# 10. Basic polyhedron 6*

The first basic polyhedron is octahedron 6*, with the graph symmetry group G = [3,4] of order 48, generated by 4-rotation S = (1)(2,3,5,6)(4), 2-rotation T = (1,3)(2,5)(4,6) and inversion Z = (1,4)((2,5)(3,6). It is the graph of Borromean rings, the first nontrivial Brunnian 3-component link, with the link symmetry group G' = [3+,4] [4]. After introducing orientation, the obtained antisymmetry group contains a rotational antireflection, from which results the amphicheirality of Borromean rings.

From 6* we derive source links substituting its vertices by digons. First we make all different symmetry choices of n-6 vertices (7 £ n £ 12), i.e. all different vertex bicolorings of octahedron. Their number we could find using Polya Enumeration Theorem (PET) [44]. For G = [3,4], ZG = 1/48(t16+3t14t2+9t12t22+6t12t4+7t23+6t2t4+8t32+8t6), and by the coefficients of ZG(x,1) = 1+x+2x2+2x3+2x4+x5+x6 is given, respectively, the number of different choices of n-6 vertices for 6 £ n £ 12. For 7 £ n £ 12, that vertex bicolorings are: {1}; {1,2}, {1,4}; {1,2,3}, {1,2,4}; {1,2,4,5}, {1,2,3,4}; {1,2,3,4,5}; {1,2,3,4,5,6}, and to them correspond, respectively, source links of the form .a; .a.b, .a:b; .a.b.c, a:b:c; .a.b.c.d, a.b.c.d; a.b.c.d.e; a.b.c.d.e.f, given in Conway notation. After that, in every chosen vertex we make one of two possible substitutions (2 or 20), having in mind the symmetry of vertex bicolored octahedron. In the terms of colorings, this is a next bicoloring: the bicoloring of chosen vertices. For n £ 12, the source links obtained from 6* by the vertex substitutions, are given in Table 7. Among them, for n = 11, there is 3-component link 2.20.2.20.2, omitted in [15].

Table 7

 n=7 .2 n=11 2.2.2.2.2 2.2.2.2.20 n=8 .2.2 2.2.2.20.2 .2.20 2.2.2.20.20 2.20.2.2.20 .2:2 20.2.2.2.20 .2:20 2.20.2.20.2 n=9 .2.2.2 2:2:2 n=12 2.2.2.2.2.2 .2.2.20 2:2:20 2.2.2.2.2.20 .2.20.2 2:20:20 2.2.2.2.20.20 20:20:20 2.2.2.20.2.20 2.2.20.2.2.20 n=10 .2.2.2.2 2.2.2.2 2.2.2.20.20.20 .2.2.2.20 2.2.2.20 2.20.2.20.2.20 .2.2.20.20 2.2.20.2 .2.20.2.20 2.2.20.20 2.20.2.20 20.2.2.20 20.2.20.20

The PR *-subworld of  P-world consists of links obtained replacing digons in the source links by R *-tangles. From the symmetry of the source links, we conclude that all further derivation represent the series of corresponding partitions with the given permutation group P. Two permutation groups are equivalent iff their permutation representations are isomorphic. Equivalent permutation groups produce the same number of  P-partitions, mutually corresponding according to the mentioned isomorphism. Hence, we will classify source links from Table 7 with respect to P-equivalence, and then derive generating links from one representative of each class. For 7 £ n £ 11, we have the following classes: .2 with P @ {(1)}; .2.2, .2.20, .2:2, .2:20 with P @ {(1,2)}; .2.2.2, .2.20.2, 2:2:20, 2:20:20 with P @ {(1,3)(2)}; .2.2.20 with P @ {(1)(2)(3)}; 2:2:2, 20:20:20 with P @ {(1,2,3)}; .2.2.2.2 with P = {(1,2,4,5)}; .2.2.2.20, 2.2.2.20 with P @ {(1)(2)(4)(5)}; .2.2.20.20, .2.20.2.20 with P @ {(1,2)(4,5), (1,4)(2,5)}; 2.2.2.2, 20.2.2.20 with P @ {(1,4)(2,3)}; 2.2.20.2, 2.2.20.20, 2.20.2.20, 20.2.20.20 with P @ {(1)(2,3)(4)}; 2.2.2.2.2, 2.2.2.20.20, 2.20.2.2.20 with P @ {(1,2)(3)(4,5)}; 2.2.2.2.20, 2.2.2.20.2 with P @ {(1)(2)(3)(4)(5)}; 20.2.2.2.20, 2.20.2.20.2 with P @ {(1,2)(3)(4,5), (1,5)(2,4)(3)}. Taking as the representative of each class its first link, we obtain the list of generating PR *-links (Table 8) derived from that representatives for 7 £ n £ 12. The complete list of generating PR *-links derived from 6* for 7 £ n £ 12 we could directly obtain from Table 8, using the mentioned isomorphism, and including in the list the source links for n = 12 (Table 7). After that, by replacing every ~k * by k * we could obtain all such links. The sign à has the same meaning as *, but it is used to denote mutually equivalent (commuting, interchangeable) partitions. For example, .~4à.~4à denotes .22.22 and .22.211 (=.211.22), .3à.3à denotes .3.3, .3.21(=.21.3) and .21.21, 3à:3à:3à denotes 3:3:3, 3:21:21(=21:3:21=21:21:3) and 21:21:21, etc.

