Introduction

These ar e th e note s o f a CBM S series o f lecture s I gav e a t Annapoli s i n

the sprin g o f 1988. Th e lecture s wer e addresse d t o a n audienc e consistin g

of low-dimensional topologist s an d operator algebraists . I tried t o make the

material comprehensibl e fo r bot h groups . Thi s mean s tha t ther e i s a n ex -

tensive introduction t o the theory of von Neumann algebras , and another t o

knot theor y an d th e braid groups . Th e materia l presente d i n thes e note s i s

more o r less exactly what wa s covered i n th e lectures. On e exceptio n i s the

definition o f the knot polynomial V(i) . I n the lectures I began with Kauff -

man's bracket a s a definition an d i n the note s I end with it . Thu s the note s

are ordered historicall y i n this respect .

It was a pleasure to give lectures where both knot theory and von Neumann

algebras wer e treated , a s wel l a s som e elementar y materia l fro m statistica l

mechanics an d conforma l field theory . Sinc e th e sprin g o f 1988 the whol e

area has undergone tremendou s development , mos t notabl y i n term s of th e

deepening connections with physics. Witten' s topological quantum field the-

ory an d hi s invariant s fo r thre e manifold s hav e bee n th e mos t visibl e par t

of thi s work . I t wa s temptin g t o rewrit e th e note s t o incorporat e som e o f

the new developments, but I decided to leave them exactly as they were afte r

the lectures, only adding occasional footnotes with indications of subsequen t

progress.

Thus som e part s o f th e tex t see m a littl e naive , fo r instanc e th e veile d

implication tha t inde x fo r subfactor s an d centra l charg e o f Virasor o repre -

sentations are directly related. Muc h progress on these connections has been

made by Wassermann .

The choice of topics was, of course, highly personal. Thu s the reader will

not find much on the detailed classification o f subfactors. Thi s is also because

the situatio n wa s stil l somewha t unclea r i n 1988, ther e bein g n o availabl e

proofs o f the main results.

I would like to thank G. Price, M. Kidwell, B. Baker and all others respon-

sible for organizin g this CBMS series.

Vaughan Jone s

August 1991

XI