Starting with the diagram of the Culprit knot, I am trying to find a way of showing that I can form the unknot without increasing the crossing number.
I do so by placing the knot in placement space. The the locations are free to move in a space of locations of locations. We keep the locations connected in the knot as they were originally connected.
Specifically, they can move around in a loop along the path of the knot.
Once we decompose two crossings into a joining (alpha-beta’s), we can double up the knot diagram again. One diagram is still and the other I can move the locations around again along the path of the new knot diagram. Then I can decompose again, etc. This means I can move one alpha-beta past another.
I then decompose completely and look for another way to put the knot back together. This new diagram is obtainable from the original diagram by the usual Reidmeister moves.
Abstract: Here I introduce concept sharing. In uncovering extended space, I show a new way 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..
What if a point could be expressed as two items of no extent which were not points? Why does there only have to be one entity which has no extent?
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=exe, e is not equal to p so that exe is not equal to p and also exe=is not equal to e as that 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 p=pxp. This is not as simple as taking out a point out of a given subset of points of the plane as I have to be able to put something back in that is truly different.
This means I need a space of places coexisting with the original places, so that I can take out the place p=pxp and replace it with the new places, exe=exe. Then we give up the notion of a fixed plane of points, instead we can have three planes, coexisting with one another.
The most basic new plane is in a sense at a lower level than the usual plane. This is a plane of places of new places .Any e in the usual plane can move off in any direction into this new plane, leaving its partner behind. Most basically, the entire plane can move, as shown above.
That means exe are not 2e’s at the same place, as is usually thought of as place but two e’s at the same place in new places. A new level to place. Now we have more room. Since they are in this sense not in the same place, 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, They are the same in that they share a place and a place of new and original places.
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 two e’s are sharing a place and a place of new 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 different as any other e, yet they originally shared. 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 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. Since for e(1,m) and e(1,n) the place is the same, any point that is bound to e(1,m) is also bound to e(1,n). Just not to both at the same time. We may have a closed loop of e’s which can move off and the shape could be altered if we have different distances associated with each e.
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=exr where e and r are two different types of entities as well. 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=exe.. 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=exe 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.
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. The exe=exe 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 if we have only two e’s at the origin and I move one e off up and to the right.: The notation is (()) are places of new places and () are places.
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 shown in the diagram above. As well, this loop could be knotted, if instead of a plane we consider a three dimensional space..
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. In the above case, we can think of them as the number of shadows at the center. They have exact boundaries and some way of showing we have two there or three there, ect.
Then borrowing from the notion of overlapping shadows we should be able to hide numbers together and they would be “two hidden as one” as well. (concept sharing) if the mathematical objects represented by the numbers had the same boundaries, like the shadows at the center.
Other than the further darkness of the overlapping shadows, we cannot see or imagine that there are two separate shadows there. Two or more numbers can be hidden as one since natural numbers represent exact positions. Similarly with two numbers hidden as one we can not see or imagine them together. Yet our logic tells us this can be so.
Then to this end let us create another number dimension, a dimension of number of numbers. This must already exist since it should be possible to put two or more numbers together at a beginning. 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 1, let us take away the original number 1 (since we have another underlying dimension, we can do this) and replace it with two new numbers 1’(1) and1’(2). These are together like the two shadows but do not form one number.
The two numbers together can be notated ((1’(1)((1’(2)). 1’(1) is “peeking out” from behind 1’(2). Shown by the use of a half parentheses. Seen more clearly here: (a(a).
Keep in mind that these numbers are different. They do not represent two obviously separate objects, but represent two mathematical objects, also concept sharing, hidden as one.
The objects are concept sharing as well and are somehow different from each other. We give the two hidden objects two new numbers 1’(1) and 1’(2).
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. Then we need the concept sharing of a number to indicate how many objects we wish to be there.
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 that we have two points. (0 is indicating an origin)
An object of no extent created together with another object of no extent, would still have no extent- but there could be two objects here, under another mathematical system.
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 underlying 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 interest of the usual plane can be removed and we can replace it with a “sharing” of two points. So that we have a subset of sharing 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.
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..
What if a point could be expressed as two items of no extent which were not points? Why does there only have to be one entity which has no extent?
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=exe, e is not equal to p so that exe is not equal to p and also exe=is not equal to e as that 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 p=pxp. This is not as simple as taking out a point out of a given subset of points of the plane as I have to be able to put something back in that is truly different.
This means I need a space of places coexisting with the original places, so that I can take out the place p=pxp and replace it with the new places, exe=exe. Then we give up the notion of a fixed plane of points, instead we can have three planes, coexisting with one another.
The most basic new plane is in a sense at a lower level than the usual plane. This is a plane of places of new places .Any e in the usual plane can move off in any direction into this new plane, leaving its partner behind. Most basically, the entire plane can move, as shown above.
That means exe are not 2e’s at the same place, as is usually thought of as place but two e’s at the same place in new places. A new level to place. Now we have more room. Since they are in this sense not in the same place, 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, They are the same in that they share a place and a place of new and original places.
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 two e’s are sharing a place and a place of new 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 different as any other e, yet they originally shared. 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 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. Since for e(1,m) and e(1,n) the place is the same, any point that is bound to e(1,m) is also bound to e(1,n). Just not to both at the same time. We may have a closed loop of e’s which can move off and the shape could be altered if we have different distances associated with each e.
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=exr where e and r are two different types of entities as well. 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=exe.. 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=exe 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.
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. The exe=exe 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 if we have only two e’s at the origin and I move one e off up and to the right.: The notation is (()) are places of new places and () are places.
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 shown in the diagram above. As well, this loop could be knotted, if instead of a plane we consider a three dimensional space..
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:
A number is an amount, as in a counting number, or a position on a number line, or a label.
It is point-like in that it has no existence in physical reality, it is a mathematical object, not a physical object. Therefore we can make a correspondence between the idea of sharing in points and an idea of sharing in numbers.
So that means the concept of a number can be extended downwards so that we have a number of original numbers space and this number of sharing numbers after we take out the original number. So, for example, with the number 1; we have a number of numbers space, let the number of original numbers be 2, instead of 1. Take out the number 1, then we can have 1(1) and 1(2) sharing.
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.
