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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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##### References:
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