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 198 8 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 199 1 XI

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