### explanation of parts as relates to diagrams

The goal of these four pages is to find a way to define the most simple knot in its most simple form-but this is done in a different space than it is used to being defined in. This may open up new moves which we could apply to more complex knots.

In ordinary maps two points may be mapped to a single point, and then we have the inverse “mapping”: a single point may be mapped back to both different points.

But here we want to be able to work with knot diagrams defined in a space made out of “parts” and placements and not points. “Parts” and placements are discussed elsewhere on this website. By parts we mean locations.

The central idea is that instead of points which combine by coincidence (+) into a single point, parts can combine by joining(*). So that two parts brought together as in two lines of parts coming together to form a joining(*), may be written as for example (e(1,1)*e(1,2)). Meaning at one placement we have the first part e(1,1) and the second part e(1,2) sharing the placement.

In usual geometry a point has no extent and no parts or pieces. Since it has no parts or pieces it can’t be separated or divided into two or more parts or pieces. If we place two points together by moving two lines together until we have a coincident (+) point, we superimpose two wholes with no parts or pieces and the result is a single whole with no parts or pieces.

But in the geometry of placements and parts we have another entity “e” which has no extent and no parts or pieces and is also a whole. E is a group of moving locations in the space of placements and locations.

“e” is different from a point p, in the usual geometry in that we can place two e’s together by joining(*) and have the result not be a single whole-like with points-but two e’s in the same placement-similar to two overlapping shadows. “e” has no extent like a point p, but is not like a point p-we regard it as being a part of a combination as in e*e as well as a whole in itself.

The two e’s would have to be distinguished in some other way then their placement. “e’s” and points are different in this manner. P-type parts are also different from points, in the same manner and are usually fixed in the space of placements and locations.

There is a subtle difference between division and separation here. By division we mean the ability to form two or more entities like the original, from the original, smaller than the original. We therefore cannot divide a point or the entity (e(1,1)*e(1,2)) into pieces. But we can separate (e(1,1)*e(1,2)) into two parts. Then there is the difference between parts and pieces.

We let p denote usually fixed parts and e denote moving parts. e’s are not distinguished by their placement, but can be distinguished by some other rule. In the case of knot diagrams the first number indicates which location we are at and the second number indicates whether the e was originally below, 1 or above, 2.

So we start then with a three dimensional space which consists of placements and parts. We can first define a mapping of a trefoil knot in three-dimensional space of regular points to the one in the space of placements and parts. Then I can move the e’s of the trefoil until three sets of two are joining at the placements shown in the diagram close to the center of the page.