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September 27, 2017

Clarifying letter to Lou

August 3, 2016

Placements

The idea is that a location may have another place other than it usually occupies. Places where we might put a location somewhere else-relative to other locations. This can be done if locations were sharing the same points of the plane with other objects, locations of locations, or placements (l*l).

We know what we mean by a moving point. But here this is not the case, here we are moving the location. We can imagine that a location could have another l*l if the meaning of a location is kept. We can do this by changing back and forth from the usual plane to an alternative plane only when all locations are back in their usual places. To do this we will need to invent a new, alternative plane where a location could be in other places.

 

A physical object can’t be in two places at the same time. but a mathematical object could be if we consider the following:

  1. A point is an undefined object meant to express the concept of an entity with no extent.
  2. Since the object has no physical reality to it, that is it is not a concrete object we can go forward and suggest that the point may be duplicated.
  3. This can be expressed in this way p–>p+p

The concept for an isolate point is that although p is not extended in any concept of extended space, it is duplicated since we can’t tell if there are two or one there.

Our imagination fails us if we try to think of two points being there but there could be two there nevertheless as the duplication of two entities of no extent still possess no extent.

Once we duplicate a point at a placement we can leave an original copy where it is and move a copy to another placement.

 

Start with the usual plane with every point given a location (a,b) where a and b are numbers.  Then form an alternative plane with l*l- a(     )b a and b are numbers- at the left the a is the x-co-ordinate of the l*l and the b at the right is the y co-ordinate of the l*l- placed together with the locations we can form the plane: a((a,b))b. The l*l is written outside of the location only to indicate that the l*l are fixed and the locations can move between them. There may be other geometries where the l*l are not fixed but we do not consider these here.

The double bracket is meant to indicate that this is a one step ‘deeper’ level of location. ‘deeper’ in the sense that now locations have a location called l*l.  For brevity the location of a location is called an l*l. Places where locations can be.

To give a location another location we have to be able to remove it from where it is and move it to another l*l. This means there has to be some other mathematical objects at the points where we have locations. This can be l*l, a(     )b. Two mathematical objects, the location and the l*l exist at the same point.  In the beginning then we have a((a,b))b for all a,b. There are higher levels such as a((a((a,b))b))b but we only consider two levels for now.

We can think of this as if we have two overlapping shadows in one plane, each shadow cast from a separate light source. See the teacup shadow diagram below. The two shadows are independent as when one light source is shut off we still have the other shadow from the other light source. One shadow can be thought of as consisting of locations and the other shadow can be thought of as made of placements.

a(     )b and (a,b) are two different mathematical objects.  Since both a(     )b and (a,b) have no extension, they can be together yielding a result of no extension. This can be done with two different locations also ex. (c,d)*(e,f) and we call this a ‘joining’.

The notation: a((a,b))b is called a second-co-ordinate and is meant to indicate this combination of two objects. The outer part is the l*l. The inner part can take the form of a single location, a void, or a finite series of joined locatons ex. (c,d)*(e,f)*…..

We can start simply, with each location where it usually is but label these places. As an example suppose the location (1,1) starts at the l*l- 1(     )1 in an alternative plane. In this double bracket alternate plane the locations can be placed anywhere.  Then create a new set of axis centered at 1(     )1 and use the notation 1((1,1))1 to indicate that at the placement 1(     )1 we can place the location (1,1).

Similar to the Tangent Plane, a(     )b acts a center of  a((a,b))b.  (a,b) can move off with an axis centered at a(     )b.

In the example below, if we start at 4((4,4))4 and move the location (4,4) to the l*l 7(     )7 the ‘vector’ grows as (4,4) is moved along.

So we can start with the plane (a,b) and then transform to the alternative plane a((a,b))b. Then we can move any location. As long as we move all locations back to there original spots we can transform back to (a,b).

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July 17, 2016

moving circle of locations, translation of locations and addition of locations

Filed under: knot idea outline,the link between l*l and e and p — Rob @ 7:24 pm

We say that a point has a location. But I want to say that that specific point or location can have another ‘location of location’ in relation to the plane of other locations.

To do this I put a plane of l*l a(     )b in so that every location (a,b) has an l*l- a(     )b. We stop at only two levels for now. So every location of the plane has a ‘location’ itself. These two together, the location of the location, and the location itself are called the second co-ordinates and take the form a((c,d)*(e,f)*…))b.

