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5. Amphicheiral rational knots

A knot is amphicheiral (or enantiomorphic) [22-29] if its "left" and "right" form are equivalent, this means, one could be transformed to the other by an ambient isotropy (sequence of Redemeister moves). If a knot k could be represented by an antisymmetrical vertex-bicolored graph on a sphere, it is amphicheiral [22-24]. In this case, for the oriented knot k there exists a symmetry transposing orientations of vertices, i.e. mutually exchanging vertices with the signs +1 and -1. As its antisymmetries, they may occur sense-reversing antisymmetries: rotational antireflection or anti-inversion, and a sense-preserving antisymmetry: 2-antirotation. If the corresponding antisymmetry group of the vertex-bicolored graph of an amphicheiral knot contains at least one such sense-reversing antisymmetry, we have an invertible amphicheiral knot; otherwise, it is non-invertible. For the knot 22, 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 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, having a symmetrical Conway symbol, and it is clearly visible from their rigidly achiral [22-24, 29] presentations. Hence, a rational knot is amphicheiral iff its Conway symbol is symmetrical and it has an even number of crossings [15]. For 4 £ n £ 12, amphicheiral rational knots are: 22, 2122, 42, 3123, 24, 4124, 3143, 2322, 212212, 2162, 62, 5125, 4224, 34, 2422, 321223, 312213, 26, 221422, 21212212, 2182, and all of them are invertible. The period of the knot 22 is 2, and the same holds for all of the family pq  (p ³ q ³ 2). For the invertibility of amphicheiral knots we have one more criterion: an amphicheiral knot is invertible if its period is 2; otherwise, it is non-invertible. 

 

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