Page 233 - J. C. Turner "History and Science of Knots"

224 History and Science of Knots

and his collaborators, and of physicists who had been working in an atmo-
sphere of `applied mathematics'. Their work had provided sufficient concepts,
terminology and knot-diagrams to enable the development of a formalized the-
ory to begin. Tabulating knots had two goals. Completeness of the list was
the first. Distinctness of all tabled structures the other. Generally speaking
the first goal could be achieved via (cumbersome) combinatorics. The second
required methods for dealing with problems involving isotopy; and for that,
new mathematical methods were required. Powerful knot invariants had to be
discovered.
As the emphasis in the theory of knots turned away from enumeration,
under the awareness of the problems due to isotopy, the hunt for good knot
invariants began. The ensuing period of transition showed great quantitative
and qualitative differences in knot research, as compared with the early and
rather empiric work in enumeration. In fact the changes effectively caused the
census problem to cease to be the theory's major research topic for the next
six decades; although its importance continued to be recognized. Its goals and
achievements served as testing grounds for new invariants and other important
tools which began to be discovered, in algebraic topology, group theory and
other mathematical fields. Much mutual interplay took place between the old
and the new approaches.
Knot theory as attempted from the purely topological side became possi-
ble only after the development of the required mathematical machinery. This
was pioneered by Henri Poincare (1854-1912) around the turn of the century.
Poincare was a professor at the university of Paris, and a leading mathemati-
cian of his day. It has been claimed that he was the last man to possess a
universal knowledge of mathematics and its applications. His prime motiva-
tion for mathematical research invariably sprang from scientific problems. He
was the first person to make a systematic and general attack on the combi-
natorial theory of a special type of geometric figures called complexes. Due
to this work he is usually regarded as the founder of combinatorial topol-
ogy. He decided that a systematic study of the analysis situs of general or
n-dimensional figures was not only desirable but also necessary. After some
notes which appeared in the Comptes Rendu of 1892 and 1893, he published
a basic paper in 1895; this was followed by 5 lengthy supplements running in
various journals, appearing in the years up to 1904. He did not regard his work
on combinatorial topology as a study of topological invariants, but rather a
systematic way of studying n-dimensional geometry. However, the influence of
his work on subsequent knot theory was to reverse these priorities-the study
of topological invariants came to the fore.
Poincare introduced a number of topological tools, such as the so-called
fundamental group of a complex, also known as the Poincare group in his
honour; it was the first in the string of homotopy groups [72]. It came to play