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.

I have a new student through ‘ Tutor by Numbers’ he is in grade 12 and is trying to get into Durham College for Automotive Engineering. I have met with him 4 times and will likely be reviewing with him in the summer.

1. Instead of changing to l*l space before creating the joining, we can also change to it as we flatten the crossings. There can be a two to two map from two locations in 3 space to two locations together at an l*l plane. Also we can consider an entire diagram. Diagrams 4 and 5. I use the double crossing loop as an example.

2.This creates a new joining type Z(n) n=1,2,3..

3.The advantage of doing this is that now we can say that we are at the level l*l but we can take advantage of the way movements of points around the double self-intersecting loop are defined in l. Since points and locations are both zero dimensional.

We can have a type of movement of locations 1+ and 1- around the loop while the locations 2+ and 2- at Z(2) stay fixed. Locations move through locations.

4. In particular I can separate Z(1) into 1+, 1- and D(1) while keeping Z(2) as is. Diagram 6.

5. 1+ and 1- can then move through Z(2).

1. Start with a crossing in regular 3 space. Diagram 1.

2. By “freezing” the crossing in regular space (the dotted lines) we can then move the crossing to l*l (location of location) space. The locations still cross, one over the other. Diagram 2.

3. We can move the two strands of the crossing to intersect at the “joining” D. Here at the vertex at the location of the locations, we have the two locations. We know which location is associated with which strand of the crossing. Diagram 3. 1+ and 1- represent two parts of the original crossing.

If we “freeze” the knot in usual space then we can move it in l*l space.

There are new crossing types. I can have two other deeper types of diagrams.

This is the creation of Z joining. If we have a crossing in l and a crossing in l*l then we can have a Z joining. If we don’t have a crossing in l but do have one in l*l we have a Q joining.

This is the description of a D joining coming from a Z joining.

I thought it might be simpler to access the third level l*l*l then the alpha and beta parts could separate out. Then there would be three levels. Since we don’t label anymore once we reach the circular form, it must be that there is only a rearrangement of the elements which leads to the two different representations.