We can also use another trick. Knowing that a generating KL and all KLs derived from it have the same number of components, we can calculate the numbers of components only for generating KLs and extend that result to all KLs derived from them. For example, 2 2 is a generating 1-component KL (i.e., a figure-eight knot), and all KLs belonging to the class (2p) (2q) (p ³ 2, q ³ 2, p ³ q) will be knots. On the other hand, the generating KL 3 3 is a 2-component link, and all KLs in its family (2p+1) (2q+1) (p ³ 1, q ³ 1, p ³ q) will be 2-component links. This is just the first and simplest property that can be generalized in the case of generating KLs and their families.
A link is oriented if each of its components has an orientation. The LinKnot function fOrientedLink (webMathematica fOrientedLink) calculates Gauss codes for a link projection given by the Conway symbol, Dowker code, or P-data. The Gauss codes obtained correspond to different orientations of components (the first part of the data obtained). The second part gives the orientations of components (where + is denoted by 1, and - by 0), and the third part is the (signed) linking number of the corresponding oriented link.
As we can conclude from their definition, rational KLs are all alternating, because they do not contain negative entries. The product of tangles is the only operation used for obtaining rational KLs. Introducing negative elementary tangles into the Conway notation of rational KLs gives nothing new, because all rational KLs with mixed signs can be reduced to Conway symbols containing only positive or negative entries. For this reduction we can use continued fractions. In particular, we have integral tangles n0 = 1+...+1 - bigons or chains of bigons, and rational tangles n1...nk. To every rational tangle corresponds the continued fraction
Every continued fraction is a rational number. For rational tangles the following theorem holds:
Theorem Two rational tangles are equivalent iff their continued fractions yield the same rational number.
The proof of this theorem can be found in G.Burde and H.Zieschang (1985), or in the papers by J.R.Goldman and L.Kauffman (1997), and L.Kauffman and S.Lambropoulou (2003). For the further reading on rational tangles and rational KLs we recommend the papers by L.Kauffman and S.Lambropoulou (2002, 2002a).
The LinKnot function RatReduce (webMathematica RatReduce) reduces the Conway symbol of a rational KL by using continued fractions. In this way, from any Conway symbol of a rational KL with entries containing mixed signs (that is a non-alternating rational representation of a rational KL) we can obtain its minimal alternating representation. For example, from a non-alternating representation of a rational link given by 2 -3 1 2 -7 we obtain its minimal alternating representation 6 1 2 1 2.
The LinKnot function MSigRat (webMathematica MSigRat) calculates the value of a continued fraction and Murasugi signature (see: Murasugi, 1996, pp. 192) for every rational KL given by its Conway symbol.