How do you prove a knot is achiral? Do you just show swap the under crossings and the over crossings?
2026-03-27 20:27:03.1774643223
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Knots isotopic to their mirror image
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Achiral knots must have self-conjugate HOMFLY polynomial. This does not prove that a knot is achiral, though, but it is a good criterion to test for chirality. Unfortunately it is not a complete invariant - there are also other knots having self-conjugate polynomial. For a program in Mathematica finding the Reidemeister moves (in some easier cases), to show achirality, see here.
Yes, it suffices to give a sequence of Reidemeister moves to transform the knot into its mirror image. Proving that a knot is chiral is much more difficult.