2.7  KL tables 

The first (and still the best) program for knot theory is Knotscape with the tables of knots with n £ 16 crossings, giving possibilities to compute their various invariants (Alexander, Jones, HOMFLYPT and Kauffman polynomials, hyperbolic invariants, signature, symmetries, etc.) There are several web sources providing knot data bases where all basic polynomials of knots are calculated, and some other knot invariants are given. Almost all of them are based on Hoste-Thistlethwaite Knotscape tables of knots, using only the part of them (knots with n £ 11 or n £ 12 crossings). The best available source of that kind is the Table of Knot Invariants by C. Livingston and J.C. Cha that contains knots with n £ 12 crossings (and not links), giving to a reader the widest list of knot invariants and some limited possibilities for an interactive computation. More possibilities for interactive use provides Knot Theory by Dror Bar-Natan that contains the tables of knots with n £ 11 crossings and links with n £ 11 crossings, which can be used for further computations in the Mathematica-based program Knot Theory. Thanks to conversion functions, the program KnotTheory if fully compatible with LinKnot, and can be loaded and used together with it. In the graphical sense, our favorite is KnotPlot by R. Scharein. Almost all graphics in this book are created by exporting Mathematica data to KnotPlot. For presenting 3-D images of KLs is used the program JavaView by K.Polthier. The program Knotilus by S.Rankin contains the largest tables of alternating knots with n £ 22 crossings and provides wide possibilities to work with Gauss codes and virtual knots. All the programs and data bases mentioned are based on the classical notation of KLs and their Dowker and Gauss codes.

The main difference between the other and LinKnot data bases is the different approach to KLs: LinKnot works with Conway notation, families and classes of KLs. Even when it works with data bases of KLs or basic polyhedra, they can be easily extended. Beginning with any KL given in Conway notation, e.g., 6*2.3, we are able to recognize its family 6*(2k).(2l+1). Moreover, beginning from any basic polyhedron belonging to the corresponding data base of basic polyhedra, we are able to write different KLs derived from it. This way, all LinKnot data are not just a fixed data base: they represent the source of the infinite classes of KLs.

The program LinKnot (K2KC) includes the data base KnotLinkBase.m of KLs given in Conway notation as Mathematica strings. This data base contains the complete lists of alternating KLs with n £ 12 crossings, and lists of non-alternating KLs with n £ 10 crossings, as well as the list of non-alternating knots with n=11 crossings. There are two functions working with the lists and particular elements of that data base: NumberOfKL (webMathematica NumberOfKL) and GetKnotLink (webMathematica GetKnotLink). As an input, the function NumberOfKL uses the Mathematica string. To choose a list of alternating KLs with k crossings, you need to write the string "ak" where a stands for alternating KLs, and k is a number of crossings. For a list of non-alternating KLs with k crossings you need to write "nk", where the letter "n" stands for "non-alternating", and k is the number of crossings. The list "n11" contains only non-alternating knots with n=11 crossings. The output of the function NumberOfKL is the number of alternating (non-alternating) KLs with a specified number of crossings. Input of the function GetKnotLink is a Mathematica string "aN" or "nN", where N is an integer that represents the number of the desired KL in the list. As the output, the function GetKnotLink returns its Conway symbol that can be used for further calculations. The structure of the database KnotLinkBase.m corresponds to the classification of KLs proposed by A. Caudron (1982). According to it, the "worlds" or their ßubworlds" (e.g., rational, stellar, arborescent, polyhedral, etc.) in the file KnotLinkBase.nb are denoted by different colors. In each particular list of alternating or non-alternating KLs given in the file KnotLinkBase.m, generating KLs whose Conway symbols contain only single vertices and chains of 2 or 3 bigons, are given in the second part of each particular list aN or nN (N=1,2,...,12) and emphasized with different colorings. From each of them a family of KLs can be obtained by adding 2 to the chains of bigons. For example, from the generating rational link 3 1 2 with n=6 crossings we obtain 5 1 2, 3 1 4 for n=8, then 7 1 2, 5 1 4, and 3 1 6 for n=10, etc. As we pointed out many times, all important properties of KLs (including their Dowker codes, polynomial invariants, minimum braids, signatures, unlinking numbers, linking numbers, symmetry properties, etc.) are well ordered according to families.

Unfortunately, the data base KnotLinkBase.nb is perhaps the weakest part of the program LinKnot because it is created manually and probably contains some errors or misprints. As may be recalled from the history of knot theory, all existing knot tables are based on the more then 100 year old results of T.P. Kirkman (1885, 1885a), P.G. Tait (1876/77a,b,c, 1883/84, 1884/85), and C.N. Little (1885, 1890, 1892, 1900), and have been corrected several times. Computer derivations of KLs have only appeared in the last few decades; these are mostly restricted to knots (Thistlethwaite, 1999), and links with a lower number of crossings (Doll and Hoste, 1991) in Dowker notation. Based on recent results obtained by the program LinKnot, it is reasonable to expect that soon it will be possible to derive KLs in Conway notation by computer, or at least alternating KLs. We are already able to generate all rational and stellar KLs without restriction on the number of crossings, as well as all alternating KLs obtained from any source link by replacing bigons by rational tangles. As it was already explained, derivation of alternating KLs is basically a series of replacements in previously generated source KLs, mostly based on specific partitions or decompositions of numbers. Symmetry plays an important role in reducing the number of possibilities and recognizing in advance possible repetitions and duplications of KLs. Because we have no general algorithmic solution for implementation of symmetry and its numerous particular cases in a computer program for KL derivation, it is necessary to create all possible Conway symbols, and then select those with different minimal Dowker codes calculated by the LinKnot function MinDowAltKL (webMathematica MinDowAltKL).

Non-alternating KLs of the polyhedral world are a special problem, because the same non-alternating KL can be generated from different basic polyhedra. For example, the Conway symbols 8*2:-2 0 and 9*-2 0 represent the same non-alternating knot. We can derive non-alternating KLs given by Conway symbols by using the following algorithm:

  1. in a Conway symbol of a KL make all combinations of bigon chains (including single vertices) with different signs;
  3. reduce every KL obtained by using the K2K function Reduction KnotLink;
  5. calculate different polynomial invariants of reduced KLs and delete repeated ones.