Page 253 - J. C. Turner "History and Science of Knots"

244 History and Science of Knots

At first it seemed to be yet another polynomial invariant in the long
line of such objects which had been proposed previously. Some of them were
truly exotic in their definitions, while others were plain monsters in their time
complexities; but V did something different. Knot theorists had begun to
understand that symmetries , such as handedness and orientation, which are
not caught by the Alexander/Conway polynomial, belong to entirely different
regions of the mathematical universe. The reason that Alexander's polynomial
does not cater for invariance under mirroring is due to the fact that AK- (t) _
AK(±t'). A key property of V is that (like the twist spectrum):

VK•(t) = VK(t -1)

Simple examples show that VK(t) need not be invariant under t -f t-1, so
that it can sometimes distinguish knots from their mirror image. For example,
in the case of the two Trefoils Ti and T, we have:

I VTr (t) = t + t3 - t4 = VTt (t-1)
VT'* (t) = t_1 + t-3 - t-4 = V ,.(t-1)

V is pretty good at detecting this sort of symmetry, though not infallible. Its
merits for this are first dashed with knot 942 from the tables by Reidemeister
and Rolfsen [74], [77]. But even so, this property shows that V is not a knot-
group invariant like A or V.
It is a remarkable fact that V's appearance on the knot scene , some 40
years after 0's, did not trigger any generalizing activity in the mathematical
community. Whereas the appearance of Jones' polynomial seemed to present
an explicit invitation to do so. It immediately led to an outburst of discoveries
of knot polynomials with more than one variable . Moreover, time since then
has shown that V was to be generalized in two quite distinct ways.
The first general polynomial to appear was to be known as Homfly; it is
also called Homfly-PT, and also Thomflyp . The names are acronyms which
are derived from the initials (of 6) of the 8 people who discovered it [36],
independently and almost simultaneously. Four of them happened to submit
an article on their work to one and the same major mathematical journal
on virtually the same day! Although they reached their results via different
routes, the main idea was inspired by generalizing the coefficients in the skein
relation. They were invited to pool their ideas , and present a single paper,
jointly under all their names; this they did. Their two-variable polynomial,
denoted by P, has the following skein relation:

2PK+(f, m) + f-1PK_ (P, m) + mPK0 (2, m) = 0, with PU = 1