Virtual knot diagrams equivalent by second Reidemeister moves

Question

Is there a way to decide if two virtual knot diagrams are equivalent using only second Reidemeister moves and virtual moves (i.e. the Reidemeister moves where at least one crossing is virtual)? Some algorithm?

Answer 1

Yes: reduce the diagram to a form where no destructive RII move is possible.

Claim: that reduced form uniquely determines the equivalence class of the knot.

Proof (sketch): think in terms of Gauss diagrams, which allows you to forget about the virtual moves since these don't change the diagram by definition. From the reduced form defined above, add crossings via a sequence of creative RII moves. After the sequence, if you want to perform a destructive move, the only option is to use at least one arrow that was just created and therefore the other arrow used is either the very one that was created simultaneously, or another one "parallel" to that. In any case the only option is to undo one of the moves that was just performed.