Incompressible and pairwise incompressible surfaces in alternating link complements
For a non-split, prime alternating link \(L\) in the \(3\)-sphere, \textit{W. Menasco} [Topology 23, 37--44 (1984; Zbl 0525.57003)] showed that any closed connected incompressible and pairwise incompressible surface \(S\) which intersects \(L\) transversely in a finite number of points has a strong constraint. More precisely, he showed that if the number of points of \(S\cap L\) is less than \(8\), then \(S\) is the \(2\)-sphere, and there are only finitely many isotopy types of \(S\) after fixing the number of points of \(S\cap L\).NEWLINENEWLINEThe main purpose of the paper under review is to examine the case where \(S\cap L\) consists of \(8\) points. The result claims that such a surface \(S\) is either the \(2\)-sphere or a torus. Furthermore, under a mild condition, such surfaces are classified into 16 patterns. In the appendix, two gaps in \textit{W. Menasco} [Pac. J. Math. 117, 353--370 (1985; Zbl 0578.57002)] are filled.
scientific article; zbMATH DE number 6059354