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December 2, 2020

Quick trip through the discussion-page 3

Filed under: knots,Mathematics — Rob @ 6:35 pm

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.

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