem for virtual braids and for analogs of this structure including flat virtual braids and welded braids. The virtual braid group is the natural companion to the category of virtual knots, just as the Artin braid group is the natural companion to classical knots and links. In this paper we follow L-move methods to prove the Virtual Markov theorems. One benefit of this approach is a fully local algebraic formulation of the theorems in each category.