Page 255 - J. C. Turner "History and Science of Knots"

246 History and Science of Knots

then took the weighted sum over all the resulting states, obtaining what is
now called the Bracket polynomial of K. This is denoted by:

< K > = E <crK> (-A2-A-2)Ilall-1

The Bracket can be generalized to yield a two-variable polynomial invari-
ant, named the L-polynomial. By using L, divided by a factor involving one
of the variables raised to the sum of the knot's crossing parities (the so-called
writhe w(K)), one can construct yet another polynomial invariant, which is
customarily denoted by F, and is known as the Kauffman polynomial. Thus:

FK(a, z) = a-wiKiLK(a, z)
In [60] Lickorish shows that the Jones and Kauffman polynomials, V and F,
are related by the following equation:
VK(t) FK(-t 2, t 4 + t4 )

The F-polynomial is quite good at detecting chirality. It is probably a little
better for this purpose than P, because it originates from four instead of
three terms; but it has its shortcomings too. However the Bracket polynomial
is a truly amazing construct. For instance, it was used to confirm the old
conjecture (made by Tait) that the number of crossing points in a connected,
reduced, alternating projection of a link is a topological invariant. Several
other long-standing problems were also dealt with quickly by means of this
and the other new knot polynomials [69], [85].
In spite of its more powerful generalizations, VK(t) has retained its interest
and value in knot research and applications. Ironically, one reason for this is,
simply, that it has only one variable; which makes it easier to work with!
Amongst all the questions asked after the discovery of the new knot poly-
nomials, the dominating one was: Granting its existence, how may these poly-
nomials be placed in a sound mathematical setting? Many researchers have
tried to shed light on this question, in the past decade, working from a va-
riety of viewpoints, moving in uncharted territories of topology, algebra and
statistical mechanics. The schematic diagram on the next page indicates the
need to underpin satisfactorily the array of aew knot invariants with a unifying
theoretical base. And the next section summarizes some of the work that has
been done in this `terra incognita' since the invariants appeared.

14. Charting Terra Incognita
Jones' discovery implied that statistical mechanics must hold clues for
an understanding of his polynomial. Therefore the discovery of V, F and