J.H. Conway in 1967 introduced a fundamentally new idea. After introducing the basic concept of a tangle (proposed before by Tait), a portion of a knot diagram with four free-end strands, he gave a concise generic geometrical notation for describing KLs in terms of their construction from tangles. He discovered a remarkable connection between rational KLs and continued fractions, and defined skein relations making possible a recursive computation of polynomial KL invariants. By using skein relations, polynomial KL invariants can be computed recursively, by a kind of "unknotting process" where one systematically obtains diagrams of each succeeding generation and thereby reduces the number of crossings. J. Conway checked and extended the knot tables of Tait, Kirkman and Little. 

The first attempt to classify KLs in certain larger classes (called worlds) came from A. Caudron. In his fundamental work, a 361-page preprint Classification des noeuds et des enlancements published in 1982, he corrected and extended Conway's results, and made the first "periodical tables of KLs". His results marked the end of the era of hand calculation. 

In 1982 H. Dowker and M. Thistlethwaite computerized the exhaustive process for derivation of all knot diagrams by using Dowker algorithm. Until now, knot tables were extended by M. Thistlethwaite, J. Hoste, and G. Weeks to n£18 crossings. In their knot tables, included in the program Knotscape, every knot is denoted by its minimal Dowker code. S. Rankin derived alternating knots for n £ 23 crossings. His tables contain, for example, 25182878921 knots with n = 23 crossings.

Thanks to their complexity, the results for links are much more restricted: in 1991 H. Doll and J. Hoste tabulated oriented links with n £ 9 crossings. In his still unpublished results, M. Thistlethwaite made computer tabulation of links with n £16 crossings.

Today, with the development of computers, the notation and enumeration of knots and links is very similar with the situation occurring in different structures with hardly recognizable ordering principles: prime numbers, polyominoes etc., resisting attempts of classification. We have attempted to present a consistent geometrical, combinatorial and graph-theoretical approach to the derivation and classification of KLs following the laed of Kirkman, Conway and Caudron. Links have always played a subordinated secondary role to knots; our approach has been to elevate links to an equal status with knots. We wrote the computer program LinKnot to enable a user to work with links in the same way as it woks with knots, by using Conway notation. The very name of the program (proposed by R. Sazdanovic) underlines the important role given to links in that program. It is primarily dedicated to an experimental work with a large series of KLs (i.e., families). Thanks to computers, we are now able to check large numbers of KLs and make some new conjectures. Therefore the program LinKnot is a tool for experimental mathematics. 

In the 1980s V. Jones discovered new polynomial and established connections between von Neumann algebras, statistical mechanics, braid theory and knot invariants. The Jones polynomial not only introduced a refined invariant for distinguishing and analyzing KLs, but related knot theory to other fields of mathematics and theoretical physics, in particular statistical mechanics and quantum field theory. That revolutionary discovery was followed by new more powerful polynomials: HOMFLYPT and Kauffman polynomials, also able to detect chirality, and to distinguish KLs. However, none of these polynomials is a complete invariant able to distinguish all KLs, or to completely recognize chirality. 

The introduction of Vassiliev's invariants gave rise to the hope that a complete KL invariant could be found. Working in the space of knots, Vassiliev invariants are essentially different from all other previously mentioned KL invariants: instead of associating to each KL an mathematical quantity or polynomial, they assign to a KL a numerical value depending on a set of initial conditions. Many of the invariants introduced before, such as Alexander, Jones, skein, and Kauffman polynomials, are Vassiliev invariants. On the other hand, none of the classical geometric KL invariants: the minimal crossing number, unknotting (unlinking) number, signature, bridge number, braid index and genus of a KL, are Vassiliev invariants. 

We hope to implement some of the recent approaches (e.g., Vassiliev invariants, and virtual knot theory by L. Kauffman) in LinKnot. This computer program will also enable us to derive by computers KLs with a higher number of crossings in Conway notation.