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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.

As an entry into Concept Sharing 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.

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 two books packed tightly together, on a shelf.

Then the point of contact can be separated into two points, one for each item. Mathematically a single point is replaced by two distinct points, with a small gap, then this gap can be increased..

Concept Sharing is a term used in education. But here I am giving it another definition.

As an entry into Concept Sharing 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. These are “little nothings”.

What if something of no extent could be expressed as two items of no extent, placed together. Since an item of no extent placed together with another similar item of no extent would still have no extent, yet there could still be two items here and not one.

Two little nothings still add up to nothing, yet there can be two little nothings there. The two items would just be hidden as one. Let’s hide!-they say. Who can see us? The two items would not be points, for points have a singularity to them. That is, when I combine two points, it leads to one point, but this doesn’t have to be so with another possible entity. See the teacup shadow diagram, below. Why does there only have to be one entity which has no extent?

These two items could share the concept of a point (concept sharing). Let’s share the concept of nothing, the two little nothings say! This is how we can hide! In order to do this we would have to take out the usual concept of a point and replace it with this new conception. This is because a point is already there as the idea of no extent. We need some room! Kick out what’s already there!

What does it mean to take out the concept of a point? It is not the same as removing a point from a given subset of points of the plane. We wish to open up a new possibility, so we have to reject the usual concept of a point and accept this new possibility. 

Yet to reject the concept of a point and replace it, it means we have to define a space where a point can be truly taken out of. We need something like another plane! Places of original places. Let’s let the points have another level to them, where they can move off into another plane, coexisting with what is already there. That way it is possible to take them out, otherwise, how can we remove them? I can’t take them out of nothing! A place where the original place can be taken out of. Another level to places, coexisting with existing places. That is, a concept space. 

If you accept the notion of an entity with no extent you have to accept this new possibility as well. If you want a point, you have to have us too! The new entities say! The notion of no extent leads naturally to concept sharing!

To remove the concept of a point, we need to have a concept space, consisting of different nested levels of the concept. This must exist because I must have this further concept of a doubled item of no extent somewhere and I need to take out the concept of a single item of no extent to have it appear by itself (unconfused). Also we must take out any other possibility of any number of items of no extent, except 2.

Furthermore, all math concepts are point-like in that they are exact ideas which have no existence in physical reality. They are mental constructions which are not available to the senses. They are in thought only. So number, set, group, function, etc. can all have concept spaces.

It seems like the two items should be the same. But not if they can be also further defined in a new plane as separate( in the case of points).

With numbers, they represent a position, an amount or a label.

Think about position and consider a race where 2 runners are tied for 10th place. We can say the runners are tied for tenth place giving the number 10 to both runners. But suppose instead of a foot race, two points are together in a race along the number line. Then in 10th place two points could be together as has been seen. Then two number 10’s can be given one to each point. These numbers could go with the points as they are mapped into the new plane which has been mentioned.

A circle which is the 10th circle to be formed could contain the two e’s with the two number 10’s. Then 10(1) and 10(2) could also show an amount of points, 2. As well these could be the labels we are giving to these points, or the position in a number line, 10.

If sets are made of points, the sets could have concept spaces. If sets were made of numbers, the numbers could be associated with points and we could have concept spaces. Similarly if groups are made of numbers or diagrams which are made of points they too could have lower concept spaces.

In the equation x^2+y^2=1 the point (0,0) is mapped to the two points (0,1) and (0,-1). This can be thought of as two points at (0,0) moving to the two locations (0,1) and (0,-1). 

This can be notated ….p(1)*p(2)*p(3)=p(1)*p(2)= p(0)=p(1)*p(2)=p(1)*p(2)*p(3)…

Where * is of different points coming together or separating apart in the usual plane.

When mapping to points there is a many to many map, a many to one map or a one to one map. Yet there is another possibility with e’s. There can be, for example, a two to two map. Where two separated e’s are in the new plane and combine to form two points at a single location.I say two points at that location now because they are defined in two different places of original places in the new plane. Therefore there are two here as in the case of the overlapping shadows.

