# Interactive Online Tutoring Services

## December 21, 2017

Filed under: clarification of demonstration — Rob @ 8:56 pm
1. I have to have all the elements back in place after a series of moves to go back to the second level from the third. Only joinings could be rearranged in that the order of separation could be altered (up or down at the joining)
2.  To get back to 3-space I need to move from z-joinings to normal crossings. So these have to be made up of 1+, 1-, 2+,2-,3+,3- but they don’t have to be in the same arrangement as they were formed. Then it is possible we could end up with the unknotted loop.

Filed under: clarification of demonstration,knots — Rob @ 8:33 pm
1.  I can duplicate a point in the plane.
2.  If I have a plane of locations, I can’t move a location to another place.
3.  Consider a plane of locations of locations co-existing with the plane of locations
4.  Then I can move one copy of the location into the second plane where it joins with the location that is already there. (We can keep the rest of the locations still).
5.  Then each point of the plane has a location and the location of the location of it’s duplicated point
6.  Suppose we have a set of connected locations which form a loop in 3-space. We can move from the plane to 3-space.
7.  Then we can imagine transforming to the second level and twisting the loop so that the loop joins itself in the centre. We have the location from the still locations in 3-space and the two moving locations joining at the centre.

## December 16, 2017

### The third level of location

It seems a feature of this geometry is that we can have two different locations at one location of locations.

So if a circle is moved to ‘intersect’ itself, I can separate the ‘intersection’ as there are two different locations at this ‘intersection’.

If these two locations are labelled as the ‘intersection we can move these two labels out along the path of the circle and bring

two locations together which are not the intersection.

But this has to separate into a different space than would the two locations A and B separate into a space of locations of locations.

Let the distance between A and B be zero and some distance at a third level. A and B then move in a space of locations of locations of locations.

Then D(1) is a joining of C and D. A virtual joining.

Since we are at the third level we move the diagram to the circle again by separating C and D at D(1) like A and B are separated at Z(1)