In this paper, new operations which generalize virtualization are introduced. The operation of virtualization on a virtual or welded link diagram consists in replacing a classical crossing by a virtual one. This is equivalent, under Reidemeister moves, to taking two arcs which do not cross one another and adding a virtual crossing together with a classical crossing in the other direction.
The generalization introduced in this paper is to add additional classical twists instead of just a single classical crossing. In the version denoted by \(V(n)\), \(n\) classical half-twists are added after the virtual crossing. In the version denoted by \(V^n\), \(n\) classical half twists are added with a virtual crossing in between each of them.
It is a well-known result that any virtual diagram is equivalent to an unlink under virtualization. One of the main results in this paper involves an invariant \(\lambda_{ij}(L)\), with the property that:
Theorem 1. If \(n\) is even, then any welded link is equivalent to an unlink under \(V(n)\) operations. If \(n\) is odd, then two welded link diagrams are equivalent under \(V(n)\) operations if and only if they have the same \(\lambda_{ij}\) invariant modulo \(n\).
The paper also shows that
Theorem 2. \(V^n\) is not an unknotting operation for welded knots \(n\geq 2\).
The bulk of the proofs utilize the arrow calculus, which involves using arrows between arcs to indicate classical crossings. This is rather like the Gauss code presentation of a virtual link diagram, and it seems likely that these results might be obtained by using Gauss codes directly.
The bulk of this paper depends upon the arrow calculus. For more information on the arrow calculus, see [J.-B. Meilhan and A. Yasuhara, Algebr. Geom. Topol. 19, No. 1, 397–456 (2019; Zbl 1419.57019)].