Recent developments in braid and link theory. (English) Zbl 0713.57002

In this excellent expository paper the author relates the braid groups, the Jones polynomial and the Yang-Baxter equation, describing each with a minimum of assumptions of knowledge on the part of the reader.
Reviewer: L.P.Neuwirth


57M25 Knots and links in the \(3\)-sphere (MSC2010)
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
20F36 Braid groups; Artin groups
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[1] Alexander, J. W., A lemma on systems of knotted curves, Proc. Nat. Acad. Sciences USA, 9, 93-95 (1923)
[2] Artin, E., Theorie der Zöpfe, Hamburg. Abh., 4, 47-72 (1925) · JFM 51.0450.01
[3] Baxter, R. J., Exactly Solved Models in Statistical Mechanics (1982), London: Academic Press, London · Zbl 0538.60093
[4] Bennequin, D., Entrelacements et equations de Pfaff, As-térisque, 107, 87-161 (1983) · Zbl 0573.58022
[5] J. S. Birman, Braids, links and mapping class groups.Ann. of Math. Studies No. 82, Princeton Univ. Press (1974).
[6] J. Birman and W. Menasco, Closed braid representatives of the unlink, preprint, 1989.
[7] —, On the classification of links that are closed 3-braids, preprint (1989).
[8] J. S. Birman and H. Wenzl, Braids, link polynomials and a new algebra.Trans. AMS, to appear, preprint, New York: Columbia Univ.
[9] Burde, G.; Zieschgang, H., Knots (1986), Berlin: de Gruyter, Berlin
[10] L. Crane, Topology of 3-manifolds and conformai field theories, preprint, Yale Univ. (1989).
[11] Fadell, E.; Neuwirth, L., Configuration spaces, Math. Scand., 10, 111-118 (1962) · Zbl 0136.44104
[12] Freyd, P.; Hoste, J.; Lickorish, W.; Millett, K.; Ocneanu, A.; Yetter, D., A new polynomial invariant of knots and links, Bull Amer. Math. Soc., 12, 257-267 (1985) · Zbl 0572.57002
[13] Jimbo, M., Quantum R-matrix related to the generalized Toda system: an algebraic approach, Lecture Notes in Physics, 246, 335-361 (1986)
[14] Jones, V.; Braid groups; algebras, Hecke; Araki; Effros, Proc. US Japan Seminar Kyoto (1973), New York: John Wiley, New York
[15] Jones, V.; Braid groups; algebras, Hecke, Hecke algebra representation of braid groups and link polynomials, Ann. of Math., 126, 335-388 (1987) · Zbl 0631.57005
[16] Kauffman, L., States models and the Jones polynomial, Topology, 26, 395-407 (1987) · Zbl 0622.57004
[17] —, An invariant of regular isotopy,Trans. Amer. Math. Soc, to appear.
[18] Kohno, T., Linear representations of braid groups and classical Yang-Baxter equations, BRAIDS, Contemp. Math., 78, 339-364 (1988)
[19] King, A.; Rocek, M., The Burau representation and the Alexander polynomial, preprint (1988), Stony Brook: SUNY, Stony Brook
[20] Morton, H., Threading knot diagrams, Math Proc. Camb. Phil. Soc., 99, 246-260 (1986) · Zbl 0595.57007
[21] Przytycki, J.; Traczyk, P., Invariants of links of Conway type, Kobe J. Math., 4, 115-139 (1987) · Zbl 0655.57002
[22] Reidemeister, K., Knotentheorie (1948), New York: Chelsea Pub. Co., New York
[23] Reshetiken, N., Quantized universal enveloping algebras, the Yang-Baxter equation, and invariants of links I and II (1987), Leningrad: Steklov Institute of Math., Leningrad
[24] Rolfsen, D., Knots and Links (1976), Berkeley: Publish or Perish, Berkeley · Zbl 0339.55004
[25] Tait, P. G., On Knots I, II, III. (1898), London: Camb. Univ Press, London
[26] Turaev, V., The Yang-Baxter equation and invariants of links, preprint (1987), Leningrad: Steklov Institute of Math, Leningrad · Zbl 0648.57003
[27] E. Witten, Quantum field theory and the Jones polynomial, preprint, Institute for Advanced Study (1988).
[28] Yamada, S., The minimum number of Seifert circles equals the braid index of a link, Invent. Math., 89, 347-356 (1987) · Zbl 0634.57004
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