### Simpler explanation of knottedness using concept sharing

Concept Sharing and a new way to understand knots

Abstract: Here I introduce concept sharing. In uncovering extended space, I develop new ways of understanding knots.

Overlapping shadows:

An interesting notion can come about from something observed in nature, the overlapping of shadows. Consider a cup of tea placed on a table and lit with two lights from above, one from the left and another from the right, as illustrated below:

Three areas of shadows are formed. The one in the center is the overlapping of two shadows.

Consider two points intersecting from the areas of the overlapping shadows. We may make an analogy of two points being placed together with the overlapping of the shadows at the centre. Think of items you see around you, some touching other items, some like a book on a table, by the force of gravity. Others like books placed together on a shelf. Here is where two points of the different items, one from each item, can be seen as being together.

Now consider a point on one of the shadows which doesn’t intersect another shadow. This too has no extent. But as we’ve seen, something of no extent( two points placed together) can be expressed as the combination of two items which also have no extent( two separate points). Can this one point be seen as multiple as well as singular? Think of a hand of bananas. Each banana is connected to the others at the top of the hand.

Duplication also exists in nature, we can think of cell division.

Seeing this duplication with the shadows is possible. Consider two teacups and four lights. Creating sets of shadows as shown above.

The shaded area is where two shadows from teacup 1 and two shadows from teacup 2 intersect. Labeling points with “p”, I can associate p(1)(1) and p(1)(2) two duplicates and p(2)(1) and p(2)(2) the two other duplicates with this area. Then the original points p(1) and p(2) can be groups of these two. Then we may think p(1) and p(2) as two points which may be separated as in the analogy of the items touching each other but not p(1)(1) and p(1)(2) or p(2)(1) and p(2)(2).

In the case of a mathematical point, a duplication would have to be located in the same position.Since points are defined as having position only, a duplicate would have to take the same position. But I already have a concept of no extent without duplicates. If two duplicates are together it should lead to a single point. The usual case of a point is singular. So this is not a duplication in the usual sense.

Duplication exists in the form of concept sharing. The idea that concepts can be multiple. This duplicates the concept. Yet since the duplicate is identical we must remove the initial concept so there is room for this new concept.

The case of a point being only singular could be removed. Then I have room for this concept sharing idea. The concept sharing of point.

we must be able to then remove the concept which was there originally and replace it with the shared concept. It must be possible for shared concepts to exist somehow, as we can imagine it..

A point and its partner must be multiple so must take up the space of one point only. Yet we already have this notion of no extent without extra points.

All other possibilities for the number of shared concepts are removed.

There must be a “container” of concepts to take the concept out of. A concept of concept space. Here the two shared concepts can be seen as different. That is, I can separate them. I have another dimension of the concept, in concept space. So for points or places I have “places of places”

So I need the concept of a “concept container”

It is the same concept as the concept in question, yet it contains the concept in question. Meaning that it exists at a lower level so that the concept in question could be removed by two or more sharing concepts (equivalent to the concept in question).

We cannot have both the initial concept and sharing concepts at the same time. We need to remove the initial concept which means we need the lower level. Since we can always have the same concept as the concept in question and another dimension of the concept is possible since I can have the sharing of two or more of the initial concept and the concept can be extended using the extender “of” as in concept of concept. You can think of this as a nesting of concepts.

We can imagine that concepts can be multiple. Since it must be somehow possible to have sharing concepts, this lower level must also exist.

Akin to the overlapping shadows, concepts are not concrete-as they are abstract ideas, two or more of the same concept can co-exist. They share and fit into the concept which was there originally.

A symbol may represent two different concepts, such as “8” representing an amount, a measurement or a label.

In the case of points or places, the lower concept level is a place of places or location of locations. The two shared points may move off to two different places of places and we can then see how they can be made different (they can be defined at two different places of places).

When they separate, they do not leave a void, there must be a place of places. They may be separated as a place can take on a new place of places. See the diagram below. I am using a jagged line to show the opening into the new place of places space. This can be made two dimensional, leading to a new plane.

The concept sharing of a number:

Numbers are exact concepts. We can think of them like the shadows, in the sense that they have exact boundaries and some way of showing one was twice another, (in area) ect.. Then as in the notion of overlapping shadows we should be able to place two numbers together and they would be “two hidden as one” if they had the same boundaries, like the shadows at the center.

Then to this end let us create another number dimension, a dimension of number of numbers. Let the usual case be that the number of numbers is only 1. But now let us expand into the next dimension and allow the number of numbers to be 2. So for example with the number 2, let us take away the original number 2 (since we have another dimension, we can do this) and replace it with two new numbers 2’(1) and 2’(2). These are together like the two shadows but do not form one number (like 2). Keep in mind that these are somehow different. They are different from the number 2 and also different from each other.

A new plane:

Points are also exact concepts. In the Euclidean plane they are places, with the notion of no extent, in the plane. We should be able to place two together using two new numbers 0’(1) and 0’(2) identifying the two points. (0 is indicating an origin)

An object of no extent placed together with another object of no extent, would still have no extent- but there could be two objects here, under another setup.

The two points 0’(1) and 0’(2) can be different by first uncovering a new place dimension, a place of places. This must already exist because there must be some way to have two points exist together and still be two points.

