2.10 A dream- new KL tables
We hope that future KL tables (and not only knot tables) will be well organized in order to follow a vertical structure (families of KLs), and not only the "horizontal" structure (minimal crossing number of KLs). The concept of new KL tables given in Appendix is now still restricted to KLs with n £ 9 crossings, but we hope that very soon it will be extended to KLs with larger number of crossings.
An almost complete derivation of alternating knots with n £ 11 crossings and non-alternating knots with n £ 10 crossings was given by P.G. Tait, T.P. Kirkman, and C.N. Little at the end of 19th century. All knot tables are based on this derivation. The first classical knot tables appeared in K. Reidemeister's book Knotentheorie (1932), where each knot was represented by one projection. All other knot tables are just copies of Reidemeister's tables with some minor changes in knot projections. In order to compare them, the reader may consult the books by D. Rolfsen (1976), G.Burde and H. Zieschang (1985), L.A. Kauffman (1987a), C. Adams (1994), A. Kawauchi (1996), and K. Murasugi (1996).
All knot tables contain polynomial knot invariants: Alexander polynomials, Jones polynomials (Adams, 1994), Kauffman polynomials (Kauffman, 1987a), and data about some other knot invariants and properties- hyperbolic volumes (Adams, 1994), signatures (Burde and Zieschang, 1985), unknotting numbers (Kawauchi, 1996), chirality and invertibility (Burde and Zieschang, 1985; Kawauchi, 1996), symmetry groups of knots (Kawauchi, 1996), etc..
Usually, knots are denoted by their ordering numbers in the classical notation as
31, 41, 51, 52, 61, 62, 63, 71 - 77, 81 - 821, 91 - 949, 101 - 10166,
without any geometrical or topological "vertical" ordering principle connecting knots with n and n+1 crossings. The classical notation gives no information about a KL (except its place in knot tables), but it has been preserved up to the present time in most knot theory books.
Most knots are alternating, and non-alternating knots appear only for n ³ 8:
819 - 821, 942 - 949, 10124 - 10166, etc.
The most complete knot tables published in books contain knots with n £ 10 crossings. The only tables containing links with n £ 9 crossings are given by D. Rolfsen (1976).
The development of computers enabled us to construct all permutations of n even numbers, check their realizability as knot projections, and find minimal Dowker codes for all knots with n £ 16 crossings (Dowker and Thistlethwaite, 1983). In the Dowker notation every knot is given by its (minimal) Dowker code and the signs of crossings (necessary only in the case of non-alternating knots). Because a Dowker code depends on a projection and the choice of a beginning point, the mapping between knots and their Dowker codes is one-to-many, so it is necessary to find a minimal Dowker code for each knot. Since minimal Dowker codes are just minimal permutations representing certain knot projections, without displaying any other geometrical or topological information about knots, they are not very useful in attempt of knot classification.
Using computer enumeration and Dowker algorithm, M.B. Thistlethwaite (by using the program Knotscape, and H. Doll and J. Hoste (1991), obtained tables of knots with n £ 16 crossings and tables of links with n £ 9 crossings. A similar program for deriving knot projections with n £ 10 crossings was developed by S. Jablan and V. Velickovic in 1995. S. Rankin derived alternating knots up to n £ 23 crossings. Still unpublished results of M. Thistlethwaite contain link tables with at most n=16 crossings.
Continuing the "geometrical" line (Kirkman-Conway-Caudron) and the classification of KLs proposed by S. Jablan (1999), we introduced new KL tables, ordered according to KL families.
New tables for prime knots with n £ 8 crossings were completed in 2002, and now we extend this result to all KLs with n £ 9 crossings (Appendix).
In the first version of new knot tables, based on knot families every family is defined by its general Conway symbol. For each family, in the "Notation" subsection are given Conway symbols of knots with n £ 10 crossings belonging to certain family and their corresponding symbols in the classical notation. The list of their particular Alexander polynomials is followed by a general formula for the Alexander polynomials of the family. For every family the symmetry group, symmetry type, signatures, and unknotting numbers are determined in general form. All the corresponding particular data are first calculated for the knots with n £ 19 crossings using the program LinKnot. The results obtained are extrapolated to whole families in order to derive general formulas for the Alexander polynomials, symmetry groups, symmetry types, signatures, and unknotting numbers.