I am studying virtual knot theory.
A virtual knot is a knot diagram with real or virtual crossing information.
The equivalence relation includes generalized Reidemeister moves.
There are premitted virtual Reidemeister moves, and forbbiden moves.


Question


Do the forbidden moves have some validity?
How are generalized Reidemeister moves made? 
For example, If we admit the moves the virtual knot is trivial.
Therefore, we make some of the moves forbidden.
The moves have two real crossings and one virtual crossing.
However, instead of these moves, why the other moves are not selected?
For instance, may we exclude classical Reidemeister I move instead of the forbidden moves?
I am confused!



Validity of generalized Reidemeister moves for a virtual knot
A virtual knot is the same thing as a Gauss diagram. The virtual crossing doesn't really exist. So moving something across it makes no sense. Another way of seeing this is that, given that the object of study is Gauss diagrams modulo Reidemeister 1, 2, and 3, the virtual Reidemeister moves don't change the Gauss diagram, and therefore are valid moves on Gauss diagrams modulo Reidemeister moves; but the forbidden moves do.


The [first google result](http://www.esotericka.org/cmc/vknots.html) on "virtual Reidemeister moves" fleshes this out.


Of course you could include one of the forbidden moves- and you would get a different theory. The theory of w-knots (see Dror Bar-Natan's [talks on the topic](http://www.math.toronto.edu/drorbn/Talks/)) allows one forbidden move but not the other.


