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Franks, John; Williams, R. F.

Braids and the Jones polynomial. (English) Zbl 0647.57002

Trans. Am. Math. Soc. 303, 97-108 (1987).
Summary: An important new invariant of knots and links is the Jones polynomial, and the subsequent generalized Jones polynomial or two-variable polynomial. We prove inequalities relating the number of strands and the crossing number of a braid with the exponents of the variables in the generalized Jones polynomial which is associated to the link formed from the braid by connecting the bottom ends to the top ends. We also relate an exponent in the polynomial to the number of components of this link.

Cited in 10 Reviews
Cited in 91 Documents

MSC:

57M25 Knots and links in the \(3\)-sphere (MSC2010)

Keywords:

number of components of link; generalized Jones polynomial; number of strands; crossing number of a braid; link
PDFBibTeX XMLCite
\textit{J. Franks} and \textit{R. F. Williams}, Trans. Am. Math. Soc. 303, 97--108 (1987; Zbl 0647.57002)
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References:

[1] D. Bennequin, Entrelacement et structures de contact, Thesis, Paris, 1982.
[2] Joan S. Birman, Braids, links, and mapping class groups, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1974. Annals of Mathematics Studies, No. 82. · Zbl 0297.57001
[3] Joan S. Birman and R. F. Williams, Knotted periodic orbits in dynamical systems. I. Lorenz’s equations, Topology 22 (1983), no. 1, 47 – 82. · Zbl 0507.58038 · doi:10.1016/0040-9383(83)90045-9
[4] P. Freyd, D. Yetter, J. Hoste, W. B. R. Lickorish, K. Millett, and A. Ocneanu, A new polynomial invariant of knots and links, Bull. Amer. Math. Soc. (N.S.) 12 (1985), no. 2, 239 – 246. · Zbl 0572.57002
[5] Vaughan F. R. Jones, A polynomial invariant for knots via von Neumann algebras, Bull. Amer. Math. Soc. (N.S.) 12 (1985), no. 1, 103 – 111. · Zbl 0564.57006
[6] W. B. R. Lickorish and Kenneth C. Millett, A polynomial invariant of oriented links, Topology 26 (1987), no. 1, 107 – 141. · Zbl 0608.57009 · doi:10.1016/0040-9383(87)90025-5
[7] H. R. Morton, Closed braid representatives for a link and its Jones-Conway polynomial, preprint, 1985.
[8] Dale Rolfsen, Knots and links, Publish or Perish, Inc., Berkeley, Calif., 1976. Mathematics Lecture Series, No. 7. · Zbl 0339.55004
[9] Lee Rudolph, Braided surfaces and Seifert ribbons for closed braids, Comment. Math. Helv. 58 (1983), no. 1, 1 – 37. · Zbl 0522.57017 · doi:10.1007/BF02564622
[10] J. M. van Buskirk, private communication.
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