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]: