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So starting with the standard diagram for the Trefoil knot. I place this in the new space and I have the sharings D1,D2,D3. These have the associated moving parts a1,b1,a2,b2,a3,b3. Then I move the outer loops inward (green and red) forming the sharings Q1,Q2,Q3,Q4,Q5,Q6. These have the associated moving labels r1,s1,r2,s2,r3,s3. Then I form the joinings (a1(r1), (b2(s2), (a3(r3), (b1(s1), (a3(r2), (b3(s3) (all red). Now this can be further decomposed to eventually form the “circle”.

Showing the third move. After reduction, only Q-types are involved.

Here I show why we must keep to the same orientation when we reform a joining after we undo it in negative space.

Here is the achirality of the knot 4(1) shown in negative space.

Here is a better post on simplifications.

I place the trefoil, partially back together then move the four joinings through the reformed 2 and 3 sharings until they can possibly form a new diagram. I show it is not possible to form this diagram in positive space so that the trefoil is knotted.