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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. |