Unknotting the Trefoil

(Using the Forbidden Moves)

This is a demonstration of why the two "forbidden moves", two Reidemeister-like moves on virtual knot diagrams, are not allowable as virtual moves. In particular, we demonstrate that allowing the moves allows us to unknot the trefoil, showing that allowing these moves would render the theory non-viable as a generalization of ordinary knot theory.

Here we start with a diagram of the trefoil knot which looks like a figure-eight knot with one crossing switched:

The Trefoil

Next, we apply a type VII move, then our first instance of a forbidden move (moving a strand with two real crossings past a virtual crossing, marked in red):

The Trefoil <--VII--> by VII <--Fh--> by Fh (forbidden)

Next, we do another type VII move, followed by another forbidden move:

from above <--VII--> by VII <--Fh--> by Fh (forbidden)

Next, another type VII followed this time by a type V move:

from above <--VII--> by VII <--V--> by V

Now we can do a type VII to get rid of two virtual crossings, then do another type V move:

from above <--VII--> by VII <--V--> by V

Next, we get rid of another two virtual crossings with a type VII move, followed by another type VII:

from above <--VII--> by VII <--V--> by V

Next, we do a another type VII followed by another forbidden move:

from above <--VII--> by VII <--Ft--> by Ft (forbidden)

Now we remove a real crossing with a type I move, then do another type V move to get us into position for our last application of a forbidden move:

from above <--I--> by I <--V--> by V

Now our last forbidden move and the knot is actually unknotted, i.e. equivalent by legitimate moves to a simple uknotted circle with no crossings, real or virtual. Getting to the trivial diagram of the knot, however, will still take us a few steps, so we'll abbreviate in a couple of instances...

from above <--Ft--> by Ft (forbidden) <--I--> by I

Here we do two type VII moves and a type V:

from above <--VII x 2--> by VII x 2 <--V--> by V

Now we get rid of some more virtual crossings by two applications of type VII moves, then two type VI moves:

from above <--VII x 2--> by VII x 2 <--VI x 2--> by VI x 2

Getting closer, we use a type I move to get rid of another real crossing, then a type VII:

from above <--I--> by I <--VII--> by VII

Finally, a single type I move gets us to the trivial diagram of the unknot:

from above <--I--> by I

Whew! That was a fair amount of work. Fortunately, it's much simpler to do in terms of Gauss Diagrams:

Gauss Diagram of the 
Trefoil <-Fh x 2-> After two applications of 
one forbidden move <-Ft x 2-> After two 
more forbidden moves <-I x 4-> After four type 
I moves

Ordinary experience tells us that the trefoil is not the same as the unknot, and for virtual knot theory to be a true extension of knot theory it cannot allow the forbidden moves, since to do so would mean that as a virtual knot, the trefoil would be trivial and the theory using the forbidden move would not be a generalization of knot theory, but a different (and not very interesting) theory. Fortunately, virtual knot theory can and does make sense without allowing the forbidden moves.

Copyright © 2000 Sam Nelson