Each placement of a location initially acts as a center for its specific location. Any location can move away from its center. There can be different types of movement. Sometimes, in order for a location to move we have to create a ‘joining’ of two locations. Locations can ‘join’ with other locations. This is discussed a little later.

One thing we can do is move a circle of locations through a coincident circle of l*l. Underlined in red are locations, moving around 90 degrees clockwise through the coincident level of l*l. The plane can be divided into two types of locations. e-type parts and p-type parts. The circle of e-type parts can move through a circle of p-type parts defined in a plane of fixed p-type locations coincident to a plane of placements.

Also I can translate, for example the location (0,0) at the l*l-0(     )0 to the l*l-1(     )1. But (0,0) is not in isolation. It has to be in combination with other locations in order to get to and to be at 1(     )1.

Let it be in ‘joining’ with other locations. Two locations can be in joining if they share the same l*l. Notate this with a *. Then when (0,0) reaches ((1,1)) we can notate this (0,0)*(1,1). This is an expanded notation. Both locations (0,0) and (1,1) are at the l*l- 1(     )1. To express this, I cannot write them on top of one another, so I use this notation. This can be notated 1((0,0)*(1,1))1.

Both locations are in the same place, but there are two locations there and not one. Since l*l exist as a background level, we have a place to put (0,0)*(1,1). Whereas before there was no background level where I could put one location relative to another. Now we can put two in the same place, before we couldn’t identify two locations to the same place.

(0,0)*(1,1) can be thought of as sharing placement akin to two overlapping shadows appearing on a plane surface, the plane surface being a plane of l*l and the overlapping shadows thought of as overlapping locations-think of the shadows as locations themselves. As described elsewhere in this weblog, if the shadows come from two light sources, and I remove one light, one shadow remains.

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July 13, 2016

moving the circle into three dimensions

I want to move this moving circle of locations into the third dimension. In order to do this let the locations join with other locations in the z-plane. For each l*l in the z-plane I have one location from the moving circle and one ‘”fixed” location which was already in the z-plane. Call the moving locations, “e-type” and call the fixed locations, “p-type”. I can create a twist. Shown in the picture below, the dotted line represents where the moving circle used to be, after the circle is moved into the third dimension and twisted once.

 

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July 10, 2016

Tutorial for knot idea outline

Filed under: knot idea outline — Rob @ 3:58 pm

IMG_20160710_0001The goal of this post is to show a few diagrams related to the knot idea outline and to use them to further explain and clarify the outline.

The goal of the outline is to come up with a new knot diagram given the new whole “e”. “e” can be thought of as the group of locations in a space of locations and placements which are allowed to move through the placements. The circle of e’s is in joining with a circle of p-type parts in joining with the plane of placements.  I can create a geometry in which we have only one circle of moving locations. The rest of the locations can be “fixed” for now.

We have that we can bring two e’s together to form e*e. But perhaps this can be talked about in detail. I can start with a space of p-type parts “p” and “e”. “p” can be thought of as the group of locations which is fixed in a space of locations and placements. “e” are those locations which are allowed to move through the placements.

 

At the joining let e(1,1) and e(1,2) not represent one specific location each, but let them represent any locations which could be brought together at that placement-p(0,0). Keep the twist fixed otherwise-just rotate the diagram upon itself.

Then e(1,1)*e(1,2) is thought of as a “general” joining. We can separate it if we ‘freeze’ the diagram.

Then what is left when we take e(1,1) and e(1,2) away from the central joining is a joining of two parts which do not join in the original diagram. This is an ‘altered’ diagram. See the diagram at the bottom of the page with printing.

Before the diagram is frozen, the strands move so that crossings move along as expected with the moving strands, but after the diagram is frozen, the crossings are separated. We move the coincident p-type parts along with the e-type parts.

The diagram then ‘shifts’ instead of ‘moves’. Anytime the strands are then brought together there is a joining of two parts which do not join in the original diagram.

Usually with ordinary geometry points move through locations.

In placement and parts geometry there is an opportunity to perform both ‘moves’ and ‘shifts’

In a ‘move’ e-type parts move through fixed p-type parts placements and this is similar to a move in the theory of knots.

In a ‘shift’ the diagram is first ‘frozen’ then the combination of both p and e-type parts which make up the diagram are both moved together, leaving a void-set of placements.

 

 

 

 

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The benefit of this is then to be able to “unfold” a diagram into a new circle.

I can move e(1,1) and e(1,2) away from the center along the strands, as shown in the next diagram at left.

