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A factorization of the Conway polynomial and covering linkage invariants. (English) Zbl 1119.57005
The article deals with the Conway polynomial of a link in the 3-sphere. J. P. Levine [Comment. Math. Helv. 74, 27–52 (1999; Zbl 0918.57001)] proved that the Conway polynomial of a link \(L\) is the product of two factors: one is the Conway polynomial of a knot \(K_L\), obtained by banding together the components of \(L\), and the other one is a power series depending on the choice of the bands and could be expressed in terms of the \(\bar{\mu}\)-invariant of the string link representation of \(L\) associated to the chosen bands. In this paper the authors give another description of this last factor by viewing the choice of the bands as a choice of a Seifert surface for \(L\). More precisely, this factor is obtained as the determinant of a matrix whose entries are linking parings in the infinite cyclic covering space of the complement of \(K_L\) and which takes values in the quotient field of \(\mathbb{Z}[t,t^{-1}]\). Moreover they describe the Taylor expansion of the linking pairing around \(t=1\) in terms of the derivation of links introduced by T. D. Cochran [Comment. Math. Helv. 60, 291–311 (1985; Zbl 0574.57008)] and give an algebraic method in order to compute it. Finally, they prove that the first non-vanishing coefficient of the Conway polynomial is determined by the linking numbers, generalizing a result of J. Hoste [Proc. Am. Math. Soc. 95, 299–302 (1985; Zbl 0576.57005)].

MSC:
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
57M25 Knots and links in the \(3\)-sphere (MSC2010)
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[1] DOI: 10.1007/BF02567416 · Zbl 0574.57008 · doi:10.1007/BF02567416
[2] DOI: 10.1007/BF01388868 · Zbl 0589.57005 · doi:10.1007/BF01388868
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[4] DOI: 10.1090/S0002-9939-1985-0801342-X · doi:10.1090/S0002-9939-1985-0801342-X
[5] DOI: 10.1007/s000140050075 · Zbl 0918.57001 · doi:10.1007/s000140050075
[6] DOI: 10.1142/S0219199701000299 · Zbl 0989.57005 · doi:10.1142/S0219199701000299
[7] DOI: 10.1007/BF01390230 · Zbl 0404.57004 · doi:10.1007/BF01390230
[8] DOI: 10.1090/S0002-9947-04-03423-3 · Zbl 1056.57007 · doi:10.1090/S0002-9947-04-03423-3
[9] DOI: 10.1016/0166-8641(84)90012-9 · Zbl 0561.57012 · doi:10.1016/0166-8641(84)90012-9
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