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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.

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