Page 230 - J. C. Turner "History and Science of Knots"

A History of Topological Knot Theory 221

with the respective crossing-parities (i.e. `over' and `under' crossing symbols),
encodes the structure. These symbol-strings, however, were of no significance
to Tait. Very much like Listing before him, at first he only considered knots
which were represented by alternating diagrams (i.e. diagrams where `over'
and `under' crossings alternate throughout, in a traverse around the string).
He did so because he erroneously believed that all knots, without changing
their minimal number of crossing points, could be made to fit such a diagram
[13].
An example will illustrate the procedure. For the knot with 5 crossing-
points, as displayed below, we find the following scheme.

ACBECADBEDIA
The IA indicates the starting point of the sequence . Note that the letters
A, B, C, D, E occur at the crossing positions 1,3,5,7,9, respectively as met dur-
ing the knot traverse.
As said, Tait's procedure was based on a reversal of this process. Thus
he tried to find , empirically, all sequences which could yield such schemes. It
is easy to see that this task would quickly lead, with increasing values of n, to
a massive undertaking. An upper bound for the number of schemes, when n
crossings are involved , can be obtained by solving the following combinatorial
problem:

How many arrangements are there of n letters, when A cannot be
in the first or last place, B not in the second or third, and so on,
and the sequence must have length 2n?

With the help of Cayley and Muir it was soon found that the number
of combinations rises sharply, into thousands, even with a modest number of
crossing points [32]. To complicate matters: many combinations do not even
represent knots. The hopelessness of manually finding knotted structures by