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On the Fox-Milnor theorem for the Alexander polynomial of links. (English) Zbl 1080.57013
A knot \(K\) in the 3-sphere \(S^3\) is said to be slice if \(K\) bounds a smooth disk in the 4-ball \(B^4\). In [Osaka J. Math. 3, 257-267 (1966; Zbl 0146.45501)], R. H. Fox and J. W. Milnor showed that the Alexander polynomial of a slice knot is a product \(f(t)f(t^{-1})\) for an integral polynomial \(f\). A. Kawauchi generalized this result to a ‘strongly slice’ link in [Osaka J. Math. 15, 151-159 (1978; Zbl 0401.57013)]. The proof mainly used the Blanchfield duality theorem. The author of the present paper generalizes the result of Fox and Milnor to another direction, namely, to the case of the multivariable Alexander polynomial of links bounding a smooth surface with Euler characteristic 1 in \(B^4\). A link \(L=L_1\cup \cdots \cup L_{\mu}\) in \(S^3\) is called an algebraic splitting if, for \(k=1,\ldots , \mu,\) \(L_k\) is a sublink of \(L\) with lk\((L_k,L_{k'})=0\) for all \(k\neq k'\). The main theorem of this paper is the following: if \(L\) bounds a smooth compact oriented surface \(F=F_1\cup \cdots \cup F_{\mu}\) in \(B^4\), with \(\mu\) connected components, such that \(\partial F_k=L_k\) and \(\chi(F)=1\), then there is \(p\in \mathbb Z [t_1,\ldots , t_{\mu}]\) such that \[ \Delta_{L}(t_1,\ldots ,t_{\mu})\prod_{k=1}^{\mu} (t_k-1)^{-\chi(F_k)}\doteq p(t_1,\ldots , t_{\mu}) p(t_1^{-1},\ldots ,t_{\mu}^{-1}), \] where \(\Delta_L(t_1,\ldots ,t_{\mu})\) is the Alexander polynomial of \(L\) associated to the splitting, and \(\doteq\) means equality up to a unit in \(\mathbb Z[t_1^{\pm 1},\ldots , t_{\mu}^{\pm 1}]\). The proof is based on the duality theorem due to W. Franz [J. Reine Angew. Math. 173, 60–64 (1935; Zbl 0012.12702)] and J. W. Milnor [Ann. Math. (2) 76, 137–147 (1962; Zbl 0108.36502)].

MSC:
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
57Q60 Cobordism and concordance in PL-topology
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