## 1.8  Rational world and KL invariants

The origin (or O-world) is the basic polyhedron 1*, the 4-valent graph with one vertex, the usual symbol of infinity (¥). The first, linear world (or L-world) contains only one source link: a Hopf link 2 (212 in the classical notation). We use it to derive the infinite family p of alternating KLs, with KL shadows represented by p-gons with digonal edges (p ³ 2). For odd p we have the infinite series of alternating knots 3, 5, 7, ... (or 31, 51, 71,...), and for even p the infinite series of alternating 2-component links 2, 4, 6, ... (or 212, 412, 612, ...) (Fig. 1.43). For n > 1 the members of the linear world can be included in the next, rational world.

Rational world (or R-world) consists of rational links. In Conway notation a rational KL is any sequence of natural numbers not beginning or ending with 1, where each sequence is identified with its inverse. From this definition we can compute the number of rational KLs with n crossings.

Theorem The number of rational KLs  is given by the formula

 2n-4+2[n/2]-2

that holds for every n ³ 4.

Proof: Think of n as a linearly ordered set of, say, stars; then choosing a composition amounts to choosing a subset of the set of n-1 spaces between the stars. E.g.,

 * *  |  *  |  * * *  |  *
(choices of spaces indicated by bars) is the composition 2 1 3 1 of 7.
Suppose that n ³ 3, and let bn denote the number of compositions of n with no 1 in either first or last position, and where a composition is identified with its reverse.
Compositions of n not having 1 as either first or last part correspond to sets of spaces between n ordered dots not containing either the first or last space, so there are an = 2n-3 of these.
If sn of these are symmetric (i.e. equal to their reverses) then we have

 bn = an - sn 2 + sn = an +sn 2 .
Choosing a symmetric composition of n without 1 in first or last place corresponds to choosing a subset of the set of spaces up to and including the middle space (if there is one) but excluding the first space. (The rest of the spaces are determined by symmetry). There are [[(n-2)/2]] such spaces, and thus sn = 2[[(n-2)/2]].
Hence

 bn = 2n-3 + 2[[(n-2)/2]] 2 = 2n-4+ 2[[(n-2)/2] - 1] = 2n-4+2[n/2]-2.

This very simple formula, derived first by C. Ernst and D.W. Sumners (1987) in a different form, and later independently by S. Jablan, is probably one of the first combinatorial results in knot theory, giving the exact number of KLs belonging to a particular class, and not only its approximation. For n ³ 4 we can compute the first 20 numbers of that sequence. The result is the sequence: 2, 3, 6, 10, 20, 36, 72, 136, 272, 528, 1056, 1080, 4160, 8256, 16512, 32986, 65792, 131328, 262656, 524800, ... This sequence is included in On-Line Encyclopedia of Integer Sequences as the sequence A005418. The number of rational knots with n crossings (n ³ 3, sequences A018240 and A090596) is given by the formula

 2n - 3 + 2[n/2] - 2(n - 1) mod 2 + (-1)(n - 1)[n/2] mod 2 3

so we can simply derive the formula for the number of rational links with n crossings as well.