Table 8

 n=7 .2 n=8 .~3* .2.2 n=9 .~4* .~3*.2 .2.2.2 .2.2.20 2:2:2 n=10 .~5* .~4*.2 .~3*.2.2 .~3*.2.20 ~3*:2:2 .~3à.~3à .2.~3*.2 .2.~3*.20 .2.2.~3*0 n=11 .~6* .~5*.2 .~4*.2.2 .~4*.2.20 ~4*:2:2 .~4*.~3* .2.~4*.2 .2.~4*.20 ~3à:~3à:2 .~3*.~3*.2 .2.2.~4*0 .~3à.2.~3à .~3*.~3*.20 .~3*.2.~3*0 .2.~3*.~3*0 n=12 .~7* .~6*.2 .~5*.2.2 .~5*.2.20 ~5*:2:2 .~5*.~3* .2.~5*.2 .2.~5*.20 ~4*:~3*:2 .~4à.~4à .~4*.~3*.2 .2.2.~5*0 ~3à:~3à:~3à .~4*.2.~3* .~4*.~3*.20 .~3*.~4*.2 .~4*.2.~3*0 .~3à.~3*.~3à .~3*.~4*.20 .~3*.2.~4*0 .2.~4*.~3*0 .2.~3*.~4*0 .~3*.~3*.~3* n=10 .2.2.2.2 .2.2.2.20 .2.2.20.20 2.2.2.2 2.2.20.2 n=11 .~3*.2.2.2 .~3*.2.2.20 .~3*.2.20.20 ~3*.2.2.2 ~3*.2.20.2 .2.~3*.2.20 2.~3*.2.2 2.~3*.20.2 .2.2.~3*.20 2.2.20.~3* .2.2.2.~3*0 n=12 .~4*.2.2.2 .~4*.2.2.20 .~4*.2.20.20 ~4*.2.2.2 ~4*.2.20.2 .~3*.~3*.2.2 .2.~4*.2.20 .~3à.~3à.20.20 2.~4*.2.2 2.~4*.20.2 .~3à.2.~3à.2 .2.2.~4*.20 .~3à.2.~3à0.20 ~3*.~3*.2.2 2.2.20.~4* .2.2.2.~4*0 .~3à.2.20.~3à0 ~3*.2.~3*.2 ~3*.~3*.20.2 .~3*.~3*.2.20 ~3à.2.2.~3à ~3*.2.20.~3* .~3*.2.~3*.20 2.~3à.~3à.2 2.~3à.~3à0.2 .~3*.2.2.~3*0 2.~3*.20.~3* .2.~3*.~3*.20 .2.~3*.2.~3*0 .2.2.~3*.~3*0 n=11 2.2.2.2.2 2.2.2.2.20 20.2.2.2.20 n=12 ~3*.2.2.2.2 ~3*.2.2.2.20 ~3*0.2.2.2.20 2.~3*.2.2.2 2.~3*.2.2.20 20.2.~3*.2.20 2.2.~3*.2.2 2.2.~3*.2.20 2.2.2.~3*.20 2.2.2.2.~3*0