On the swap move of letters in words and the forbidden moves in virtual diagrams

Abstract

In the theory of topology of words, the move to swap two adjacent letters in words is a simple operator that makes an [Formula: see text]-homotopy class of a nanoword into the [Formula: see text]-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.