For every even n (n³ 6) we have an ([n/ 2])-component source link 2,2,...,2. Its shadow is an n-gonal prism with colored lateral edges denoting bigons. The symmetry group G = [2,n] of that graph is generated by an n-fold rotation, vertical, and horizontal plane reflection (Grünbaum and Shephard, 1985). Hence, we conclude that any such source link 2,2,...,2 remains invariant after cyclic permutations of bigons (cyclic rotations), where every permutation is identified with its reverse (because of a vertical reflection), or if all bigons are reverted (thanks to a horizontal reflection). We can obtain all KLs belonging to the stellar world from the source KLs mentioned by replacing bigons by chains of bigons and using symmetry. In fact, in the source KLs of the stellar world we replace bigons by rational tangles 3, 4, ..., belonging to the linear world (L-world). Because the L-world is a subworld of the rational world (R-world), we will continue with the derivation of stellar-rational KLs, treating stellar-linear KLs as a subworld of the RS-world. In this way, the complete S-world will be included in the SR-world.

The next world is the arborescent world. Its members are multiple combinations of KLs belonging to the preceding worlds. For example, we have the following combinations: stellar-rational subworld (S(R)-subworld) obtained by replacing bigons in stellar source KLs by rational tangles not beginning by 1, rational-stellar subworld (R(S)-subworld) obtained by replacing the first and last bigon in rational source KLs by stellar KLs, (R(S))(R)-subworld obtained by replacing in the R(S)-subworld bigons belonging to stellar parts of the source KLs by rational tangles not beginning by 1, stellar-stellar subworld (S(S)-subworld) obtained by replacing bigons in stellar source KLs by stellar KLs, etc. In a certain sense, the complete system of KLs looks like Chinese nested spheres, where every sphere is placed inside the preceding one.

Every additive expression of a natural number n as an ordered sequence of natural numbers is called a decomposition of n (or an ordered partition of n). We denote by [`n] the set of all decompositions of n not beginning with 1. For example,

[`3] is the set {3,2 1},

[`4] is {4,3 1 2 2,2 1 1},

[`5] is {5,4 1,3 2,3 1 1,2 3,2 2 1, 2 1 2, 2 1 1 1}, etc.

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