Symmetry Type

If a knot is viewed as the oriented diffeormorphism class of an oriented pair, K = (S3, S1), with Si diffeomorphic to Si, there are 4 oriented knots associate to any particular knot K. In addition to K itself, there is the reverse, Kr = (S3, -S1), the concordance inverse, -K = (-S3,-S1), and the mirror image, Km = (-S3,S1). A knot is called reversible if K = Kr, negative amphicheiral if K = -K and positive amphicheiral if K = Km.

Of course a knot possessing any two of these types of symmetry has all three. Thus, in the table, a knot is called reversible if that is the only symmetry it has. Similarly for negative amphicheiral. If it has none of these symmetries it is chiral, and if it has all three it is called fully amphicheiral.

It should be pointed out that for prime knots with less than 12 crossings, all amphicheiral knots are negative amphicheiral.