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December 15, 2020

A quick sketch of the conclusion

Filed under: knots,Mathematics — Rob @ 5:25 pm

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.

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