On the Alexander polynomials of certain three-component links

Kidwell, Mark E.

doi:10.1090/S0002-9939-1978-0482737-X

On the Alexander polynomials of certain three-component links
HTML articles powered by AMS MathViewer

by Mark E. Kidwell

Proc. Amer. Math. Soc. 71 (1978), 351-354

DOI: https://doi.org/10.1090/S0002-9939-1978-0482737-X

PDF | Request permission

Abstract:

Let L be a three-component link all of whose linking numbers are zero. Write the Alexander polynomial of L as $\Delta (x,y,z) = (1 - x)(1 - y)(1 - z)f(x,y,z)$. Then the integer $|f(1,1,1)|$ is a perfect square.

References

E. Artin, Geometric algebra, Interscience Publishers, Inc., New York-London, 1957. MR 0082463
Joan S. Birman, Braids, links, and mapping class groups, Annals of Mathematics Studies, No. 82, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1974. MR 0375281
R. H. Fox, Some problems in knot theory, Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961) Prentice-Hall, Englewood Cliffs, N.J., 1962, pp. 168–176. MR 0140100
Fujitsugu Hosokawa, On $\nabla$-polynomials of links, Osaka Math. J. 10 (1958), 273–282. MR 102820
Dale Rolfsen, Knots and links, Mathematics Lecture Series, No. 7, Publish or Perish, Inc., Berkeley, Calif., 1976. MR 0515288
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