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4. Rational links

The second, rational world (or R-world) consists of rational links. Its basic links are 2l  (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) = 2n-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]:

1) for every sequence of the form xa0 ( a Î {0,1}),  xa0 = x
2) for every sequence of the form xa1 (a Î {0,1}), a1 = 1-a

If the Boolean function f satisfies the conditions:

f(0) = 0, f(1) = 1, f(xy) = 1-f(x)f(y),

then l is a knot if f(l) = 1, and 2-component 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 
 
 

n=4 22
n=5 212
n=6 23 2122
n=7 2212 2132
n=8 24 21212 2142
22122
n=9 22122 22132 2152
2312 212122
n=10 25 221222 2122122 2162
212212 22142
23122 212122
221212
n=11 2412 221322 22152 2172
23122 2121212 212142
23132 2122132
2212122
2212212
2122122
n=12 26 2212122 221422 2132132 2182
212312 21212212 22162
24122 21222122 212152
231212 23142 2122142
231222 2212132
2122122 22122122
2213212
2122132
21212122

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+ak means a0+a1t+¼+aktk+¼+a2kt2k, and for 2-component links a0+a1+¼+ak means a0+a1t+¼+a2kt2k+a2k+1t2k+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

æ 
è 
n-l-1 
l-k-1
ö 
ø 
links, etc. 
 

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