Page 229 - J. C. Turner "History and Science of Knots"

220 History and Science of Knots

atoms at high energies could be interpreted as the cutting and recombining of
knots.
Even though Thompson's vortex theory of atoms stood for only about
two decades, Tait's fascination for knots had been aroused. Thus the study of
vortices stands as the starting point of a highly important pioneering study of
the topology of knots [87].
The vortex theory led Tait immediately to the problems at the heart
of the subject: with insufficiently developed mathematics to help out, knots
and links could not be characterized. As already indicated, the accessible
work on knot models was very scattered and fragmentary; and results on
knots had usually been arrived at almost simultaneously, and independently,
by mathematicians ignorant of each other's work. At that time, it was not
even clear (to Tait) whether or not there were finitely many knots. Therefore
his first self-appointed task, which gradually became his main occupation, was
one of enumeration, in which he tried to find and classify knotted structures.
Tait called this the census problem. The main thrust of his work was how
to find all possible (distinct) knotted structures which can be represented by
plane diagrams of continuous curves having n crossing-points. He studied
ways by which such diagrams could be `reduced', a reduction corresponding to
a topological change in a knot which led to fewer crossing-points in the plane
diagram. He called the minimal number of crossing-points achievable for a
diagram of a given knotted structure, the degree of knottiness of that structure.
Tait gave several methods for making the reductions. During his attempts to
develop these ideas, he followed (roughly) two lines of combinatorial attack.
The first involved the development of a method which he called the Scheme-
method, one which seems to have been known to Gauss [86, p. 13]. The second
attempt he came to develop was partially inspired by information gleaned from
Johann Listing's work. He termed it the Partition-method.

5.1. The Scheme-Method

The central problem, in Tait's case, was to find the combinations of symbol
sequences which would encode a connection relation for n points in the plane.
He introduced a tool, for which he coined the name scheme, which was basi-
cally a symbolic shorthand for a connected graph. For ease of demonstration,
we shall describe this method in reverse.
Given a diagram D of a 1-link L, with at least one crossing. Choose any
crossing from which to start. Call it A. Move from A in either of the two possi-
ble directions. Traverse the knot diagram and name the crossings encountered
in the odd places respectively B,C,D and so forth until all crossings have been
assigned a letter. Traverse the structure over again from any starting-point
and jot down the sequence of crossing-points visited. This symbol-string, along