Page 246 - J. C. Turner "History and Science of Knots"

A History of Topological Knot Theory 237

relation became the invariant's definition. Its well-definedness could be proved
by showing its invariance under the Reidemeister moves.
More so than with Alexander polynomials, which require definition and
computations of certain determinants, the preferred way for defining polyno-
mial invariants obtained via skein relations is to proceed from knot-diagrams.
A polynomial is computed recursively, by a kind of 'unknotting process' when
one systematically obtains diagrams on reducing numbers of crossings, making
use of a given skein relation. When diagrams with already known polynomials
are arrived at, the process can be retraced, and the polynomial for the original
knot is arrived at.
Success with, and increased use of, this procedure caused knot-diagrams to
become notational devices at the same level as other symbols in mathematical
writings.
Incidentally, as we noted above, Conway did expand the knot-tables; and
his work was later continued by Thistlethwaite and Perlio [86], [71]. The latter
completed the census problem for 10-fold knottiness in 1974, and detected some
errors in Little's 1885 table of 11-fold crossing knots. Now we have complete
listings of knots with up to 13 crossing-points [86]. And researchers are working
to enumerate knots on 14 and 15 crossings [8]. There is an estimate that there
exist over 150 000 different prime alternating knots on 15 crossings.
The following table shows the totals of prime alternating ltnots which have
from 3 to 13 crossing-points. The top row gives the numbers of crossing-points,
and the bottom row the corresponding frequencies of knots.

In the early 80s, John Turner [89] studied knot-graphs, and experimented
with operations similar to Conway's. He obtained various ltnot invariants,
working from both non-oriented and oriented diagrams. One idea he pursued
was to take an alternating knot-diagram, and reduce it systematically by a
process he called twinning. Crossings were 'deleted', one at a time, and two
new knots (the 'twins', each with one fewer crossing than the original knot)
were formed at each 'deletion' (see the example in Fig. 18). He continued this
twinning, producing a binary tree of knots, &nd stopping the deletion process
whenever a twist ltnot was arrived at.
The ultimate result, from any given starting knot, was a collection of twist
ltnots (situated at the tree leaf-nodes), each of which had well-defined twist-
senses, labelled plus or minus. He saw these as being fundamental building
units of the original ltnot, and was able to prove, subject to one of Tait's many
conjectures being true, that the end-collection of twists was independent of the
order of reduction by 'deletions' of crossings, and that it was a knot invariant.