The support of global graph links (Q5902733)

In this note the author states (without proof) the results of his Ph. D. dissertation [Univ. of Tokyo], which concern knots and links in (connected, compact, orientable, three-dimensional) manifolds. Such a knot or link is said to be (quasi-)local if it is contained in some imbedded (homotopy) three-ball, and inessential if its components are all homotopically trivial. The author gives a characterization of graph links analogous to (though somewhat more complicated than) that of graph knots mentioned by \textit{C. McA. Gordon} [Trans. Am. Math. Soc. 275, 687-708 (1983; Zbl 0519.57005)] and \textit{M. Ue} [J. Fac. Sci., Univ. Tokyo, Sect. I A 30, 334-352 (1983; Zbl 0542.57010)], and deduces from it a condition that is necessary for a graph link to be inessential; a corollary is that in graph-manifolds with torsionfree fundamental groups whose prime decompositions include no factor \(S^ 1\times S^ 2\), the class of local graph links coincides with the class of inessential graph links. The author also mentions the more general result that in a closed manifold M whose prime decomposition includes no \(S^ 1\times S^ 2\), a link is quasi-local if and only if the fundamental group of its exterior is a free product \(\pi_ 1(M)*G\). For knots the same result holds for closed manifolds generally (even those that have \(S^ 1\times S^ 2\) as a factor); a consequence is a generalization of the well-known characterization of the unknot in \(S^ 3:\) a knot in a closed manifold is unknotted if and only if the fundamental group of its exterior is \(\pi_ 1(M)*{\mathbb{Z}}\).