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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 '13 at 1:56
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Potentially. 8_20 allows a reidemeister III move on a minimal crossing diagram. Here is a link to such a diagram from knotinfo.

http://www.indiana.edu/~knotinfo/diagram_display/diagram_display_8_20.html

Certainly if a minimal crossing diagram is alternating then there is no reidemeister III available simply because the move requires either two successive over or under crossings, so then if a minimal crossing diagram allows a reidemeister III it must be nonalternating.

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