Classification of string links up to self delta-moves and concordance
\textit{J. W. Milnor}'s \(\overline{\mu}\)-invariant of \(n\)-component links is defined for each sequence \(i_1i_2\cdots i_m\), \(i_j\in \{1,2,\dots , n\}\), cf. [Ann. Math. (2) 59, 177--195 (1954; Zbl 0055.16901)]. It is a concordance invariant. Let \(r(I)\) be the maximum number of times that any index appears. For example, \(r(1123)=2\), and \(r(1231223)=3\). Then, it is known that if \(r(I)=1\), the \(\overline{\mu}\)-invariant is a link-homotopy invariant. Similarly, the Milnor \(\mu\)-invariant \(\mu_{L}(I)\) is defined for a string link \(L\). See a paper of \textit{N. Habegger} and \textit{X.-S. Lin} [J. Am. Math. Soc. 3, No.~2, 389--419 (1990; Zbl 0704.57016)]. A \(\Delta\)-move is a certain local move on a link or tangle which involves three strands, and preserves linking numbers. A self \(\Delta\)-move is a \(\Delta\)-move where the three strands belong to the same component of a (string) link or a tangle. Two (string) links \(L\) and \(L'\) are said to be self \(\Delta\)-concordant if there is a sequence \(L=L_1,\dots , L_{\ell}=L'\) of (string) links such that \(L_{i}\) and \(L_{i+1}\) are either concordant or self \(\Delta\)-equivalent. In the paper under review, the author shows that two \(n\)-component string links \(L\) and \(L'\) are self \(\Delta\)-concordant if and only if \(\mu_{L}(I)=\mu_{L'}(I)\) for any \(I\) with \(r(I)\leq 2\).
scientific article; zbMATH DE number 5530579
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