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March 13, 2024

Showing the Culprit knot is the unknot

Filed under: knots,Mathematics,unknotting the Culprit knot — Rob burchett @ 2:26 pm

Starting with the diagram of the Culprit knot, I am trying to find a way of showing that I can form the unknot without increasing the crossing number.

I do so by placing the knot in placement space. The the locations are free to move in a space of locations of locations. We keep the locations connected in the knot as they were originally connected.

Specifically, they can move around in a loop along the path of the knot.

Once we decompose two crossings into a joining (alpha-beta’s), we can double up the knot diagram again. One diagram is still and the other I can move the locations around again along the path of the new knot diagram. Then I can decompose again, etc. This means I can move one alpha-beta past another.

I then decompose completely and look for another way to put the knot back together. This new diagram is obtainable from the original diagram by the usual Reidmeister moves.