ON AMPHICHEIRAL KNOTS

Amphicheiral knots with up to 12 crossings are discussed from the perspective of their symmetry properties. By use of an algorithm that involves the development of appropriate vertex-bicolored knot graphs, rigidly achiral presentations have been found for all amphicheiral invertible prime knots with up to 10 crossings and for a selected number of such knots with 12 crossings, including 12(1994), the first example of an amphicheiral prime knot whose S2n diagram is also a reduced diagram. Characteristic properties of wire models of these presentations have been examined. The adjacency matrices of the vertex-bicolored graphs of amphicheiral knots exhibit twofold antisymmetry, and, with the sole exception of 12(427), all such knots are capable of rigidly antisymmetric presentations.