×

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

MSC:

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
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Alexander, J. W., A lemma on systems of knotted curves, Proc. Nat. Acad. Sciences USA, 9, 93-95 (1923) · JFM 49.0408.03 · doi:10.1073/pnas.9.3.93
[2] Artin, E., Theorie der Zöpfe, Hamburg. Abh., 4, 47-72 (1925) · JFM 51.0450.01 · doi:10.1007/BF02950718
[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. · Zbl 0684.57004
[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 · doi:10.1090/S0273-0979-1985-15361-3
[13] Jimbo, M., Quantum R-matrix related to the generalized Toda system: an algebraic approach, Lecture Notes in Physics, 246, 335-361 (1986) · doi:10.1007/3-540-16452-9_21
[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 · doi:10.2307/1971403
[16] Kauffman, L., States models and the Jones polynomial, Topology, 26, 395-407 (1987) · Zbl 0622.57004 · doi:10.1016/0040-9383(87)90009-7
[17] —, An invariant of regular isotopy,Trans. Amer. Math. Soc, to appear. · Zbl 0763.57004
[18] Kohno, T., Linear representations of braid groups and classical Yang-Baxter equations, BRAIDS, Contemp. Math., 78, 339-364 (1988) · Zbl 0661.20026 · doi:10.1090/conm/078/975088
[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 · doi:10.1007/BF01389082
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.