Narrow Results

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.

Nelson and Kanenobu showed that forbidden moves unknot any virtual knot. Similarly a long virtual knot can be unknotted by a finite sequence of forbidden moves. Goussarov, Polyak and Viro introduced finite type invariants of virtual knots and long virtual knots and gave combinatorial representations of finite type invariants. We introduce Fⁿ-moves which generalize the forbidden moves. Assume that two long virtual knots K and K′ are related by a finite sequence of Fⁿ-moves. We show that the values of the finite type invariants of degree 2 of K and K′ are congruent modulo n and give a lower bound for the number of Fⁿ-moves needed to transform K to K′.

Twisted knot theory, introduced by Bourgoin, is a generalization of virtual knot theory. It is well known that any virtual knot can be deformed into a trivial knot by a finite sequence of generalized Reidemeister moves and two forbidden moves $F1$ and $F2$. Similarly, we show that any twisted knot also can be deformed into a trivial knot or a trivial knot with a bar by a finite sequence of extended Reidemeister moves and three forbidden moves $T4$, $F1$ (or $F2$) and $F3$ (or $F4$).

In the theory of topology of words, the move to swap two adjacent letters in words is a simple operator that makes an $S$-homotopy class of a nanoword into the $S$-homotopy class of the empty word. In this paper, we study the relation between the swap move of letters and the forbidden moves of virtual diagrams, that is, local moves of virtual diagrams which unknot any virtual knot.