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.

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.

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.

Here I am showing the final step where the ar,bs joinings, represented by dots diagram 2, are moved into the D-sharings of the equivalent diagram. This is done by reversing the basic transformation.

The top set of diagrams show the knot in R^3. We start with one form of the trefoil, as described in diagram 1 in the posts below. Then we transform it using the basic transformation. After this we move it using the 4 Reidemeister moves to obtain some new diagram which has the same ‘shape’ as the original diagram. The crossings may or may not be differet. The direction is the same as the original. Then it might be possible to end up with an unknotted form or a mirror image form.

Below this set of moves, we can have moves which track along in P^3. First converting to the D-diagram from the original form above. The transformation is gone through and this is done to eliminate the D-joinings so that we only have Q-joinings undergoing the counterparts of the Reidemeister moves. Eventually we end up with diagram 3 which can be compared with the diagram which comes from converting diagram 2 to placement space (P^3). See post above.