Concordance to links with unknotted components
This paper addresses the question of topological (respectively, smooth) link concordance: given a pair of links \(L_1\) and \(L_2\) in \(S^3\), when is there a collection of disjoint locally flat (smooth) cylinders \(C\) embedded in \(S^3\times[0,1]\) with \(\partial C=L_1 \times 0\cup -L_2 \times 1\)? The paper examines the gap between concordance of components of \(L_1\) and \(L_2\) and concordance of \(L_1\) and \(L_2\) themselves.NEWLINENEWLINEPrevious work \textit{T. D. Cochran} [Trans. Am. Math. Soc. 327, No. 2, 641--654 (1991; Zbl 0733.57003)], \textit{T. D. Cochran} and \textit{K. E. Orr} [Ann. Math. (2) 138, No. 3, 519--554 (1993; Zbl 0828.57015)] on this topic focused on unknotted components, but the authors' present work extends to link components with arbitrary Alexander polynomial. The main theorem (Theorem 1.3) is the following: For any finite collection \(D\) of classical Alexander polynomials of knots and for any knot \(J_0\) with \(\Delta_{J_0}(t)\in D\), there are links \(K_1\cup K_2\) such that (1) \(K_1\) is smoothly concordant to \(J_0\) and (2) \(K_1\cup K_2\) is not topologically concordant to any link whose first component has an Alexander polynomial in \(D\).NEWLINENEWLINEAlthough the paper constructs specific examples of two-component links as described above, the authors note that their techniques -- mainly the theory of covering links and the \(d\)-invariants of Heegaard Floer theory -- extend to links of any number of components.
scientific article; zbMATH DE number 6035500
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