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If you have a knot which has minimal crossings, can you do a Reidemeister III move?

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I think there are examples where you can do this. I think you can't do a Reidemeister move on an alternating diagram (where if you travel along the knot you alternated over- and under-passes), but there are knots whose minimal crossing diagrams aren't alternating --- on such a knot, you should be able to do a Reidemeister III. – Gerry Myerson Jan 11 at 1:56

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