## Proceedings of the Centre for Mathematics and its Applications

- Proc. Centre Math. Appl.
- Miniconference on Geometry and Physics. Michael N. Barber and Michael K. Murray, eds. Proceedings of the Centre for Mathematical Analysis, v. 22. (Canberra AUS: Centre for Mathematics and its Applications, Mathematical Sciences Institute, The Australian National University, 1989), 18 - 49

### Jones polynomials and 3-manifolds

#### Abstract

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.

#### Article information

**Dates**

First available in Project Euclid:
19 November 2014

**Permanent link to this document**

https://projecteuclid.org/
euclid.pcma/1416406251

**Mathematical Reviews number (MathSciNet)**

MR1027860

**Zentralblatt MATH identifier**

0685.57003

#### Citation

Aitchison, Iain R. Jones polynomials and 3-manifolds. Miniconference on Geometry and Physics, 18--49, Centre for Mathematics and its Applications, Mathematical Sciences Institute, The Australian National University, Canberra AUS, 1989. https://projecteuclid.org/euclid.pcma/1416406251