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The first coefficient of the Conway polynomial. (English) Zbl 0576.57005
F. Hosokawa [Osaka Math. J. 10, 273-282 (1958; Zbl 0105.174)] proved that the reduced Alexander polynomial of a link in the three- sphere is related to the linking numbers between the various components of the link. Using essentially the same Seifert matrix argument as Hosokawa used, the author shows that Conway’s potential function enjoys an analogous, though slightly more precise, relationship with linking numbers. (The extra precision arises from the fact that the potential function has a definite sign, while the Alexander polynomial does not.) The same result appeared in a paper by R. Hartley [Comment. Math. Helv. 58, 365-378 (1983; Zbl 0539.57003)].
Reviewer: L.Traldi

MSC:
57M25 Knots and links in the \(3\)-sphere (MSC2010)
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References:
[1] J. H. Conway, An enumeration of knots and links, and some of their algebraic properties, Computational Problems in Abstract Algebra (Proc. Conf., Oxford, 1967) Pergamon, Oxford, 1970, pp. 329 – 358.
[2] Cole A. Giller, A family of links and the Conway calculus, Trans. Amer. Math. Soc. 270 (1982), no. 1, 75 – 109. · Zbl 0492.57002
[3] Fujitsugu Hosokawa, On ∇-polynomials of links, Osaka Math. J. 10 (1958), 273 – 282. · Zbl 0105.17404
[4] Jim Hoste, The Arf invariant of a totally proper link, Topology Appl. 18 (1984), no. 2-3, 163 – 177. · Zbl 0564.57003 · doi:10.1016/0166-8641(84)90008-7 · doi.org
[5] Louis H. Kauffman, The Conway polynomial, Topology 20 (1981), no. 1, 101 – 108. · Zbl 0456.57004 · doi:10.1016/0040-9383(81)90017-3 · doi.org
[6] Hitoshi Murakami, The Arf invariant and the Conway polynomial of a link, Math. Sem. Notes Kobe Univ. 11 (1983), no. 2, 335 – 344. · Zbl 0595.57008
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