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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!

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  • $\begingroup$ I've posted an answer, but this strikes me as more of a math.SE question- also, note that googling "virtual Reidemeister move" would have answered the question. $\endgroup$
    – Daniel Moskovich
    Aug 9, 2012 at 20:06
  • $\begingroup$ If you remove Reidemeister I you get the theory of framed knots. Deciding which moves to include determines which objects you are studying. $\endgroup$
    – Daniel Moskovich
    Aug 9, 2012 at 20:11

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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 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) allows one forbidden move but not the other.

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    $\begingroup$ Thanks! ''Moving something across a virtual crossing makes no sense", which made me understand geometrically! $\endgroup$
    – muta yasushi
    Aug 9, 2012 at 21:46
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    $\begingroup$ @Daniel, I think that the idea of welded knots originated with Roger Fenn and/or Colin Rourke. Welded knots give rise to knotted ribbon tori in 4-space. This idea was found by Silver and Shin Satoh, but it is no surprise given that welded knots are related to the mapping class group of circles in space. $\endgroup$
    – Scott Carter
    Aug 9, 2012 at 23:45

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