The ncomponent links
(n ³ 4)
without nontrivial sublinks were described by H. Brunn (1892). Here they
are presented by a series of illustrations.
The LinKnot function fBreakComp (webMathematica fBreakComp) in a link given by Conway symbol, Dowker code, or Pdata, and by ordering number k of a cut component calculates Pdata of a link with kth component cut. The function BreakCoAll (webMathematica BreakCoAll) in a link given by its Conway symbol, Dowker code, or Pdata, cuts all components. The result are all different KLs obtained by cutting all components, where for a split link the result is {0}. The function CuttNo (webMathematica CuttNo) calculates a cutting number of a link given by its Conway symbol, Dowker code, or Pdata. From the drawings of different Borromean links we could try to find their Conway symbols. For that purpose two LinKnot functions can be useful. The first of them is the function fFindCon (webMathematica fFindCon) that can find for us a Conway symbol of any alternating KL with at most 12 vertices given by its Dowker code with signs, or by Pdata. In order to identify any KL from its drawing in the mousetracking window, use the function GetPdataby Tracking, then apply the function fFindCon. The function fBasicPoly (webMathematica fBasicPoly) recognizes for any KL given by its Dowker code, or Pdata the corresponding basic polyhedron from which the KL in question is derived. These two functions can be very efficiently used for a recognition of different KLs coming from an ornamental or knotwork heritage. In this way we can recognize Tait's series as .(2k+1):(2k+1) 0, (k³ 1). The first link in the series of Borromean links without bigons is the basic polyhedron 1312^{*}, from which originates the complete family of Borromean links with bigons 1312^{*}.(2k+1) 0, (k ³ 0) (Fig. 1.84c). Certainly, for each of them the cutting number is 1, and you can check that by using the function CuttNo. A torus knot or link [m,n] is a simple closed curve on the torus which wraps around m times meridianally and n times longitudinally. If the integers m, n are relatively prime, it is a torus knot; otherwise, we have a torus link. Working with the function fTorusKL (webMathematica fTorusKL) that calculates for a torus knot or link [m,n] its Pdata, braid word, minimal number of crossings, unknotting number or number of components, bridge number, Alexander polynomial and (Murasugi) signature, for GCD(m,n) = 3, where GCD(m,n) is the greatest common divisor for m and n, you will obtain an infinite series of torus links. All of them are nonalternating links that originate from basic polyhedra that are mgonal antiprisms (m ³ 3). Making them alternating, it can be obtained an infinite series of Borromean links. We already mentioned that the main difference between "real" and "mathematical" KLs is that the first are openended, and the others are closed. Dealing with mathematical KLs we could do the opposite: cut our KLs, fix their ends (e.g., by holding two ends of a real knot obtained in our hands) and calculate the number of different (nonisomorphic) classes of real KLs obtained. The LinKnot function fCuttRealKL (webMathematica fCuttRealKL) computes a number of "real" cuttings of a given projection of KL given by its Conway symbol, Dowker code, or Pdata, i.e., the number of cuttings with a different cutting point in the projection and their results. The result is the number of different "real" cutting classes with preserved signs, or with preserved or reversed signs.
