Page 252 - J. C. Turner "History and Science of Knots"

A History of Topological Knot Theory 243

13. Problems in Paradise
Until this time, the beginning of the 1980s, the overall picture of the knot
theoretical arena was one of relative tranquility. Purists insisted that knots
and braids were different things. The former lived in the world of topologists,
whilst braids belonged to the algebraists. This view catered for a state of
peaceful coexistence, the result of an evolution in which both camps more or
less went on minding their own business. This changed in a radical manner
when a new knot polynomial erupted onto the scene. In the 1980s the New
Zealander Vaughan Jones, through work in von Neumann Algebras associated
with certain physics problems in statistical mechanics, had found a new way
into knot polynomials. In order to study those aspects of theoretical physics
he had constructed an algebra J, given by:

2
rl : ai = ai
(al, ... , a,,._1 : r1, r2, r3) where r2 : aiai±iai = Tai
r3:aia3=a3ai, li-jl>2
What happened is perhaps best expressed in his own words [48, p. 53]:
In my work I had been astonished to discover expressions that bore
strong resemblance to the algebraic expression of certain topological
relations among braids.
He was so struck by the resemblance between the definition of his algebra
and that of the braid group Bn, that in May 1984 he journeyed to Columbia
University to meet Joan Birman, to discuss his ideas with her. Their initial
deliberations were discouraging. But Jones found, very soon after, that a rep-
resentation of B,ti could be transformed into one of Jn; which in turn possesses
a trace map. Since trace maps are natural class functions, and Jones' trace also
supported the second of the Markov moves, he thus had effectively constructed
a link invariant!
General acknowledgement of the importance of this discovery was not
immediate; but eventually his ideas came to have immense impact on much
mathematical research in topology. In fact, he was later awarded the presti-
gious Fields' Medal for his contribution to the advancement of mathematics.
Not only did Jones' work link knots to statistical mechanics, but also it sparked
an interaction between knot-theory and braid-theory, the like of which had not
been seen since the times when Artin's and Alexander's ideas became enjoined.
The new Jones knot invariant became known as the Jones polynomial;
it is denoted by V. Jones himself had published its skein relation in his first
article [47] which documents his ideas. The relation has the form:

t-1VK+(t) - tVK_ (t) + (t- 2 - t2 )VK0 (t) = 0, with VU = 1