Page 241 - J. C. Turner "History and Science of Knots"

232 History and Science of Knots

reference the diagrams will be called `a set of Alexander crossings'.
The relationship is:

AK, (t) - AK_(t) + (t_2 - tz)OKo(t) = 0, with A, = 1

Many years later this relation, and others like it (now called skein relations),
came to have great significance in the development of recursive methods to
produce knot invariants. In spite of its early discovery, the literature has
shown a remarkable tendency to remain loyal to the calculation of AK by
means of determinants. This is rather strange, as this relation bears within
it the possibility of calculating AK(t) recursively by `untying-or splitting
repeatedly-a knot-diagram'.
The idea of unknotting was not born here, though, as Tait had already
considered the Gordian number. The relation given above is in a sense decep-
tively simple. There is no reason a priori why it should define any invariant.
It may after all depend on things like projections or properties of the plane.
Alexander did not find sufficient conditions to give any recursive process by
which to obtain his polynomial. He did however prove the well-definedness of
his proposed invariant.
The period of Alexander's work can suitably be called one of change and
crossroads. The idea that knots could perhaps be understood by studying
braids was one of the most promising ones to be pursued at that time. It
caused Emil Artin to introduce the braid group, and Alexander to make some
fundamental discoveries which bridged the gap between the two theories of
knots and braids. We shall discuss these developments later, when we come
to focus on braid theory's contribution to the study of knots.

9. Kurt Reidemeister and His Moves

In 1923 Kurt Werner Friedrich Reidemeister (1893-1971) accepted an associate
professorship in Vienna where he did research on the foundations of mathemat-
ics. In 1925 he obtained a full professorship in Konigsberg where his interests
went to the foundations of geometry. It is not surprising that he was the
person who disposed of many of the basic problems and early difficulties in
the field of knot theory. His thorough work covered fundamental treatments
of knot enumeration, projections and isotopy. Tait and his collaborators had
found many knots, but they had not catalogued them in any workable man-
ner. Reidemeister ordered and numbered them, using a notation which gave
their positions in the list and their minimal numbers of crossing points. His
notation, and tables of knot diagrams, stood for many years.
In the field of planar knot-projections, Reidemeister studied small, local
changes made to a knot and how they corresponded to changes in the diagram