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15. Amphicheirality

For n £ 12 there is 21 rational amphicheiral knot [15], divided into infinite subfamilies of amphicheiral rational knots (p,q,r ³ 2): p2, p = 0 (mod 2), (22, 42, 62); p12p, (2122, 3123, 4124, 5125); p14p, p = 1 (mod 2), (3143); p16p, p = 0 (mod 2), (2162); p18p, (2182); pq2p, (p = 0 (mod 2)) Ú(p = q = 1 (mod 2)), (24, 2322, 34, 2422, 4224); pq12qp, p = 1 (mod 2), (321223); p1q21p, (p = 1 (mod 2)) Ú(p = q = 0 (mod 2)), (212212, 312213); pq14qp, (p ¹ q (mod 2)) Ú(p = q = 0 (mod 2)), (221422); p1q12q1p, p = 0 (mod  2), (21212212); pqr2qp, (p = q = r (mod 2)) Ú(p+q+r = 0 (mod 2)) and (q ¹ r (mod   2)) Ú(p = r = 0 (mod 2)) and (q = 1 (mod 2)), (26). All rational amphicheiral knots could be derived from the knot 22, replacing both of its digons by the same rational tangle. This way, the effect of the rotational antireflection existing in 22 remains preserved, so all of them are invertible. 

The amphicheiral knots of A-world originate from the same source: they could be obtained by such replacements of digons from the rational links with an even number of crossings, given by a symmetrical Conway symbol. By replacing both digons in 22 by the same S-tangle, we obtain for n £ 12 the amphicheiral knots (3,2)(3,2), (21,2)(21,2), (22,2)(22,2), (212,2)(212,2), (3,3)(3,3), (21,21)(21,21), (3,2+)(3,2+), (21,2+)(21,2+). Because their amphichierality results from the 2-antirotation, all of them are non-invertible. Proceeding in the same manner for larger values of n, we obtain amphicheiral knots of A-world (e.g. from 2122 we derive (3,2)12(3,2), (21,2)12(21,2) for n = 14, etc.). 

The main origin of amphicheiral knots will be the polyhedrons with the symmetry group G containing rotational reflection, rotation of order 2, or inversion. First such polyhedron is 6*. For n £ 12 from it we obtain the amphicheiral knots: .2.2 for n = 8; .21.21, .2.2.20.20, 2.2.2.2 for n = 10; .4.4, .31.31, .22.22, .3.3.2.2, .21.21.2.2, .3.3.20.20, .21.2.210.20, 3.2.2.3, 2.21.21.2, 30.2.2.30, 20.21.21.20, 2.2.2.2.20.20, 2.2.20.2.2.20 for n = 12. 

For 6*, G = [3,4] and G' = [3+,4], so the amphicheiral knots derived from it will contain antirotation or rotational antireflection. This way, from 6* we derive non-invertible amphicheiral knots (e.g. .2.2, .2.2.2.2, etc.), as well as the invertible ones (e.g. .2.2.20.20). 

The next such polyhedron 8* is the invertible amphicheiral knot 8* with the antireflection. From it, we derive amphicheiral knots: 8*20.20, 8*2:.2 for n = 10; 8*3.3, 8*21.21, 8*30.30, 8*3:.3, 8*30:.30, 8*210:.210, 8*.20:2.2:20, 8*2.2.2.2, 8*20.20.20.20, 8*2.20.20.2, 8*2.2:.20.20, 8*20.20:.20.20 for n = 12. 

From the same reasons as before, some of the obtained amphicheirals are invertible (e.g. 8*, etc.), and others are not (e.g. 8*20.20, 8*2:.2, etc.). 

The third such polyhedron is the amphicheiral invertible knot 10*. From it, we derive amphicheiral knots 10*2::.2, 10*20:::20, 10*2::::2 for n = 12. 

From polyhedron 10** we derive amphicheiral knots 10**2::.2, 10**20::.20, 10**:2.2, 10**:20.20, 10**:2::.2, 10**:20::.20 for n = 12. 

Finaly, for n = 12 we have three amphicheiral knots corresponding to the polyhedrons 12B, 12K and 12L. 

The axis of 2-antirotation is projected in the perpendicular projection plane into the center of antisymmetry. Each of the amphicheial knots mentioned, except the knot H59 .21.2.210.20, contains 2-antirotation and admits one or several centro-antisymmetrical projections, i.e. possesses a discernible antisymmetry [22]. For the exceptional knot .21.2.210.20 = H59 = H60, the amphicheirality is the result of anti-inversion, and no one of its projections is not centro-antisymmetrical. Anyway, its corresponding projection polynomials d(H59(t)) = t12-4t8+2t6-t4-6t2, d(H60(t)) = t12-4t8-2t6+3t4+2t2, with the property d(t) = d(-t), or symmetrical Liang polynomials [28]:

A(H59) = -4t-6-6t-5+49t-4+18t-3-113t-2+68t-1-19+68t-113t2+18t3+49t4-6t5-4t6,

A(H60) = 108t-4-204t-3-340t-2+594t-1-19+594t-340t2-204t3+108t4 

could discover this concealed antisymmetry [22], visible from the corresponding vertex-bicolored graphs.
 

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