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December 23, 2020

Clearer explantion for the Trefoil

Filed under: knots,Mathematics — Rob @ 11:23 am

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.

December 19, 2020

Towards a more general understanding of knottedness

Filed under: knots,Mathematics — Rob @ 3:49 pm

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.

Filed under: Mathematics,set theory — Rob @ 1:23 pm

December 18, 2020

Final transformation

Filed under: knots,Mathematics — Rob @ 12:19 pm

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.

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.

December 14, 2020

Filed under: Mathematics,set theory — Rob @ 1:20 pm

Filed under: Mathematics,set theory — Rob @ 1:18 pm

December 12, 2020

Filed under: Mathematics,number theory — Rob @ 7:50 am

Filed under: Mathematics,number theory — Rob @ 7:48 am

December 2, 2020

Set theory idea

Filed under: Mathematics,set theory — Rob @ 9:16 pm
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