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May 4, 2021

Central ideas

Filed under: knots,Mathematics — Rob @ 7:55 pm

The main idea is simple, that two ‘points’ can exist together and still be two ‘points’. Since ‘points’ have no extent, this is possible. We have the concept of the point being an exact location in the plane. Two new ‘points’ can be created together if they are different from the usual idea of points. One may say why two? Why not more? But we can state at the beginning that we have only two.

We have to have a new co-existing plane different plane so that we can take away one ‘point’ that already exists. This plane is a plane of ‘locations of locations’. Once this plane is in place, the usual points are altered only in that they can be then mobile. I need to take the one ‘point’ away otherwise I have both one and two ‘points’ there when I go to create the two ‘points’ together. Needing to be able to take this one ‘point’ away creates the need and existence of this new level.

I need to take away the concept of no extent being a point and replace it with a new concept of no extent. This can be an object which has no length, breadth or height but does have parts to it. In this special case the number of parts is two. (this is not like removing a point from a set of points, I am removing the concept of a point).

I can’t take a ‘point’ away without a new level. I need this space to be able to put two ‘points’ in. If If this new plane isn’t here then the two points are coincident and we have just one point.

I want to remove a ‘point’, leaving a ‘location of a location’ and then I can have the two new ‘points’ created together. Then this means a plane of location of locations exists. The usual locations are fixed in this plane. This new plane of ‘locations of locations’ is considered fixed, for now.

These new entities(‘points’) I call ‘parts’ (as in parts of the whole). Parts are different from points, as they are mobile in this new plane.

We may imagine the empty space in a three-dimensional container. We can extend the point concept stated above to spaces as I can have a set of points be the three-dimensional space. This empty space can be moved to take up the same “space of spaces” as the still spaces, if we move two empty spaces together. Then two empty spaces take up the same “space of spaces” left. . The spaces may be full of something such as the string which makes up a knot. Then the two full spaces can still be at one “space of spaces”. Two items are not together at the same space.

This may be the reality we live in. I am working on some physics in another section of this website. It may be for the far off future, but if we can move space itself then we may be able to travel to one of the exoplanets. Since the speed of light is limited by space but we may not be limited by this speed if we move space itself.

The new level can be what I call placement space. It is a space where a location may have another ‘location of a locations’. Or a small piece of space can have another “space of spaces”. Basically a location or section of space is mobile. At the beginning it is stated how many locations are at any given placement. For the work on knots this is two, and the resulting entity I call a “doublet”. Then two different locations may share the same placement (location of locations) and I call this a “sharing”. A special type of sharing may exist in the manipulation of diagrams which I call a “joining”.

I use the knottedness and chirality or achirality of knots to demonstrate the usefulness of these ideas.

March 1, 2021

Filed under: knots,Mathematics — Rob @ 8:24 pm

February 18, 2021

The achirality of 4(1)

Filed under: knots,Mathematics — Rob @ 4:01 pm

February 15, 2021

Filed under: knots,Mathematics — Rob @ 9:02 pm

February 13, 2021

Filed under: knots,Mathematics — Rob @ 4:27 pm

This note is showing the two final diagram reached after the Reidemeister moves in R^3. These diagrams also have to be equivalent in P^3. Then I can set up a congruency of locations between the two diagrams.

January 14, 2021

Filed under: knots,Mathematics — Rob @ 7:15 am

This post is describing the push move (named by Murasugi), which is an extra move in knots which doesn’t effect the knot. In R^3, this is seen as the curve being mapped to itself by a rotation of the curve upon itself. In P^3 we see this as a movement of joining pairs around the curve. The cyclic order of the joining pairs gives us the shape of the knot diagram.

January 8, 2021

Filed under: knots,Mathematics — Rob @ 3:39 pm

This page is showing the different moves possible in the two different spaces. The top space is R^3 and the bottom space is placement space (P^3). Placement space is described in the posts below. The idea that leads to placement space is described in the quick view, below.

There is a link between R^3 and P^3. Also in P^3 we can create a D-sharing- a type of crossing. In turns out in placement space there are two types of basic crossings. I’ve labelled these D and Q. These are shown in some detail in the diagrams below.

Applying moves S1,S2,S3 counterparts to the Reidemeister moves to take us from one diagram to the next. I use Reidemeister moves to go from one diagram to the next, but do not use these to prove knottedness or chirality.

These moves all involve Q-sharings only and I’ve drawn them above. In order that I only apply these moves to Q-sharings a transformation is applied to the trefoil. This transformation is described in the posts below. I’ve shown it in a diagram above.

Then these S1,S2,S3 have counterparts in the space R^3 above. This then leads us to a diagram with the same shape as the original, but with possibly an unknotted case or a mirror image case.

To rule these out we put back in the cyclic permutable set {a(1)r(1),b(2)s(2),a(3)r(3),b(1)s(1),a(2)b(2),b(3)s(3)} which was removed during the transformation in placement space. It might be possible to obtain a mirror image or an unknot because I am applying the moves generally.

That is I allow all possible paths by Reidemeister moves which lead back to the diagram with the same shape. It turns out I can only reform the original configuration, not the unknotted or mirror image case.

For future work I can say that the shape of the knot need not be one shape alone, also in larger knots such as the Perko pair, I can remove the D-sharings and put new D-sharings in, changing the writhe of the knot which is the number of positive crossings minus the number of negative crossings.

December 29, 2020

Quick view-abstract

Filed under: knots,Mathematics — Rob @ 9:29 pm

As a quick way to understand my starting point think of something which can be called two disjoint points (named by Kauffman), but not two points separated, but two points together. Then to have this I must first remove the notion of the single point, otherwise I would have both one and two points here. So some other structure is here. Then this is placement space. For a detailed explanation of this see the beginning of all the posts below.

What I’m trying to show is that in placement space P^3, I can move the joining pairs cyclicly and these can reform to make D-sharings. The joining pairs come from an original set of D-sharings. These two sets of D-sharings are related by a set of Reidemeister moves, in R^3 so these are equal knots.

Then for the Trefoil, I can start with the simple representition and show that there are no reformations that lead to the unknotted case and also do not lead to the mirror image, in the same shape, after any number of Reidemeister moves.

I also consider future work which could be done.

December 23, 2020

Clearer explantion for the Trefoil

Filed under: knots,Mathematics — Rob @ 11:23 am

This is showing a more clear exposition of the final conclusion for the trefoil, as well I show the way to finding a minimum diagram with a different shape than the original knot diagram.

December 19, 2020

Towards a more general understanding of knottedness

Filed under: knots,Mathematics — Rob @ 3:49 pm

All the below posts serve to prove the Trefoil knot is knotted and Chiral using a new method. But can the method be applied to more general knots? The above is the rough outline of some steps which could be applied.

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