This type of motion can be chosen from the possible ways of separating. If we look at the diagram at the bottom of the page  with 1+ and 1-. We can notate e(1,1) as 1- and e(1,2) as 1+. As we move any part through a joining we move them in the way that they go through in a straight line. They don’t have to, as we could go off to the left as well. But for these diagrams we go straight through.

Bringing these ideas together, The self-joining loop can be separated as shown in the diagrams for it. As well then the trefoil knot can be started to be separated as shown in the diagram below the self-joining loop.

 

 

 

July 2, 2016

explanation of parts as relates to diagrams

Filed under: knot idea outline — Rob @ 1:08 pm

The goal of these four pages is to find a way to define the most simple knot in its most simple form-but this is done in a different space than it is used to being defined in. This may open up new moves which we could apply to more complex knots.

IMG_20160702_0001In ordinary maps two points may be mapped to a single point, and then we have the inverse “mapping”: a single point may be mapped back to both different points.

But here we want to be able to work with knot diagrams defined in a space made out of “parts” and placements and not points. “Parts” and placements are discussed elsewhere on this website. By parts we mean locations.

The central idea is that instead of points which combine by coincidence (+) into a single point, parts can combine by joining(*). So that two parts brought together as in two lines of parts coming together to form a joining(*), may be written as for example  (e(1,1)*e(1,2)).  Meaning at one placement we have the first part e(1,1) and the second part e(1,2) sharing the placement.

In usual geometry a point has no extent and no parts or pieces. Since it has no parts or pieces it can’t be separated or divided into two or more parts or pieces. If we place two points together by moving two lines together until we have a coincident (+) point, we superimpose two wholes with no parts or pieces and the result is a single whole with no parts or pieces.

But in the geometry of placements and parts we have another entity “e” which has no extent and no parts or pieces and is also a whole. E is a group of moving locations in the space of placements and locations.

“e” is different from a point p, in the usual geometry in that we can place two e’s together by joining(*) and have the result not be a single whole-like with points-but two e’s in the same placement-similar to two overlapping shadows.  “e” has no extent like a point p, but is not like a point p-we regard it as being a part of a combination as in e*e as well as a whole in itself.

The two e’s would have to be distinguished in some other way then their placement. “e’s” and points are different in this manner. P-type parts are also different from points, in the same manner and are usually fixed in the space of placements and locations.

There is a subtle difference between division and separation here. By division we mean the ability to form two or more entities like the original, from the original, smaller than the original. We therefore cannot divide a point or the entity (e(1,1)*e(1,2)) into pieces. But we can separate (e(1,1)*e(1,2)) into two parts. Then there is the difference between parts and pieces.

We let p denote usually fixed parts and e denote moving parts. e’s are not distinguished by their placement, but can be distinguished by some other rule. In the case of knot diagrams the first number indicates which location we are at and the second number indicates whether the e was originally below, 1 or above, 2.

So we start then with a three dimensional space which consists of placements and parts. We can first define a mapping of a trefoil knot in three-dimensional space of regular points to the one in the space of placements and parts. Then I can move the e’s of the trefoil until three sets of two are joining at the placements shown in the diagram close to the center of the page.

 

moving into a new circle

Filed under: knot idea outline — Rob @ 1:07 pm

Then one thing  we can do is unfold the trefoil into a new circle with e-labels as shown in the diagram. The circle is not just formed by going around the trefoil and noting which e’s are met, but this is a movement in the space of placement and parts.IMG_20160702_0002

separation of the joining

Filed under: knot idea outline — Rob @ 1:06 pm

IMG_20160702_0003To look for one of these new moves, consider the joining (e(3,2)*e(3,1)). These e’s do not represent specific parts, but these represent what were crossings in the original knot diagram. That means when sections of the curves move in the plane, under movement of the e’s without the coincident p’s. the e’s move with the joining, they do not separate unless we want them too.

But now since the joining is made up of two parts I can freeze the diagram and separate them with the coincident p’s  along the two different curves leaving an joining of two parts which do not join in the original diagram-see bottom diagram.

 

movement of an e through a joining

Filed under: knot idea outline — Rob @ 1:04 pm

Now having done this we can move for example a separated e(3,1) to the joiningIMG_20160702_0004 e(2,2)*e(2,1) to form a “triple” e(2,2)*e(2,1)*e(3,1) and then move e(3,1) past the joining. Then this is a move in which half of the crossing is moving through another crossing.

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