Incompressibility of closed surfaces in toroidally alternating link complements (Q1392657)

The author studies incompressible closed orientable surfaces in toroidally alternating link complements. The class of toroidally alternating links has been introduced by \textit{C. C. Adams} in [Topology 33, No. 2, 353-369 (1994; Zbl 0839.57004)], and it contains in particular the alternating links, almost alternating links, pretzel links and Montesinos links. In [Pac. J. Math. 117, 353-370 (1985; Zbl 0578.57002)] \textit{W. Menasco} has shown in particular that there are no closed incompressible and pairwise incompressible surfaces in non-split prime alternating link complements. The author uses the notion of standard position, defined by W. Menasco in that same paper, and he proves the following Theorem: Let \(M\) be the 3-sphere \(S^3\) or a lens space which is not \(S^2\times S^1\), let \(L\) be a non-split toroidally alternating link in \(M\) and let \(F\subset M-L\) be a closed orientable surface in standard position. Then \(F\) is incompressible and pairwise incompressible in \(M-L\).