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Jones polynomials and 3-manifolds. (English) Zbl 0685.57003
Geometry and Physics, Proc. Miniconf., Canberra/Aust. 1989, Proc. Cent. Math. Anal. Aust. Natl. Univ. 22, 18-49 (1989).
[For the entire collection see Zbl 0678.00018.]
There are several excellent surveys of the Jones knot polynomials and their generalizations - five are listed at the start of the paper under review - but here the author seeks to provide “a useful supplement for non-topologists interested in Witten’s recent preprint” which has now been published [E. Witten, Quantum field theory and the Jones polynomial, Commun. Math. Phys. 121, 351-399 (1989; Zbl 0667.57005)].
After introductory material, Jones’ original definition of the Jones polynomial using Hecke algebras is reviewed. The many two-variable generalizations are discussed, with emphasis given to their connections with statistical mechanics via state-models. In a section treating computations and applications, the author surveys what is known (or, more often, unknown) about the knot polynomials, with regard to such topics as computability, relation with cyclic covers and torsion invariants, higher-dimensional generalizations, Seifert surfaces, fibred knots, Seifert forms and signatures, Casson’s invariant, satellites and cabled knots, band sums and unions, and alternating knots. The theory of 3- manifolds is reviewed, stressing the constructions of 3-manifolds using knots and links: surgery presentations, and branched coverings and universal knots and links. The case is successfully made that very little is understood about what information the polynomial invariants of a knot give about the manifolds obtained from it by these constructions, although as the author warns, “since the subject is evolving rapidly, some questions raised here may be resolved in the near future”. Most physicists will find these later sections heavy going, but the reader with some experience in low-dimensional topology should find them an enjoyable survey of a large body of results.
Throughout the paper, extensive references are given, including several in the physics literature which may have eluded many topologists.
Reviewer: D.McCullough
57M25 Knots and links in the \(3\)-sphere (MSC2010)
82B10 Quantum equilibrium statistical mechanics (general)
57N10 Topology of general \(3\)-manifolds (MSC2010)