Chapter 3 

History of Knot Theory and Certain Applications of Knots and Links

 

3.1  History of knot theory

In the history of mankind, the discovery of knots probably predated that of fire or wheel. Ropes, cords, and knots needed to secure them played an important role in the early technological development of humans. The main reason for the lack of discovery of such artifacts is that they have been made from organic materials (vegetable fibres, sinews, thongs, hair, etc.), thus subject to decay. However, even certain wild gorillas are able to make complete knots, primarily Granny and Reef knots, so that the beginning of knot tying most likely preceded the evolution of mankind. The indirect testimony for an early use of cordage and knots are perforated objects, beads or pendants, dating some 300 000 years ago, and spherical stones found in Africa and China (about 500 000 years old), probably used as bola weights in hunting. More recently bows and arrows that required well-made cordage and secure knots, as well as Paleolithic figurine in soft limestone from Kostenki (Russia, 24 000 B.C.) show belts made from multiple twined flexible elements. Some actual Neolithic knots are preserved in North Zealand and Denmark. Sophisticated plaits made with strips of date palm leaf originate from Ancient Egypt (Turner and Van De Griend, 1995). Arrangements of knots served as a basis for mathematical recording systems in the Peruvian quipus or Zuni knots from the New Mexico, where the knots functioned as symbolic and mnemonic devices. Various examples of knot-art can be found in all ancient civilizations, in Japanese and Chinese art, Celtic art, ethnic Tamil and Tchokwe art, in Arabian, Greek or Smyrnian laces... In their knot-art Celts made extensive use of knot-work pictures created for decorative and religious purposes (G. Bain, 1973; I. Bain, 1990). For that they needed a high level of mathematics, to geometrically create knotted curves even with zoomorphic ornaments. 

In his essay on orthopedic knots, a Greek physician named Heraklas (first century A.D.) described and explained, giving step-by-step instructions, eighteen ways to tie orthopedic slings. This essay, that survived because it was included (without drawings) in Medical Collections by Oribasius of Pergamum, was recovered, re-illustrated, and translated to Latin during the Renaissance. It is the oldest testimony of a scientific application of knots. 

The idea to consider knots from the point of view of combinatorial topology (this means, Analysis Situs or Geometria Situs, introduced by G.W. Leibnitz in 1679) was first proposed by A.T. Vandermonde in 1771. Describing braids, nets, or knots fashioned by craftsmans, he mentioned that there are no important questions of measurement, but those of position, the manner in which the threads are interlaced. The possibility to study knots from the mathematical point of view was for the first time proposed by C.F. Gauss. One of his oldest documents is a sheet of paper dated 1794, containing thirteen sketches of knots with English names written beside them, probably an excerpt he copied from some English book. 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. He also defined a linking number (1833). His work was continued by J.B. Listing. He represented knots as closed circles by their projections (diagrams) and made an attempt to derive and classify all projections having fewer than seven crossings. Listing defined reduced diagrams of knots, diagrams with a minimal number of crossings, and proposed the first invariant named Complexions-Symbol for knots with such diagrams. Although his Complexions-Symbol had too many serious defects to be acceptable as a knot invariant, it posed a challenge to other researchers to try to find better invariants. He showed that the figure-eight knot, in honor of his accomplishment to knot theory called Listing knot, is equal to its mirror image (this means, that figure-eight knot is achiral), recognized that the left trefoil is different from the right trefoil knot (this means, that trefoil knot is chiral), and introduced a writhe of a knot, a signed sum of all crossings in a knot diagram. 

The main stimulus for the enumeration of knots come from Sir William Thompson (Baron Kelvin of Largs). In attempt to make a classification of chemical elements, in the mid-1860s he announced his model of the atom, called "vortex theory". After Helmholtz's paper on vortex motion based on an experiment with vortex smoke rings, Kelvin developed the theory of particles as tiny topological twists, i.e. knots in the ether. He believed that the variety of chemical elements could be explained by the kinds of different knots. He also thought that the ability of atoms to transform into each other, transmutation, was related to the cutting and recombining of knots. That theory was taken seriously until the Mendeleev's discovery of the periodical system. 

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