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June 26, 2020

The knottedness and chirality of the trefoil

Filed under: knots,Mathematics — Rob @ 2:32 pm

This is an excerpt from my larger work. You can access this at the left under the categories mathematics, knots.

Basically what I do is look at two starting/ending diagrams of the trefoil which have the same “shape” but could have different crossings. That is, at corresponding crossings instead of one arc being over another the arcs could be reversed and one arc would be under the other.

Then if the diagrams are to be equivalent they must be related through a sequence of the three basic knot moves.

However, I start with a different space and take the diagrams out of the space they are in.

I create a more basic space of “locations of location”. This then means that locations can be mobile but not in the sense of a moving location in calculus. Locations can move, freed up from there fixed positions in the new space.

Then the diagrams move to fill back in the void left when the counterpart diagrams left.

Then can I form any unknotted or mirror image knots? If it turns out I can’t then the trefoil is knotted and chiral.

The knottedness and chirality of the trefoil page 2

Filed under: knots,Mathematics — Rob @ 2:31 pm

This is an excerpt from a larger work found at the left under the categories mathematics, knots.

Basically what I do is look at two starting/ending diagrams of the trefoil which have the same shape but could have different crossings. That is, at corresponding crossings instead of one arc being over another the arcs could be reversed and an arc would be under another.

Then if the diagrams are to be equivalent there must be a sequence of the three basic knot moves leading from one to the other.

However, I start with a different space and take the diagrams out of the space they are in. I create a more basic space of “locations of location”. Locations are then no longer at fixed positions but are able to move.

Then the diagrams move to fill in the voids left by their counterparts.

Then can I form any unknotted or mirror image knots? If it turns out I can’t then the trefoil is knotted and chiral.

It is also possible that the ar and bs joining pairs can move past one another. This is because they are parts. We could have a D-sharing forming and then an ar or bs joining could move through the sharing, then the D-sharing can reform and we can move one joining pair past another, if we allow another level to the geometry. Then it might be possible to have a different shape appearing. This can be some future work.

An extended space, continued, supplemental

Filed under: knots,Mathematics — Rob @ 2:25 pm

June 22, 2020

An extended space, further conclusion, page 1

Filed under: knots,Mathematics — Rob @ 2:31 pm

An extended space, further conclusion

Filed under: knots,Mathematics — Rob @ 2:29 pm

June 15, 2020

An extended space, conclusion page 1

Filed under: knots,Mathematics — Rob @ 1:50 pm

An extended space, conclusion page 2

Filed under: knots,Mathematics — Rob @ 1:49 pm

An extended space, conclusion-page 3

Filed under: knots,Mathematics — Rob @ 1:46 pm

June 9, 2020

An extended space continued page 1

Filed under: knots,Mathematics — Rob @ 8:48 am

An extended space continued page 2

Filed under: knots,Mathematics — Rob @ 8:47 am
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