### New points of view in knot theory

Bulletin of the American Mathematical Soc. (N.S.), Vol. 28, no. 2 (1993), 253-287

Introduction

In this article we shall give an account of certain
developments in knot
theory which followed upon the discovery of the Jones
polynomial in
1984. The focus of our account will be recent glimmerings
of understanding of
the topological meaning of the new invariants. A second
theme will be the
central role that braid theory has played in the subject.
A third will be the
unifying principles provided by representations of simple
Lie algebras and
their universal enveloping algebras. These choices in
emphasis are our own.
They represent, at best, particular aspects of the
far-reaching ramifications
that followed the discovery of the Jones polynomial.

...

Our goal, throughout this review, is to present the material
in the most transparent and nontechnical manner possible
in order to help
readers who work in other areas to learn as much as
possible about
the state of the art in knot theory. Thus, when we give
"proofs", they
will be, at best, sketches of proofs. We hope there will
be enough detail
so that, say, a diligent graduate student who is motivated
to
read a little beyond this paper will be able to fill in
the gaps.