Page 257 - J. C. Turner "History and Science of Knots"

248 History and Science of Knots

point, where they may have left- or right-handed spiralling segments instead
of a single crossing. This was an idea that Conway had already explored when
he discovered V [23]. It is called cabling.
Unlike V, and to a certain extent P, the F-polynomial was discovered by
purely combinatorial techniques, and it seemed at first glance to be completely
unrelated to braids. However, so-called BWM algebras were constructed by
Joan Birman, Hans Wenzl and also, independently, by Jun Murakami [17],
[68]. Geometrically speaking, they extended the braid groups with U-turns,
making them into (braid) monoids. The BWM algebras are quotients of the
complex group algebra CBn, and they support a 2-parameter family of Markov
traces whose associated link invariant is the Kauffman polynomial. Each of
these algebras contains Hn as a direct summand, and the Markov trace that
associates to Homfly is the restriction to Hn of the Markov trace that defines
F.
Each of the algebras just described supported a Markov trace, and so
determined a link-type invariant. In this way a uniform picture of the old and
new link invariants gradually emerged, with the representation theory of Bn
being an important central part of the picture.
However, the various generalizations of link polynomials have been sub-
sumed under an even more general and unifying procedure, via the so-called
Yang-Baxter Equation (YBE). A Yang- Baxter operator on a vector space V
is a linear isomorphism R : V ® V -4V ® V such that the following hexagon
commutes:
V®V®V R1 V®V®V
1®R r \1®R

V®V®V / V®V®V

R®1 / R®1
1®R
V®V®V V®V®V
This is equivalent to requiring that:

(R ® Idv) o (Id® ® R) o (R ® Idv) = (Idv ® R) o (R ® Idv) o (Idv ® R)

This equation is the Yang-Baxter Equation (YBE), introduced by C. N. Yang
in 1967 in the context of the 1-dimensional quantum n-body problem as a
factorization condition on the S-matrix. Later it was used by Rodney Baxter
to obtain explicit formulae for the partition function of the 8-vertex model
by the transfer matrix method. The YBE plays a fundamental role in two
physical theories: namely the theory of exactly solvable models in statistical
mechanics, and the theory of completely integrable systems.