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Thanks to Lou Kauffman for sending me a picture of the Culprit knot.

Thanks to Lou Kauffman for showing me swing moves.

Izzy’s giving a look at the Culprit knot.

I suggest that the Poincare conjecture is a direct result of a new understanding of the basic structure of space itself.

In the concept sharing of a point, we can see that space has this new leveled structure.

Abstract: Here I introduce concept sharing. In uncovering extended space, I develop new ways of understanding Poincare’s conjecture.

Overlapping shadows:

An interesting notion can come about from something observed in nature, the overlapping of shadows. Consider a cup of tea placed on a table and lit with two lights from above, one from the left and another from the right, as illustrated below:

Three areas of shadows are formed. The one in the center is the overlapping of two shadows. 

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.

They are ideas or concepts so that they are further like the shadows in that there is no real substance to them.

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. 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. 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.

Keep in mind that these numbers are different. They do not represent two obviously separate objects, but represent two mathematical objects hidden as one.

The objects 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 placed 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 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.

So also that any loop may be considered to be a point, because the point can have an infinite number of parts, and any point considered to be a loop. Therefore any space which does not contain “holes” or twists in any dimension will be necessarily simply connected.