### Third page of notes

The ideas of this page of notes follow from the previous page. b(1) is the joining of two p’s. But then we can also rotate the intersecting loop of locations around so that the locations of the loop occupy the same placements, but the locations change placements. That is the loop of locations rotates around and we define b(1) to be the same for all of these rotations.

The 1 refers to the self-intersecting loop being one for which half of the locations are above the intersection and half of the intersections are below the intersection. Call this now the placement b(1,1). b(1,1)=e(1)*e(2) where *=joining.

We can create a b(1,2) by shifting the self-intersecting loop slightly. Still half of the locations are above the intersection and half are below, yet we move the place where the loop intersects itself along just a little bit. Then also we can create b(1,3) and so on.

These loops can all be unfolded to each create a circle where we have the two e’s that create all of the different b’s. We can add all of these circles together by coincidence to get the first diagram on the left of the first page.

Then we can think of this as happening continuously too. So that we instead of moving a small distance to get from b(1,1) to b(1,2), we move in a continuous way.

Then next, we can think of this circle of e’s folded over once to intersect itself. Here we have c(1,1) but c must be a different idea than e. Let c be the idea that we sum all of the e’s in one place, once around the loop. Then we can have another c c(1,2) this will be another sum, in a slightly different order than c(1,1) but the sum will be the same.

Then if we unfold these and add, like we did previously we can obtain a circle of sums-all c’s. This one sum can also be placed at the center placement, so we can make a many to one map of a circle to a single placement.