Milnor's concordance invariants for knots on surfaces (Q2092759)

To appreciate the paper under review, it is worth briefly tracing the evolution of the subject. \textit{K. T. Chen} [Proc. Am. Math. Soc. 3, 44--55 (1952; Zbl 0049.40402); Ann. Math. (2) 56, 343--353 (1952; Zbl 0049.40403)] investigated the quotients of the lower central series of a finitely presented group. As an application, he proved that, for polygonal links in euclidean space, fundamental group of the link complement, modulo its lower central subgroup is invariant under \textit{isotopy}. This was soon generalized by Milnor to links in 3-manifolds [\textit{J. W. Milnor}, Princeton Math. Ser. 12, 280--306 (1957; Zbl 0080.16901); \textit{R. H. Fox} (ed.) et al., Algebraic geometry and topology. A Symposium in honor of S. Lefschetz. Princeton University Press, Princeton, NJ (1957; Zbl 0077.16602)] and he extracted a numerical isotopy invariant \(\overline{\mu}\) for links in Euclidean space. Later, Stallings [stallings] showed that the \(\overline{\mu}\)~s are invariant under the coarser equivalence relation \textit{ link concordance}. \textit{L. H. Kauffman} [Eur. J. Comb. 20, No. 7, 663--690 (1999; Zbl 0938.57006)] initiated the study of virtual knots (as a generalization of classical knot). Geometrically, they could be regarded as embeddings of circles in thickened orientable surfaces \(\Sigma\times [0,1]\) [\textit{G. Kuperberg}, Algebr. Geom. Topol. 3, 587--591 (2003; Zbl 1031.57010)]. Using this perspective, the definition of concordance of virtual knots could be introduced accordingly [\textit{J. S. Carter} et al., J. Knot Theory Ramifications 11, No. 3, 311--322 (2002; Zbl 1004.57007); \textit{V. Turaev}, J. Topol. 1, No. 2, 285--305 (2008; Zbl 1166.57008)]. Concordance of virtual knots, as well as its relation to the concordance of classical knots, has become an area of active interest. \textit{H. U. Boden} and \textit{M. Nagel} [Proc. Am. Math. Soc. 145, No. 12, 5451--5461 (2017; Zbl 1397.57007)] demonstrated that the concordance group of classical knots embeds into the center of the concordance group of long virtual knots \(\mathcal{VC}\). It remains an open question whether \(\mathcal{VC}\) is abelian? Furthermore, the structure of \(\mathcal{VC}\). J. Green has produced a table of virtual knots. Determining their slice status is a fundamental problem. There are 38 virtual knots (out of the 92800 virtual knots up to six classical crossings) in Green's table where all the previously known concordance invariants are trivial, yet a slice movie has to be found. In the current paper, the author extended the Chen-Milnor theory to the realm of virtual knots. As applications, he showed: \begin{itemize} \item concordance group of the long virtual knots \(\mathcal{VC}\) is non-abelian. \item 12 of the 38 vitrual knots mentioned above are not slice, and slice movies are found for 22, reducing the undetermined cases to 4. \end{itemize}