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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): *p*^{2}, *p* = 0 (*mod* 2), (2^{2},
4^{2}, 6^{2}); *p*1^{2}*p*, (21^{2}2,
31^{2}3, 41^{2}4, 51^{2}5); *p*1^{4}*p*,
*p* = 1 (*mod* 2), (31^{4}3); *p*1^{6}*p*,
*p* = 0 (*mod* 2), (21^{6}2); *p*1^{8}*p*,
(21^{8}2); *pq*^{2}*p*, (*p* = 0 (*mod*
2)) Ú(*p* = *q *= 1 (*mod
*2)), (2^{4}, 23^{2}2, 3^{4}, 24^{2}2,
42^{2}4); *pq*1^{2}*qp*, *p* = 1 (*mod*
2), (321^{2}23); *p*1*q*^{2}1*p*, (*p*
= 1 (*mod *2)) Ú(*p* = *q*
= 0 (*mod* 2)), (212^{2}12, 312^{2}13); *pq*1^{4}*qp*,
(*p* ¹ *q* (*mod *2)) Ú(*p*
= *q* = 0 (*mod *2)), (2^{2}1^{4}2^{2});
*p*1*q*1^{2}*q*1*p*, *p* = 0 (*mod *
2), (2121^{2}212); *pqr*^{2}*qp*, (*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)), (2^{6}). All rational amphicheiral
knots could be derived from the knot 2^{2}, replacing both of its
digons by the same rational tangle. This way, the effect of the rotational
antireflection existing in 2^{2} 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 2^{2} by the same *S*-tangle,
we obtain for *n *£ 12 the amphicheiral
knots (3,2)(3,2), (21,2)(21,2), (2^{2},2)(2^{2},2), (21^{2},2)(21^{2},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 21^{2}2 we derive
(3,2)1^{2}(3,2), (21,2)1^{2}(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 H_{59}
.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 =
H_{59} = H_{60}, the amphicheirality is the result
of anti-inversion, and no one of its projections is not centro-antisymmetrical.
Anyway, its corresponding projection polynomials *d*(H_{59}(*t*))
= *t*^{12}-4*t*^{8}+2*t*^{6}-*t*^{4}-6*t*^{2},
*d*(H_{60}(*t*)) = *t*^{12}-4*t*^{8}-2*t*^{6}+3*t*^{4}+2*t*^{2},
with the property *d*(*t*) = *d*(-*t*), or symmetrical
Liang polynomials [28]:
*A*(H_{59}) = -4*t*^{-6}-6*t*^{-5}+49*t*^{-4}+18*t*^{-3}-113*t*^{-2}+68*t*^{-1}-19+68*t*-113*t*^{2}+18*t*^{3}+49*t*^{4}-6*t*^{5}-4*t*^{6},
*A*(H_{60}) = 108*t*^{-4}-204*t*^{-3}-340*t*^{-2}+594*t*^{-1}-19+594*t*-340*t*^{2}-204*t*^{3}+108*t*^{4}
could discover this concealed antisymmetry [22], visible from the corresponding vertex-bicolored graphs. |