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June 25, 2017

Filed under: clarification of demonstration — Rob @ 6:16 pm
1.In the first diagram I am showing how the crossings in a knot diagram could be put into l*l space. But seeing this is complex, I thought I could reduce the complexity.
2.In the second diagram, I’m showing that we can start instead with a self-intersecting loop with one location at the center in l*l space and then add a location, by moving one from somewhere close to the intersection and bringing it into joining there with the location that is already there.. This could be done with other such diagrams as shown.
3.In the third diagram I am showing  what I am using as an example with the joining’s labelled 1,2,3.
4.In diagram four I am showing the decomposition of this diagram, moving the locations of locations. 1-,2-,3- can be the locations that come from the outside and the +’s can come from the diagrams.
We have a condition of uncertainty with 1+,1-,2+,2-,3+,3-. We can assign the two parts of 1 to the two strands which go through 1, but the way it is created we can say that it is uncertain which part belongs to which strand. The same can be said of the other labels 2 and 3. We separate the joining pieces bringing into joining parts which do not join in the original diagram, by choosing one specific separation from the two available for each joining at random.
5.The fifth diagram shows that the final decomposition can be reformed in two more possible ways.
6.In the sixth diagram I show that the final decomposition can be put together in a way that leads to a new “shape”.
In the case shown the diagram is folded up in a different way. There maybe other self-transformations leading to new shapes which may be possible with more complex diagrams.

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