Symmetry Type

If a knot is viewed as the oriented diffeomorphism class of an oriented pair, K = (S3, S1), with Si diffeomorphic to Si, there are four oriented knots associated 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.

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 type of symmetry it has, and likewise for negative amphicheiral. If it has none of these types of symmetry it is called chiral, and if it has all three it is called fully amphicheiral.

For prime knots with fewer than 12 crossings, all amphicheiral knots are negative amphicheiral.