4. Rational linksThe second, rational world (or Rworld) consists of rational links. Its basic links are 2^{l} (l ³ 2). Every generating rational link with n crossings (n ³ 4) is uniquely defined in Conway notation as 2(n4)2, so the number of different generating rational links is d ^{*} (n) = b(n4). Every rational link with n crossings is uniquely represented in Conway notation as a n ^{**}decomposition, so their complete number is d ^{**}(n) = 2^{n4}+2^{[n/2]2}, for every n fixed (n ³ 4).
Every rational link is 1 or 2component. Let be given
a rational link l in Conway notation.Working with all numbers reduced
mod 2, we introduce the following cancelation rules [15]:
If the Boolean function f satisfies the conditions:
then l is a knot if f(l) = 1, and 2component link if f(l) = 0. Because generating links are completely sufficient, we will restrict our attention to them, and to infinite families generated by them. The complete list of generating rational links for 4 £ n £ 12 is: Table 1
Let us consider the first nontrivial infinite family of rational links, generated by 2^{2}. Its graph is the tetrahedron with two colored nonadjacent edges. From it, we obtain the infinite family pq (p ³ q ³ 2), consisting of [n/2]1 links for every n fixed (n = p+q). The joint properties of links belonging to this family and their symmetrical distribution is illustrated by the table, where every knot and 2component link is given also in standard Alexander&Briggs notation, consequently extended to the knots with more then 10, and 2component links with more then 9 crossings. For every knot is given its Alexander polynomial D(t) [16], and for every 2component link its reduced Alexander polynomial D(t,t) [9], both abbreviated thanks to their symmetry. In the case of knots, a_{0}+a_{1}+¼+a_{k} means a_{0}+a_{1}t+¼+a_{k}t^{k}+¼+a_{2k}t^{2k}, and for 2component links a_{0}+a_{1}+¼+a_{k} means a_{0}+a_{1}t+¼+a_{2k}t^{2k}+a_{2k+1}t^{2k+1}. For every knot, in the corresponding upper left corner is given its unknotting number, and amphicheiral invertible knots are denoted by "f" in upper right corner. In the family pq (p ³ q ³ 2) we have knots, and 2component links for p = q = 1 (mod 2).
Using the combinatorial results mentioned, we could enumerate
different particular families, or particular links belonging to them. For
example, for every fixed n, every asymmetrical generating rational
link represented by (l,k)bicomposition of n generates
