Incompressible surfaces in link complements (Q2731940)

For a link in \(S^3\) that has a \(2n\)-plat projection for some \(n\geq 3\), under conditions on the slopes of the rational tangles in the projection, the author constructs closed essential surfaces in the link complement (which is an irreducible \(3\)-manifold). \textit{E. Finkelstein} and \textit{Y. Moriah} [Trans. Am. Math. Soc. 352, No. 2, 655-677 (2000; Zbl 0934.57019); Topology Appl. 96, No. 2, 153-170 (1999; Zbl 0931.57004)] have previously proved the existence of closed essential surfaces under more restrictive conditions. If \(n=1,2\), then the link complement contains no essential surfaces by a result of \textit{A. E. Hatcher} and \textit{W. Thurston} [Invent. Math. 79, 225-246 (1985; Zbl 0602.57002)]. The author proves that in most cases the surfaces remain essential after any totally nontrivial surgery on the link, thereby generalizing another result of \textit{E. Finkelstein} and \textit{Y. Moriah} [op. cit.] from knots to links.