In a similar way as we uncovered the new number dimension (the number of numbers) we can uncover the new place dimension.

Take the original point out (we can do this since we have a new dimension of place, a place of places) and replace it with the two new points. This can be done for the whole plane of points.

That is, there is nothing special about the origin, so each point of the usual plane can be removed and we can replace it with a “sharing” of two points. So that we have a whole plane of doubled points co-existing with a plane of places of places.

One of the new points can be fixed, while the other one is capable of “shifting” away in this new dimension of place. In this way these two can be different. Then all of the sharings in the new plane can become new origins-one point being fixed while the other point is capable of shifting away.

Then labeling the points 0’(1) and 0’(2) we can “shift” them apart, keeping 0’(1) fixed and moving 0’(2) into the plane of places and of places of places. This can be done by shifting space itself (the space of one point). See below:

We are shifting away from an origin in place of place space. This is not a movement as usual as these do not start as two distinct points in the usual plane followed by a movement apart. This is the movement of one part of a sharing pair away from another by shifting it over a co-existing space of places of places.

In order for two different sharings to move away from one another, we have to bring 0’(1) and 0’(2) back together. Then we can treat the sharing as a single item, similar to the usual concept of a single point.

Then have another sharing a small distance away from the usual sharing. Then the sharings can move apart. The sharing takes up doubled locations as it moves away.

Then there is this combination between what we already know about space and this new knowledge for each individual sharing.

Then when we have two points at two places of places. We can have a unit of shift between them which is equal to the unit of distance in the usual plane.

Measure shift (another type of distance-this can be thought of as negative distance) between the center and 0’(1) and 0’(2) if we expand out in space. Also we can have the same value considering distance or shift (if we keep the unit the same for distance and shift). Then we can have a “mixed” space with both usual space and new space overlapping. (Some points doubled and separated, other points still together).

A connected and possibly knotted loop:

We can create a loop of these doubled points in places of places space in three dimensions of place and place of places space. Let the basic topology of this be the same as it is in the usual space. That is, we allow space to shift by isotopy, to expand or contract, to form crossings, etc. As we have another level of place, let it act basically as another dimension to space.

We can have a basic topology of space in places of places, mirroring the topology of points or closed curves, in the usual space. Additionally, there are other things we can now do.

I can shift the copy of the points away from the original loop. In this way I can compare two loops which might be knotted. Start with two knots in the space, form the doubles, then shift one

copy of each away. These may then be manipulated to see if we can form a congruence between the two, moved away copies. If these can be shown to be congruent then so are the original knots.

Creation of shifted diagrams:

- Diagrams have crossings in R^3 or S^3 (usual space)
- Label these crossings 1+,1-,ect. The locations of D-joinings/sharings (mixed space). For simplicity of the diagrams I do not include these on the diagrams.
- Place a, b at each D. An a or a b is an extra moving point which keeps track of the crossing/sharing.These can come from the surrounding space..
- A moving crossing is now a sharing as we are capable of having 2 points at a vertex with a and b always moving along with the moving crossing/sharing.
- There is a new type of Crossing/sharing possible. A Q-type. This is a crossing/sharing of two points which do not cross in the original diagram.

1+,1-,ect. Are the specific locations in R^3 where we put the other concept sharing diagram (shifted diagram) back together with it’s counterpart. But in the shifting diagram they have some freedom. They can travel along as they are labels, reachable through shifts of the diagram (that is, all shifts are reversible) or they can stop at a specific pair of locations and the a, b pairs (joinings) can shift on forward by rotation. This is still reversible as I can get back to this D, as I can reverse from forward shifting. At the end, after I go all the way back I come back to the same location.

So 1+ and 1- label the crossing 1, where the original locations can be traced back from. I am placing the labels on specific locations. I usually don’t include these labels on diagrams, for diagram simplicity.

The diagrams could indicate where the crossing is so that I can trace back to the original location which matches shifts in this space; they are fully reversible and take full advantage of the new freedoms. Then all these diagrams are equivalent and complete as long as we don’t cut the diagram or change the order of locations.

Then R1,R2,R3, isotopy have their equivalents as well. If I create a new sharing, not already present at the beginning as a crossing, I call it a Q-sharing. This is a different kind of crossing/ sharing from a D-sharing. I have rotation of the locations, creation of labels, joinings, movements of joinings and labels through sharings.I can move a joining through a sharing too. And that’s all (this is a complete list of what’s possible).

Let there be another diagram D(2) in R^3 and we wish to compare this to the original diagram. Move it to mixed space. Concept share it and one part of it shifts off. If we can make a congruence between this and the other diagram in mixed space then the two diagrams in the usual space are also congruent.

So we need to look for a match of the labels, joinings when we simplify the diagrams.

If they are the same, then the information should be contained in one diagram. That is, shifts of one diagram should be able to produce the second one. So we need to look at one diagram to see if we can produce another(using all shifts available).

The new shifts are rotation (which isn’t usually considered) and movement of labels and joinings through sharings.

In conclusion, we can use concept sharing to understand knots better. We can compare any two knots using new shifts of space in an uncovered dimension of place of places.