4. Rational links

The second, rational world (or R-world) 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(n-4)2, so the number of different generating rational links is  d ^* (n) = b(n-4). Every rational link with n crossings is uniquely represented in Conway notation as a n ^**-decomposition, so their complete number is d ^**(n) = 2^n-4+2^[n/2]-2, for every n fixed (n ³ 4). 

Every rational link is 1- or 2-component. 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:

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 2². 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 2-component link is given also in standard Alexander&Briggs notation, consequently extended to the knots with more then 10, and 2-component links with more then 9 crossings. For every knot is given its Alexander polynomial D(t) [16], and for every 2-component link its reduced Alexander polynomial D(t,t) [9], both abbreviated thanks to their symmetry. In the case of knots, a₀+a₁+¼+a_k means a₀+a₁t+¼+a_kt^k+¼+a_2kt^2k, and for 2-component links a₀+a₁+¼+a_k means a₀+a₁t+¼+a_2kt^2k+a_2k+1t^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 2-component 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