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Jones polynomials of alternating links


Author: Kunio Murasugi
Journal: Trans. Amer. Math. Soc. 295 (1986), 147-174
MSC: Primary 57M25
DOI: https://doi.org/10.1090/S0002-9947-1986-0831194-9
MathSciNet review: 831194
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Abstract: Let $ {J_K}(t) = {a_r}{t^r} + \cdots + {a_s}{t^s},r > s$, be the Jones polynomial of a knot $ K$ in $ {S^3}$. For an alternating knot, it is proved that $ r - s$ is bounded by the number of double points in any alternating projection of $ K$. This upper bound is attained by many alternating knots, including $ 2$-bridge knots, and therefore, for these knots, $ r - s$ gives the minimum number of double points among all alternating projections of $ K$. If $ K$ is a special alternating knot, it is also proved that $ {a_s} = 1$ and $ s$ is equal to the genus of $ K$. Similar results hold for links.


References [Enhancements On Off] (What's this?)

  • [1] C. Bankwitz, Über die Torsionszahlen der alternierenden Knoten, Math. Ann. 103 (1930), 145-161. MR 1512619
  • [2] R. H. Crowell, Genus of alternating link types, Ann. of Math. (2) 69 (1959), 258-275. MR 0099665 (20:6103b)
  • [3] -, Non-alternating links, Illinois J. Math. 3 (1959), 101-120. MR 0099667 (20:6105)
  • [4] D. Gabai, The Murasugi sum is a natural geometric operation, Contemp. Math. 20 (1983), 131-143. MR 718138 (85d:57003)
  • [5] V. Jones, A polynomial invariant for knots via von Neumann algebras, Bull. Amer. Math. Soc. (N.S.) 12 (1985), 103-111. MR 766964 (86e:57006)
  • [6] V. F. R. Jones and J. Birman, Seminar notes.
  • [7] L.H. Kauffman, Formal knot theory, Math. Notes, no. 30, Princeton Univ. Press, Princeton, N.J., 1983. MR 712133 (85b:57006)
  • [8] W. B. R. Lickorish and K. C. Millett, Polynomial invariant of oriented links (to appear).
  • [9] H. R. Morton, The Jones polynomial for unoriented links (to appear). MR 830630 (87e:57011)
  • [10] H. Murakami, A recursive calculation of the Arf invariant of a link (to appear). MR 833206 (88a:57014)
  • [11] K. Murasugi, On the genus of the alternating knot. I. II, J. Math. Soc. Japan 10 (1958), 94-105; ibid. 10 (1958), 235-248. MR 0099664 (20:6103a)
  • [12] -, On a certain numerical invariant of link types, Trans. Amer. Math. Soc. 117 (1965), 387-422. MR 0171275 (30:1506)
  • [13] -, On the Alexander polynomial of alternating algebraic knots, J. Austral. Math. Soc. (A) 39 (1985), 317-333. MR 802722 (87e:57012)
  • [14] -, Jones polynomials and classical conjectures in knot theory (to appear).

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1986-0831194-9
Keywords: Knot, link, Jones polynomial, alternating knot, reduced Alexander polynomial
Article copyright: © Copyright 1986 American Mathematical Society

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