Toroidally alternating knots and links
The author introduces the notion of toroidally alternating link and studies surfaces in toroidally alternating link complements using the method established by M. Menasco. Let \(M\) be an orientable 3-manifold and \(T\) a torus in \(M\). A link in \(M\) is toroidally alternating with respect to \(T\) if \(L\) has a good alternating diagram on \(T\). In case \(M\) is \(S^3\) and \(T\) is a Heegaard splitting torus, the class of toroidally alternating links contains those of alternating links, almost alternating links, pretzel links and Montesinos links. The main theorem concerns the case where \(M\) is a genus one manifold, that is, \(S^3\), \(S^2 \times S^1\) or \(L(p,q)\), and its Heegaard splitting torus. The author proves that there is no incompressible meridianally incompressible closed surface in toroidally alternating knot complements in the case where \(M\) is \(S^3\) or \(L(p,q)\) with \(p\) odd. This generalizes a result proved for alternating links in \(S^3\) by Menasco. This result is applied to show that the complement of nontrivial non-torus toroidally alternating knot in \(L(p,q)\) is hyperbolic except for the lens spaces homeomorphic to \(L(2,1)\) and \(L(4k, 2k - 1)\). Some extensions are also discussed. \textit{C. Hayashi} also studies alternating links on surfaces of arbitrary positive genus [Links with alternating diagrams on closed surfaces of positive genus, Math. Proc. Camb. Philos. Soc. 117, No. 1, 113-128 (1995)].
scientific article; zbMATH DE number 590280