Before we apply the S-moves to the trefoil in placement space we can apply a simplification so that only Q-sharings undergo S-moves. Then the two final diagrams in the previous page can be compared.

Q-sharings are different from D-sharings. In placement space we have another degree of freedom, we can choose to switch the sharing parts in Q and leave them on the same strands in D-sharings. This is shown in diagrams below.

Here I am showing a ‘circle of joinings’ which can exist after the simplification is applied.

Then we can move the parts that exist on the outside of the final diagram in the previous page into the slots created by the Q-sharings that remain after the S-moves. See diagram 4 below. We can eliminate the Q-sharings and can possibly show a different diagram in R^3 by reversing the steps and moving from placement space back to usual space.

When I write: can I in placement space…make an ‘exchange’. I mean is it possible to make new moves in placement space that will rearrange the parts so that I can form a different diagram in R^3?

Then I form a correspondence diagram, above with T(1) and T(2), a trefoil with the same ‘shape’ as T(1), that is to say if we ignore the crossings then the diagrams can be made congruent.

And the corresponding D-diagram moving in placement space, undergoing S-moves.

We start with the mathematical trefoil knot in three-dimensional space. Diagram 1 in the series of diagrams in the posts below. The movements possible in this space are known as isotopy, or one of three possible knot moves: R(0)(isotopy-movement of an arc through space),R(1),R(2),R(3). These are discussed and shown in the diagrams below the three pages. Briefly put there are four ways of moving from one given diagram of a knot to another diagram of the same knot.

The first paragraph is a description of the ending of the trefoil’s motion in regular three-dimensional space (R^3). We switch to placement space(P^3). Diagram 2 below.

This space is described in the longer discussion but can be briefly put as a space where instead of a location everywhere, we have a doublet everywhere. A doublet is the notion of two or more overlapping shadows, that is the idea that more than one location could occupy the same ‘placement’. So that placement is a second level of location.

Then start with the trefoil in it’s reduced form. It has a minimal crossing number of 3. We are trying to show that it is knotted, that is that there is no way to unknot it no matter how many of R(0),R(1),R(2),R(3) we go through. This is well known, however we are showing this through the use of the new placement space. This may have more intuitive appeal than other ways of proving the trefoil’s knottedness as well as showing the usefulness of placement space. Other unknown things may be shown after we show the way with this initial demonstation.

So when I write: 1. First we move the trefoil so that it no longer moves in R^3 I mean as is shown in one of the diagrams below, techniqually, basically we begin to exisit in placement space P^3.

When we do this, because we can have as many ‘parts’ as we wish in one placement (by taking them from the space)(parts are what I call points or locations in placement space)(they can be mobile, unlike points or locations in R^3). At the beginning of a geometry we indicate how many parts are at any given placement. ‘Fixing’ the geometry. The placements remain fixed.

To begin the demonstration there are two parts(doublet) at any given placement, allowing us to take out 1 part of any doublet to supplement any diagram. I might call them a(1) and b(1) when they are placed on a diagram to indicate their relative location. (a is above b).

So when I write: put in parts a(1), b(1), a(2), b(2), a(3), b(3)

I mean put in a whole other trefoil of parts so we have 4 parts at each placement of the trefoil. Now we can separate one of the trefoils away, leaving one behind which can act to link us to the space R^3 from which we started.

Then the other mobile trefoil can move off into placement space.

Here I can put the a(1),b(1),a(2),b(2),a(3),b(3) parts together to form Z(1), Z(2), Z(3) Sharings. (Sharings are what I call placements which have more than one part which originate from different placements).

This is what I mean when I write: move a(2), b(2) and a(3), b(3) to the placements Z(2), Z(3).

So now we have a Z diagram as shown.

When I write: re-double the Z-Sharings I mean again take parts from the geometry and move them into the three Z-Sharings.

The goal is to create another type of sharing, the D-sharing which will allow us to move the mobile trefoil in moves which are similar to R(0),R(1),R(2),R(3). I call them S(0),S(1),S(2),S(3).

When I write: The a’s and b’s of T’ can be thought of as… I mean that after we create the D-sharing. We can leave one of a(1),b(1),a(2),b(2),a(3),b(3) and then we can have a path back to the Z-diagram. The the D-diagram can move and we can perform S(0),S(1),S(2),S(3). The D-sharing is such that the a and b do not swtich arcs . This is shown in one of the diagrams below.

Later, it maybe needed to have the separtating parts be mobile around the outside of the diagram. We could double them all up again and have one copy mobile when the other stays still. Then the one which is mobile can take up a still position and the still parts could be allowed to move to them.

When I write: a ring of a(1)b(2), a(2)b(1), a(3)b(3) this could be better notated as (((a(1)b(2)))…etc as I mean by the ((, that the two parts a(1)b(2) for example are sharing at a given placement but this is a special type of oriented sharing. Both a(1) and b(1) are together but I can’t write them on top of each other and when I reform the ‘joining’ a(1) comes out first according to the recombination.

These joinings come from application of S(2),S(1) to the D-diagram and can be done in three different ways, leading to three different expressions.