Given the relationship between exactly-solved models in statistical mechanics on 2- dimensional lattices at criticality, and conformal theories in 2-dimensional quantum field theory as arising in Witten's generalization of Jones polynomials, it is appropriate to briefly describe how these models give rise, via the braid group, to polynomial invariants of classical links. Subsequently we mention some of the properties of these invariants, still in the classical context. Finally we mention some of the roles knots and links pla.y in the representation and construction of closed, orientable 3-manifolds, finishing with some remarks on Thurston's geometrization conjectures. Hopefully this talk will be a useful supplement for non-topologists interested in Witten's recent preprint [Wi].

Some excellent detailed surveys of different aspects of Jones polynomials (pre-Witten) now exist, which we recommend for further reading and references. These are Connes [Co], de la Harpe, Kervaire and Weber [HKW], Kauffman [Kau2], Lehrer [Le] and Lickorish [Li2]. Since the subject is evolving rapidly, some questions raised here may be resolved in the very near future.

The basic features of supersymmetric quantum mechanics are reviewed and illustrated by examples from physics and geometry (the hydrogen atom, and massless fields in curved space). Using a discrete approximation to the path integral in the associated supersymmetric quantum mechanics, the Atiyah-Singer Index Theorem is derived for the twisted Diraf operator. Specializations of this in four dimensions include the Gauss-Bonnet theorem, and the Hirzebruch signature theorem. The relationship of the index theorem to anomalies, and their cancellation in the standard model and beyond, is briefly discussed.

This lecture was to be a description of the work of Mendès France and of various of his collaborators on how one may felicitously and instructively attach thermodynamic quantities to plane curves. However, our interest in that project cooled somewhat as we became interested in questions arising from the preparation of the pictures that were to illustrate the ideas to be presented. As a result we give only a summary introduction to the work on thermodynamics and divert our efforts to a description and reformulation of the work of Berry and Goldberg [1] on renormalisation of certain curves containing fantastic curls and twists.