On the distribution of Alexander polynomials of alternating knots and links

Kinoshita, Shin’ichi

doi:10.1090/S0002-9939-1980-0572320-9

On the distribution of Alexander polynomials of alternating knots and links
HTML articles powered by AMS MathViewer

by Shin’ichi Kinoshita

Proc. Amer. Math. Soc. 79 (1980), 644-648

DOI: https://doi.org/10.1090/S0002-9939-1980-0572320-9

PDF | Request permission

Abstract:

In this paper we will be concerned with the distribution of the reduced Alexander polynomials of alternating knots and links. Our theorems are quite different in style from those of R. H. Crowell, K. Murasugi and R. Riley, which intend to characterize the reduced Alexander polynomials of alternating knots and links.

References

J. W. Alexander, Topological invariants of knots and links, Trans. Amer. Math. Soc. 30 (1928), no. 2, 275–306. MR 1501429, DOI 10.1090/S0002-9947-1928-1501429-1
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

Non-alternating link

3

Richard H. Crowell and Ralph H. Fox, Introduction to knot theory, Ginn and Company, Boston, Mass., 1963. Based upon lectures given at Haverford College under the Philips Lecture Program. MR 0146828
R. H. Fox, Congruence classes of knots, Osaka Math. J. 10 (1958), 37–41. MR 131871
Fujitsugu Hosokawa, On $\nabla$-polynomials of links, Osaka Math. J. 10 (1958), 273–282. MR 102820
Shin‘ichi Kinoshita, On Wendt’s theorem of knots. II, Osaka Math. J. 10 (1958), 259–261. MR 124889
Kunio Murasugi, On the Alexander polynomial of the alternating knot, Osaka Math. J. 10 (1958), 181–189; errata, 11 (1959), 95. MR 99666
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 alternating knots, Osaka Math. J. 12 (1960), 277–303. MR 137107
Robert Riley, A finiteness theorem for alternating links, J. London Math. Soc. (2) 5 (1972), 263–266. MR 312487, DOI 10.1112/jlms/s2-5.2.263
H. Seifert, Über das Geschlecht von Knoten, Math. Ann. 110 (1935), no. 1, 571–592 (German). MR 1512955, DOI 10.1007/BF01448044
Guillermo Torres, On the Alexander polynomial, Ann. of Math. (2) 57 (1953), 57–89. MR 52104, DOI 10.2307/1969726