By using symmetry, from a short symbol [`3],2,2 we obtain (3,2,2) and (2 1,2,2), from [`4],2,2 we obtain (4,2,2), (3 1,2,2), (2 2,2,2), (2 1 1,2,2), from [`3],[`3],2 we obtain (3,3,2), (3,2 1,2) = (2 1,3,2) and (2 1,2 1,2), and then eliminate duplications. Continuing in this way, we obtain all alternating KLs of the stellar-rational subworld. Thus, the stellar world is completely included in the SR-world.

Instead of repeating the complete derivation of S+ links from the source links of the type 2,2,...,2+k, (k = 1,2,...), they can be obtained directly from the stellar-rational KLs derived, by adding an appropriate number of pluses. For n = 7 we add one plus to stellar-rational KLs with 6 crossings, for n = 8 we derive S+ links from SR-links with 6 crossings by adding two pluses, and from SR-links with 7 crossings by adding one plus, etc. This way, for every n we derive KLs with k pluses from SR-links with n-k, ..., n-2, n-1 crossings (n ³k+6, k = 1,2,...).

The LinKnot functions fStellarBasic (webMathematica fStellarBasic), fStellar (webMathematica fStellar), and fStellarPlus (webMathematica fStellarPlus) calculate the number and Conway symbols of all stellar and stellar-rational KLs without and with pluses for a given number of crossings n, respectively.

Here we encountered different combinatorial problems. For example, for every fixed n we could try to calculate the number of different classes in the first column of Table 1 (beginning with series 1, 1, 2, 3, 4, 5, 7 for n£ 12), or the number of source KLs, or even the number of all KLs derived from some generating KL for every n fixed, etc. For example, the number of different classes mentioned is equal to the coefficient corresponding to qn-6 in , where is the Gauss polynomial

 

 

(1-qn)...(1-qn-r+1)
(1-qr)...(1-q)
.

All such problems belong to the theory of partitions with a given symmetry group (P-partitions). Let P be a permutation group P on k objects and n ³ k be an integer. To every object ki (1 £ki£k) a natural number ni is assigned, where åi = 1k ni = n. Two partitions defined by such signed (or weighted) objects are equal iff there is a permutation from P transforming one to another. We need to find and enumerate different P-partitions. In some special cases, we can reduce problems of P-partitions to classical partition theory, but in general, this enumeration is an open problem.

 

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