Page 231 - J. C. Turner "History and Science of Knots"

222 History and Science of Knots

means of the scheme method was realized. It was abandoned in favour of a
more effective one, to be described next.

5.2. The Partition-Method

This was also based on graph properties of knot diagrams. An n crossing-
point diagram will have (n + 2) regions, which can be denoted by Ri, 0 <
i < (n + 1). Crucial to the method is the observation that for any region
Ri the number of corresponding sides (edges of the graph) si has to fulfill
the inequality: 2 < si < n. It is known that the total number of sides in
the graph equals 2n. These facts enable one to partition the number n and,
via use of polyhedrons, to arrive at graphs which can be assigned a crossing-
coding, thereby yielding representations of knotted structures. This method
eventually was made to work well. It was also the method which Tait developed
further still, after sharing ideas with the Reverend T. P. Kirkman and C. N.
Little, an American professor. They had independently pursued the same lines.
Their communications led to a happy collaboration, which resulted in the trio
collectively listing virtually all alternating knots up to 11-fold knottiness.

5.3. The Final Results

What did Tait and his collaborators eventually achieve? They found 82 types
of knots of 9 or less crossings. An especially remarkable achievement was their
work in the class of 10-fold knottiness. A mammoth undertaking which, with
their tools, took them 6 years to complete; it resulted in some beautiful tables
of 10-crossing knot diagrams. Little continued the struggle, and published the
results of his attempts on 10-fold knottiness in 1885 [63]. They were finally able
to resolve, too, a large number of the alternating 11-crossing knots. Kirkman
provided Little with a manuscript of 1581 polyhedral drawings, from which he
distinguished 357 different knot-types.
It is impossible to summarise adequately, in a few paragraphs, the extent
of Tait's contributions to the birth of Knot Theory. His researches affected
all aspects of the subject. He empirically discovered a great number of useful
results, while experimenting with many ideas which future researchers would
take up. He worked on the so-called Gordian number, which is the minimal
number of crossing-point changes required in a knot to produce an unknot.
He made some pertinent conjectures which were not resolved for well over a
century. He had already considered knots such as Moebius braids, and he both
toyed and toiled with problems relating to symmetries such as mirroring. He
found a nice little theorem on amphicheirality (a knot is said to be amphicheiral
if it can be topologically transformed into its mirror image). He was very