### A clearer and simpler understanding of knottedness using concept sharing

As an entry into Concept Sharing it’s best to start with a specific example. Then let’s start with the concept of a point. In math this is the notion of an entity with no extent, or in Cartesian geometry the notion of something with position only.

Consider a teacup placed on a table with two lights from above. One from the left and one from the right. See below:

Now as seen in the overlap, two points of the shadows can take up the space of one point.

We have the familiar idea of two items just touching or resting upon one another as we see in everyday life. For example a book resting on a table, or books packed tightly together, on a shelf.

But there is something else that is hinted at here. That is that points could be multiple. A point could be in combination with multiple copies of itself. There could be multiple copies of the one point present within the space of one point.

Look at the turtle diagram below. Points are tangent to one another, if we represent points with circles. This represents points which are together, which could be separated.

But we could also have multiplicity of the individual circles, representing multiplicity of points.

The trouble is that this must fit into our current structure of mathematics as I am not adding any new notion in, merely clarifying. We already have this notion of a point as being solitary, we have to extend this.

The solution is that since this other case must be possible, it must be possible to take out the point that is there and replace it with the possibly multiple point. I say possibly because we allow the case where the point could be single too.

This necessitates a “lower” level of places which contain original places, as points are thought of as places in Cartesian geometry.

So we must have a plane or a space in which the ordinary places of Cartesian geometry take on other places of places.

A picture of this would look like below:

This is a movement of one piece of a doubled origin.

Not only the origin but each identified point of the new space can act as its own centre, moving away from its copy.

We could have a closed loop of these points all moving together. As well, this loop could be knotted.

Then this is also the entry into Concept Sharing as math concepts such as number, set, group, ect. Can all be thought of as point-like. That is to say they are all ideas which could have multiple expressions.

They are all exact and have no physical reality, they are just ideas.

Since they can all be multiple, there must exist lower concept spaces.

The concept sharing of a number:

A natural number is a mathematical concept. They are represented by the set of natural numbers {1,2,3,….}. In the above case, we can think of the number of shadows at the center. The shadows have exact boundaries and some way of showing we have two there or three or more there.

Numbers are further like the shadows in that they are ideas, they are positions only and not physical objects or even shadows or waves of light. They could exist as two or more, similar to the overlapping shadows but truly hidden as one, instead of the usual singular case.

Imagine a tower of such overlapping shadows. That is, one set of overlapping shadows on the first level, then a second level also with a set of overlapping shadows then a third, and so on. At the third level, for example we might have two overlapping shadows. We could label these 3(1) and 3(2). Now imagine that the shadows are positions with no physicality to them. Then we cannot tell that there are two there, but we can tell there could be two there. In order to have two there, we must first take out the usual notion of coincidence. So this means we have to be able to take out the number 3. Meaning there must exist another number of numbers level as this is possible.

Two or more of the same number (in the sense that two new numbers are in the position occupied by the first number, but the first number is removed) can “fit into” each other (concept sharing). As long as we state how many we want and that we know there could have been more or less there, this is possible.

Before you say, yes but this is just the same number, consider the possibility that we can take that solution out. Then the statement “two or more of the same number” can make sense as I remove the original number and the two new numbers both share this original position but are otherwise different in that they represent different concept sharing objects.

Then borrowing from the notion of overlapping shadows we should be able to hide numbers together and they would be “two or more hidden as one or two fitting into each other” as well. We could then number mathematical objects which have the same boundaries, like the shadows at the center. These other objects would also be concept sharing.

Because concepts are like the overlapping shadows, they could fit into each other, there needs to be a number of concepts to allow us to put a number on how many ideas we wish to be present. This must be different type of number from a number of separated concepts. Starting with a number(natural) imagine a race where we can have a tie for first or a tie for second, etc. We need to be able to say how many are tied for first. So number this 1(1) and 1(2).

But suppose that the racers are indistinguishable and also that we can’t tell how many are there. In order to proceed, assuming we can pick a certain race, we can at least give a list, a certain number of racers for a race. Since we can’t tell externally how many numbers are fitting into each other with have to designate it with a number of numbers.

These numbers can be used to number shared concepts, since there is no way to otherwise distinguish them from a single concept, similar to the racers (if they were together like the overlapping shadows). Yet we can give ahead of time a certain number of shared concepts, similar to picking a certain race.

Then there must be another number level, a level 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, lower level and allow the number of numbers to be 2.

So for example with the number 1, let us take away the original number 1 so the set becomes {2,3,4,5…}and replace it with two new numbers 1(1) and1(2). These are together with the same position as 1 and are like the two shadows but do not form one number since they refer to two different mathematical objects. In this way they are different.

