Page 284 - J. C. Turner "History and Science of Knots"

On Theories of Knots 275

for the Encyclopedia to give to the world.
However, the kinds of diagram given in the encyclopedias described above
are not suitable for topological studies of knots. In the next subsection we
describe diagrams that have been developed for that purpose.

4.6. Topological Knot Diagrams
Topological Knot Theorists use different tables of diagrams, when they wish
to refer to a particular knot. When one approaches a theory of knots, one
has to begin by drawing diagrams of them; usually these are essentially two-
dimensional (projections into a plane) with some kind of markings indicating
where the crossings occur and whether they are `overs' or `unders'. For histori-
cal interest, we give below (Fig. 3) diagrams of a few ten-crossing knots, as first
published by P. G. Tait in Transactions of The Royal Society of Edinburgh,
1885.

TENFOLD KNOTTINESS. Place Vill
0 c•

• oCTo^ a`i o x "CJe z
M
ae c` c• c» e• a e•
C
n t a ^c o a s o z s x r r c r o ^,l^Jr o r r
c•• c•• Cu 0u
^a
a'C z x 0
V.. 0 D
D` - C. + C. 7 D•
h_\
Ma
r r r r K 0 -L. L L
I
Fig. 3. Part of Tait's Table of the Ten-crossing Alternating Knots (1885)
The full page gives all possible* 10-crossing closed, alternating, prime knots,
discovered by methods developed by Tait and the Reverend Kirkman. The
reader will note the many symbols, letters and other characters, attached by
Tait to his diagrams; they relate to his various ways of classifying the knots.
T. P. Kirkman also produced tables of knot diagrams [9]. He used polygo-
nal (straight-line) figures, which didn't resemble ordinary string knots, as did
Tait's; further, he used Greek symbols and phrases to denote his classifying
concepts-making it very difficult for persons without a classical education to

*The page gives diagrams for 166 such knots. In 1979, nearly a century later, K. A. Perko
showed [12], using `modern' topological invariants, that two of these knots are the same
under topological transformation; it is now known that there are just 165 prime alternating
knots with 10 crossings. So Tait wasn't far out with his catalogue!