Since it is possible to have two entities together and still be two entities, this other case must exist somewhere. The entities must be zero-dimensional but not be points.

Yet the overlapping shadows show us that there could be a constant number of entities. This is because the shadows have somewhere to be cast onto. Rather than a notion of no extent alone, which could or could not be multiple, the surface makes it possible to show a finite number of entities of no extent.

Then postulate a place of places where ‘e” the entity of no extent which is like a point in that it has no extent but unlike a point being different in the following way:

e(0) is not equal to e(1)*e(2) is not equal to e(1)*e(2)*e(3) where again * is the idea of movement but this time in the new plane.

Then usually the idea of points can be notated pxp=p or pxpxp=p…etc. Where x is the idea of coming together or separating apart and p is a point. But what if there were another entity of no extent, call it e such that exe=p, e is not equal to p as then exe=e would be the same as pxp=p. We can call these entity equations.

It seems like exe are two identical entities of no extent and it should result in e. But consider that to have exe=p, I have to take out pxp=p. This means I need a space of places which contain the original places, so that I can take out the place p=pxp and replace it with the new places, exe=p.

That means exe are not 2e’s at the same place but 2e’s at the same place of original places, a lower level to place. Since they are not in the same place, as there is no place there, they don’t combine. Briefly we can write this exe=exe (sharing).

Take out the concept of place and put in this new concept of place. The only way it can be different is if the places don’t combine to form a single place but stay separate while being together. (sharing as in the overlapping teacup shadows)

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 different points of the shadows can take up the space of one point. This is analogous to two e’s sharing.

Then we can separate the two e’s, but the only way this can be different from the usual idea of separation in points, is as if both entities are the same and different (not just different as in points). In the case of points, there exists this gap from one common point to the case of two distinct points.

The place has been removed so we have an underlying dimension where places have other places of original places. Like a jigsaw puzzle of a landscape being taken apart. In this way they are the same place yet they are at different places.

Then this leads to a new extent, a line with two distances one being this new zero and the other being the usual concept of distance, extended.

This is a new dimension. Each e of the extent is both the same and different as any other e. This is just a new dimension in length. We can notate any two e’s as e(1,m) and e(1,n).

This extent may be considered as negative distance as we need to shrink it by adding positive distance (usual distance) to get back to the new zero and then take this out and replace it with pxp=p to get back to the usual zero.

We can set a mathematical system with exe=p or choose three e’s so that exexe=p or the number of e’s could be variable.

This must fit into our current structure of mathematics as I am not adding any new notion in, merely clarifying the concept of a point as having no extent, then adding in the necessary new entities. The notion of no extent is the same. We already have this notion of a point as being pxp=p, we have to extend this.

Additionally, there is also the case exr=p where e and r are two different types of entities as well. E is not equal to p and r is not equal to p, additionally e is not equal to r. This can be for future work.

So we have the idea that a point is an entity with no extent, and also another notion that it could be exe=p but how do these fit together?

It must be that we have replaced the usual idea of a point as being pxp=p with this new idea of a point as being exe=p.. This means there is another level to space. Since I’ve taken out the usual notion of a point, I must have taken it out from somewhere. This is the space, places of new and original places. It can be modeled after the usual idea of extent, yet the distances are negative.

Then this also means I can separate exe=p in the new space and move in a space between two of the same e, like so, the displacements from the 2e’s are shown.

of space as well as a leveled nature.

Then we can have the idea of a multiple point or two tangent points.

With the tangent point we measure the diameters from the point of tangency outward. These can be separated as usual with the usual distance appearing between them, this is the idea of pxp=p. There is a small gap before we reach p, from two distinct points. The exe=p points can be separated as well, with the new space appearing between them.This is the space of places of new places. This is the negative space.

So we must have a plane or a space in which the ordinary places e or p, take on other places.

A picture of this would look like below:

This is a movement of one piece of a doubled origin, a single e.

Not only the origin but each identified exe=p of the new space can act as its own centre, The two e’s can  move away from each other.

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 can all have sharings.

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:

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:

1. Diagrams have crossings in R^3 or S^3 (usual space)
2. 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.
3. 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..
4. 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.
5. 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.