Make your own free website on Tripod.com


0. Introduction

Knots have been used by mankind from prehistoric times up to ancient periods of European and Egyptian civilizations, even serving as the basis for mathematical numerical systems (e.g. for Mayan quipu). Their samples could be found in all civilizations, in Chinese art, Celtic art, ethnical Tamil and Tchokwe art, in Arabian, Greek or Smirnian laces... In the modern science and art, knots and links you could find in DNA, in physics, chemistry, in sculpture... From the design point of view, they belong to the modular structures.

The possibility to study knots from the mathematical point of view was for the first time proposed by C.F.Gauss. Gauss formulated the "crossing problem", by assigning letters to the crossing points of a self-intersecting curve and trying to determine "words" defining a closed curve. J.B.Listing represented knots by their projections (diagrams) and made an attempt to derive and classify all the projections having fewer than seven crossing points. The complete derivation of non-isomorphic knot projections having fewer than ten crossings was completed by P.G.Tait [43]. and T.P.Kirkman [12]. Kirkman's geometrical system for the systematic derivation of knot projections, closely connected with the enumeration of polyhedra, represented at the same time the geometrical method for the classification of knot projections. 

In the 30-ties, after the appearance of the first knot invariant, discovered by J.W.Alexander, the knot theory was established as the part of topology, completely loosing connection with it's roots - geometry. In K.Redmeister's book "Knotentheorie" (1932), each knot is represented by one projection, (randomly?) chosen from several possible ones. Today, with the development of computers, the notation and enumeration of knots and links is very similar with the situation occurring in different unordered structures: prime numbers, polyominoes etc., giving no chance for any classification. Following the "geometrical" line (Kirkman-Conway-Caudron), we will try to present the consistent (geometrical or graph-theoretical) approach to the derivation and classification of knots and links. 

 

NEXT