2 VISUALIZING AMPHICHEIRALITY

 

In the case of rational knots, all amphicheiral knots are derived from the same source: from the figure-eight knot 2 2 (or 41 in the classic notation). The amphicheirality of the figure-eight knot can be visualized by an animation consisting from a series of ambient isotopies transforming a “left” figure-eight knot to the “right”. 

For the knot 2 2, the graph symmetry group is G = [2+, 4], and the knot symmetry group G' = [2+, 4+] is generated by the rotational reflection, with the axis defined by the midpoints of colored (i.e., double) edges of the tetrahedron .                                    
Considering the sign of the vertices, it is a rotational antireflection. Its effect is preserved in all rational knots with an even number of crossings that have a symmetrical Conway symbol. As the result, we obtain a series of centro-antisymmetrical graphs corresponding to rational knots.