They are obtained as the result of the LinKnot function RatGenSourKL (webMathematica RatGenSourKL) that calculates the number and Conway symbols of all rational source KLs and rational generating KLs with n crossings. Even at that level, we can recognize a regular distribution of the source links graphs and their corresponding alternating KLs (e.g., of the type 2 ... 2, where 2 occurs l times, abreviated as 2^{l}; they are knots for even l and 2component links for odd l) (Figs. 1.501.51). If you are interested in the number of components of particular KLs, for rational KLs it is always 1 or 2. In order to find it, to each of them you can apply the LinKnot function fComponentNo that calculates the number of components, but there are some more elegant solutions. The first of them is to apply the general rules for finding a number of components of a rational KL. The rules are the following: let a rational link L be given in the Conway notation. Working with all numbers reduced mod 2, we introduce the following cancellation rules:
If the Boolean function f satisfies the conditions f(0) = 0, f(1) = 1, f(xy) = 1f(x)f(y), then L is a knot if f(L) = 1, and 2component link if f(L) = 0. For example, we conclude that 6 2 1 4 4 is a knot, because 6 2 1 4 4 = 0 0 1 0 0 mod 2, 0 0 1 0 0 = 0 0 1 (Rule 1), 0 0 1 = 0 1 (Rule 2) and f(0 1) = 1f(0)f(1) = 1. On the other hand, 4 5 1 2 3 is a link, since 4 5 1 2 3 = 0 1 1 0 1 (mod 2), 0 1 1 0 1 = 0 1 1 1 = 0 1 0 (Rule 2), 0 1 0 = 0 (Rule 1), and f(0) = 0.
