# Interactive Online Tutoring Services

## June 10, 2017

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).

### Clarification

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.

## May 27, 2017

### Page 3

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.

## May 22, 2017

## May 16, 2017

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.