Jones polynomials of alternating links

Murasugi, Kunio

doi:10.1090/S0002-9947-1986-0831194-9

Jones polynomials of alternating links
HTML articles powered by AMS MathViewer

by Kunio Murasugi

Trans. Amer. Math. Soc. 295 (1986), 147-174

DOI: https://doi.org/10.1090/S0002-9947-1986-0831194-9

PDF | Request permission

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

Carl Bankwitz, Über die Torsionszahlen der alternierenden Knoten, Math. Ann. 103 (1930), no. 1, 145–161 (German). MR 1512619, DOI 10.1007/BF01455692
Kunio Murasugi, On the genus of the alternating knot. I, II, J. Math. Soc. Japan 10 (1958), 94–105, 235–248. MR 99664, DOI 10.2969/jmsj/01010094
Richard H. Crowell, Nonalternating links, Illinois J. Math. 3 (1959), 101–120. MR 99667
David Gabai, The Murasugi sum is a natural geometric operation, Low-dimensional topology (San Francisco, Calif., 1981) Contemp. Math., vol. 20, Amer. Math. Soc., Providence, RI, 1983, pp. 131–143. MR 718138, DOI 10.1090/conm/020/718138
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. MR 766964, DOI 10.1090/S0273-0979-1985-15304-2
Louis H. Kauffman, Formal knot theory, Mathematical Notes, vol. 30, Princeton University Press, Princeton, NJ, 1983. MR 712133

Polynomial invariant of oriented links

H. R. Morton, The Jones polynomial for unoriented links, Quart. J. Math. Oxford Ser. (2) 37 (1986), no. 145, 55–60. MR 830630, DOI 10.1093/qmath/37.1.55
Hitoshi Murakami, A recursive calculation of the Arf invariant of a link, J. Math. Soc. Japan 38 (1986), no. 2, 335–338. MR 833206, DOI 10.2969/jmsj/03820335
Kunio Murasugi, On the genus of the alternating knot. I, II, J. Math. Soc. Japan 10 (1958), 94–105, 235–248. MR 99664, DOI 10.2969/jmsj/01010094
Kunio Murasugi, On a certain numerical invariant of link types, Trans. Amer. Math. Soc. 117 (1965), 387–422. MR 171275, DOI 10.1090/S0002-9947-1965-0171275-5
Kunio Murasugi, On the Alexander polynomial of alternating algebraic knots, J. Austral. Math. Soc. Ser. A 39 (1985), no. 3, 317–333. MR 802722

Jones polynomials and classical conjectures in knot theory