Concept sharing of a number can be thought of as a kind of race where the numbers are in a tie for a certain place. So for example two ones tied for first place, as above.

So here I am talking about removing the number 1, itself and not an object that the number could represent. This necessitates a another level of numbers of number 1. Usually there is only 1, but now we have 2 numbers which are not 1, but both have the position of the missing number 1.

Together they can be notated (1(1)(1(2)). Two brackets around the final number and only one bracket in front of the first number. This is to indicate one is hidden within the other-but revealed by the notation. It can be more clearly seen here: (b(a).-b is moved away from a, sharing the same position as b-b is “peeking out”, from fitting into a.

Additionally, they are different in that the number in the full parenthesis is still, whereas the number in the half parenthesis can move. This can be seen later in the case of a separable space where the numbers are representing two concept sharing objects which are connected by a continuum (seen later below).

In the case of the shadows, we can tell that there are two shadows there and give one the number 1 and the other the number two.

In the case of mathematical objects there is no external way of telling how many objects there are, previously it was assumed it was only one. We can state how many we wish at the onset thus fixing a certain mathematical system.

Thus we need the concept sharing of a number to indicate how many objects we wish to be there and that we are aware that it could have been a different number.

A new plane:

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

Two objects fitting into each other of no extent would still have no extent- but there could be two objects here, under another mathematical system.

The two places 0(1) and 0(2) can be different by first removing a place and replacing it with (0(1)(0(2)). I am talking about removing the place itself and not removing a place from a set of places.

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

This can be done for the whole plane of places. That is, there is nothing special about the origin, so each place of the usual plane can be removed and we can replace it with a “sharing” of two places. So that we have a whole plane of doubled places.

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

Then labeling the places 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 place). See below:

We are shifting away from an origin in place of place space. This is not a movement as usual as these are not two distinct points in the usual plane indicated by a distance 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 place.

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 a knot in the space, form the doubles, then shift one

copy . This may then be manipulated to see if we can form a congruence between the moved copy and the inverse or mirror image knot. If these can be shown to be congruent then so are the original knots. Also we can seek other diagrams with a different order of crossings/joinings.

Creation of shifted diagrams:

- Diagrams have crossings in R^3 or S^3 (usual space). Consider the trefoil.
- Label these crossings Z(1), Z(2), Z(3),ect. The locations of Z-crossings/sharings (mixed space).
- Place a, b at each Z. An a or a b is an extra moving point (which can be called a “part”-as in part of a whole) which keeps track of the crossing/sharing.These can come from the surrounding space..
- A moving crossing is now a crossing/sharing as we are capable of having 2 parts 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. Additionally r and s parts which make up the joining can also pass through each other. This is not allowed for a and b parts. So the knot keeps its knottedness.

Z(1), Z(2), ect. Are representing moving crossings/sharngs in P^3 where “placement space” is the name I give the larger space.(places of places). But in the shifting diagram they have some freedom. They can travel along as they are moving sharings/crossings, reachable through shifts of the diagram (that is, all shifts are reversible) or they can stop at a specific placement and the a, b pairs (joinings) example (a(1)(r(1)) can shift on forward by rotation. This is still reversible as I can get back to this Z, as I can reverse from forward shifting. At the end, after I go all the way back I come back to the same location.

Then all these diagrams are equivalent and complete as long as we don’t cut the diagram.

Then R0, R1,R2,R3, have their equivalents as well. (call them S0,S1,S2,S3) 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 Z-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 its connected parts 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 sharings, 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 by S0 (which isn’t usually considered) and movement of joinings through sharings.

So for the diagram set for the Trefoil:

- Start with the Trefoil positively oriented in R(3) or S(3) fig1
- Realize the new space with concept sharing of a point (creation of parts, placement space)
- Label the sharings/crossings with a(1),b(1), Z(1),…ect. fig.2
- Double up the parts, move one copy away.
- Move from positive space to mixed space (presence of placement space). Creation of Q sharings/crossings. Z(a,b) exists in both spaces Q(r,s) exists in only mixed space.
- S0,S1,S2,S3 can be made leading back to a final diagram. Fig. 5. I can put back the joining pairs to reform the sharings only in a specific way
- There is an invariant of the Trefoil shown. Fig. 6a. We can compare it to itself using S0 looking to see if we can have an inverse or a mirror image of the knot.
- There may be a more direct way to compare two knots. Fig 6b.
- We can look for a different shape as joinings can move past other joinings.
- In the Trefoil, in particular, there are no other shapes, as well we cannot change any orientations, as shown.

Therefore the Trefoil is knotted and chiral.

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.