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.

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.