Page 254 - J. C. Turner "History and Science of Knots"

A History of Topological Knot Theory 245

For the sake of comparison all four skein relations are listed below:
AK+ (t) - OK_ (t) + (t - ' - t z )OK0 (t) = 0, with D U = 1
V K+(t) - (t) - tV K0 ( t) = 0 , with VU = 1
V K_
t-'VK+ - tVK_ + (t-2 - t2)VKo(t) = 0, with VU = 1
2PK+(2, m) + t-1PK_ (f, m) + mPK0 (1, m) = 0 , with PU = 1

However, very soon after the public announcement of P's discovery, exam-
ples of indistinguishable pairs of mirror image knots emerged . Of course, one
counter-example is sufficient to torpedo a conjecture, but in 1986 Taizo Ka-
nenobu produced infinitely many classes of, in turn , infinitely many distinct
knots with the same P-polynomial. Using the second elementary ideal of the
Alexander module, he showed [52] that for the knots Kp,q (see Fig. 20):

P(Kp,q) = P(Kp ,qi) if and only if p + q = p' + q'.

P contains the information of A, V, V and more; but there the similar-
ity ends. And to-date all attempts to interpret V in the same topological
framework as A have failed.

25 2P

Fig. 20. The knot Kp, q used by Taizo Kanenobu
Louis Kauffman's F-polynomial was the second distinct generalization
of V to appear. He obtained his polynomial by cutting up a knot in a special
way [54]. The skein relations we have seen so far are recursive definitions
on knot diagrams which differ in the set of crossings which was shown in
Fig. 18. Kauffman's polynomial is based on the very imaginative idea of a
state model, which sees an unoriented knot diagram as a state or, and which
provides information carried by the diagram. Given a diagram (of a knot K,
say), he proposed splitting a crossing in two ways, thereby obtaining two new
states (the same operation as used to obtain Turner's twist spectrum). These
were assigned a `weight' A or A-' according to their type of split (if a new
state included a circle, a more complex weight, a function of A and A', had
to be assigned). He continued this process until no crossings were left; and