Page 286 - J. C. Turner "History and Science of Knots"

On Theories of Knots 277

He now works at home, extending his researches into braiding theory, and
publishing books and pamphlets on his findings. Since 1988 some 3000 pages
of published work have flowed from his pen (or, in this age, from his word
processor). This is a considerable body of work; and he has at least as much
again, of results stored in notebooks, waiting to be put in book or pamphlet
form. Virtually single-handedly he has laid the foundations for a `theory of
real knots', as distinct from what he calls `the classical theory of hypothetical
knots'. He makes this distinction quite clear in his pamphlet Knots-Facts
and Fallacies [15, Pamphlet 8]; in this work (which we shall quote from exten-
sively, below) he sets out his beliefs as to what knot theory should be about,
expanding on the theme that `classical knot theory is concerned with imagi-
nary closed knots only, and hence is not able to address the vast majority of
real knots.' He supports this statement not by dubious verbal argument, but
by making detailed analyses, using his own methods, of several fundamental
knots. The types of problem he addresses are quite outside the experiences of
`classical mathematical knot theorists'.

Ftg.49 F1g .50 Ftg.51 Fig.52
33=16/L .49/E=3L/EL 37= 14/L.50/E=Z4/EL 41.15/L•57/E=31/EL 45 .13/L.58/E.Z3/EL

Fig.53 Ftg.54 F1g.55 Ftg.56
34=8/L•53/E•30/EL 38=6/L.54/E=ZZ/EL 4Z-7/L.61/E-29/EL 46.5A-WE=E1/EL

Ftg.57 Ftg . 58 Ftg .59 Ftg.60
35.12/L•51 /E=28/EL 39.10/L•5Z/E•Z0/EL 43 . 11/L=59/E=E7/EL 47.9/L•60/E•19/EL

Fig 61 F/g.6Z Ftg.63 Fig.64
35 4/L = 55/E=28/EL 40=Z/L•56/E•18/EL 44 . 3/L=63/E•Z5/EL 48 =1/L.64/E•17/EL
Fig. 4. 16 grid diagrams of 4-crossing knot forms, from 112 diagrams given in [15,
Pamphlet 8]

To give just one example here, he discusses all possible forms of the four-
crossing knot which are of interest to a braider. A classical knot theorist is
generally satisfied that there is only one such form, namely 41i or Listing's