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July 10, 2016

Tutorial for knot idea outline

Filed under: knot idea outline — Rob @ 3:58 pm

IMG_20160710_0001The goal of this post is to show a few diagrams related to the knot idea outline and to use them to further explain and clarify the outline.

The goal of the outline is to come up with a new knot diagram given the new whole “e”. “e” can be thought of as the group of locations in a space of locations and placements which are allowed to move through the placements. The circle of e’s is in joining with a circle of p-type parts in joining with the plane of placements.  I can create a geometry in which we have only one circle of moving locations. The rest of the locations can be “fixed” for now.

We have that we can bring two e’s together to form e*e. But perhaps this can be talked about in detail. I can start with a space of p-type parts “p” and “e”. “p” can be thought of as the group of locations which is fixed in a space of locations and placements. “e” are those locations which are allowed to move through the placements.

 

At the joining let e(1,1) and e(1,2) not represent one specific location each, but let them represent any locations which could be brought together at that placement-p(0,0). Keep the twist fixed otherwise-just rotate the diagram upon itself.

Then e(1,1)*e(1,2) is thought of as a “general” joining. We can separate it if we ‘freeze’ the diagram.

Then what is left when we take e(1,1) and e(1,2) away from the central joining is a joining of two parts which do not join in the original diagram. This is an ‘altered’ diagram. See the diagram at the bottom of the page with printing.

Before the diagram is frozen, the strands move so that crossings move along as expected with the moving strands, but after the diagram is frozen, the crossings are separated. We move the coincident p-type parts along with the e-type parts.

The diagram then ‘shifts’ instead of ‘moves’. Anytime the strands are then brought together there is a joining of two parts which do not join in the original diagram.

Usually with ordinary geometry points move through locations.

In placement and parts geometry there is an opportunity to perform both ‘moves’ and ‘shifts’

In a ‘move’ e-type parts move through fixed p-type parts placements and this is similar to a move in the theory of knots.

In a ‘shift’ the diagram is first ‘frozen’ then the combination of both p and e-type parts which make up the diagram are both moved together, leaving a void-set of placements.

 

 

 

 

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The benefit of this is then to be able to “unfold” a diagram into a new circle.

I can move e(1,1) and e(1,2) away from the center along the strands, as shown in the next diagram at left.

This type of motion can be chosen from the possible ways of separating. If we look at the diagram at the bottom of the page  with 1+ and 1-. We can notate e(1,1) as 1- and e(1,2) as 1+. As we move any part through a joining we move them in the way that they go through in a straight line. They don’t have to, as we could go off to the left as well. But for these diagrams we go straight through.

Bringing these ideas together, The self-joining loop can be separated as shown in the diagrams for it. As well then the trefoil knot can be started to be separated as shown in the diagram below the self-joining loop.

 

 

 

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