Page 245 - J. C. Turner "History and Science of Knots"

236 History and Science of Knots

Conway gave a notation for describing knots in terms of their construction
from tangles: using this notation, he was able to give rules for determining
equivalences between knots. His methods were much simpler than previous
ones, and lent themselves to programming for computer analysis. In a 1970
paper [23], Conway presented these ideas, and also listed knot-types in his
notation for the following: all the prime alternating and nonalternating knots
with crossing numbers n = 3, ... ,11; the 2-links up to n = 8; the 2-, 3-, and
4-links for n = 9; and all links for n = 10. In addition, for most of the knots in
his tables, Conway gave values for several classical, and also new, invariants,
obtained by his methods.
Thus, in a very short time*, using his newly devised methods of tangles,
Conway had checked and extended the tables of Tait, Kirkman and Little,
produced so laboriously about eighty years previously.
Certain manipulations on the Conway tangles gave rise to polynomials.
Sample calculations with these were made, and they revealed certain algebraic
relations between the polynomials (which in fact obviated the use of a com-
puter for calculating his tables). It was natural to think of tangles as elements
in a vector space, in which certain identities became linear relations. There
were many natural questions to be asked about these spaces, and study of
these led Conway to his discovery of a polynomial knot-invariant. Initially,
Conway only wanted to further the enumeration and tabulation of knot-types,
which task had been at a standstill for the past six decades when he began
his attack upon it. But his contribution turned out to be a major one, in the
hunt for knot invariants. Even though his find, in a sense, was Alexander's
polynomial disguised in a normalized form, it was obtained by totally new
methods. It became known as the Conway polynomial, often denoted by VK.
It was also a polynomial which could be calculated directly from a diagram by
means of a recursive method, not requiring the evaluation of any determinant.
John Conway wanted to call the relationship between three links whose
diagrams differ only in a set of Alexander crossings a potential function; but
instead, this relationship went on to lead its own life in knot research, acquiring
the name of skein relation. In Conway's original work this potential function
had the form:
V K+ (t) - V K_ (t) = tV K0 (t) , with V = 1

It relates to the Alexander polynomial AK via the equation:

AK(t) =VK(t2 - t 2)
The important idea, which set new trends in knot research, was that the skein

*In [23] Conway claims that he could check in a mere afternoon much of the work that Tait
and Co. took six years to complete!