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December 29, 2020

Quick view-abstract

Filed under: knots,Mathematics — Rob @ 9:29 pm

As a quick way to understand my starting point think of something which can be called two disjoint points (named by Kauffman), but not two points separated, but two points together. Then to have this I must first remove the notion of the single point, otherwise I would have both one and two points here. So some other structure is here. Then this is placement space. For a detailed explanation of this see the beginning of all the posts below.

What I’m trying to show is that in placement space P^3, I can move the joining pairs cyclicly and these can reform to make D-sharings. The joining pairs come from an original set of D-sharings. These two sets of D-sharings are related by a set of Reidemeister moves, in R^3 so these are equal knots.

Then for the Trefoil, I can start with the simple representition and show that there are no reformations that lead to the unknotted case and also do not lead to the mirror image, in the same shape, after any number of Reidemeister moves.

I also consider future work which could be done.

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