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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 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 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: 2^{2}, 21^{2}2, 4^{2}, 31^{2}3, 2^{4},
41^{2}4, 31^{4}3, 23^{2}2, 212^{2}12, 21^{6}2,
6^{2}, 51^{2}5, 42^{2}4, 3^{4}, 24^{2}2,
321^{2}23, 312^{2}13, 2^{6}, 2^{2}1^{4}2^{2},
2121^{2}212, 21^{8}2, and all of them are invertible. The
period of the knot 2^{2} 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. |