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New invariants in the theory of knots. (English) Zbl 0657.57001

This is an excellent introduction to invariants of knots a la Kauffman, i.e., a diagrammatic approach. Beginning with teaching us how to present the knot in three-dimensional space, the author explains the recently discovered polynomial invariants and some applications, and also shows how the ideas of the polynomial invariants are related to statistical physics and graph theory. In Sect. 1, the Reidemeister moves, the fundamental tool in this article, and some elementary invariants such as linking number, writhe, etc. are explained, and lastly the most elementary proof that the trefoil is knotted is given. In Sect. 2, the bracket polynomial invariant is constructed and it is shown how it gives rise to the Jones polynomial and to chirality for the trefoil. Sect. 3 uses the bracket polynomial to prove some old conjectures about alternating knots due to the author, Murasugi, and Thistlethwaite. Sect. 4 gives more discussion of the bracket polynomial and its relation to braids and the algebra of Jones’ original representation. Sect. 5 discusses 2-variable generalized polynomials, the Homfly and the Kauffman polynomials, and the historical background of Alexander and Conway polynomials. Sect. 6 shows how the bracket polynomial is directly related to the Potts model in statistical physics. Sects. 7 and 8 explain and generalize a relation with the Tutte polynomial in graph theory originally due to Thistlethwaite. Sect. 9 discusses the knot theory of graphs embedded in three-dimensional space. Sect. 10 has speculations and problems.
Reviewer: T.Kanenobu

MSC:

57-02 Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes
57M25 Knots and links in the \(3\)-sphere (MSC2010)
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