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January 8, 2021

Filed under: knots,Mathematics — Rob @ 3:39 pm

This page is showing the different moves possible in the two different spaces. The top space is R^3 and the bottom space is placement space (P^3). Placement space is described in the posts below. The idea that leads to placement space is described in the quick view, below.

There is a link between R^3 and P^3. Also in P^3 we can create a D-sharing- a type of crossing. In turns out in placement space there are two types of basic crossings. I’ve labelled these D and Q. These are shown in some detail in the diagrams below.

Applying moves S1,S2,S3 counterparts to the Reidemeister moves to take us from one diagram to the next. I use Reidemeister moves to go from one diagram to the next, but do not use these to prove knottedness or chirality.

These moves all involve Q-sharings only and I’ve drawn them above. In order that I only apply these moves to Q-sharings a transformation is applied to the trefoil. This transformation is described in the posts below. I’ve shown it in a diagram above.

Then these S1,S2,S3 have counterparts in the space R^3 above. This then leads us to a diagram with the same shape as the original, but with possibly an unknotted case or a mirror image case.

To rule these out we put back in the cyclic permutable set {a(1)r(1),b(2)s(2),a(3)r(3),b(1)s(1),a(2)b(2),b(3)s(3)} which was removed during the transformation in placement space. It might be possible to obtain a mirror image or an unknot because I am applying the moves generally.

That is I allow all possible paths by Reidemeister moves which lead back to the diagram with the same shape. It turns out I can only reform the original configuration, not the unknotted or mirror image case.

For future work I can say that the shape of the knot need not be one shape alone, also in larger knots such as the Perko pair, I can remove the D-sharings and put new D-sharings in, changing the writhe of the knot which is the number of positive crossings minus the number of negative crossings.

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