Some closed incompressible surfaces in knot complements which survive surgery. (English) Zbl 0577.57002

Low dimensional topology, 3rd Topology Semin. Univ. Sussex 1982, Lond. Math. Soc. Lect. Note Ser. 95, 179-194 (1985).
[For the entire collection see Zbl 0561.00016.]
A closed, orientable, incompressible, non-\(boundary\)-\(parallel\) surface S embedded in the complement of a knot k in \(S^ 3\) is called an m-surface (2m-surface) if it contains a loop that is parallel to a meridian curve (it carries two such loops that are non-isotopic on S, resp.). For example, complements of composite knots contain m-surfaces and complements of a sum of an incompressible tangle and a prime tangle contain 2m-surfaces. A closed, orientable, incompressible, non- \(boundary\)-\(paralle\) surface S in the complement of k is a d-surface if there is an embedding of \(B^ 2\times I\) in \(S^ 3\) such that \(B^ 2\times I\cap S=\partial B^ 2\times I\) and such that \(B^ 2\times I\cap k\) consists of two linked arcs \(a_ i\) with \(\partial a_ i\) consisting of two points in \(B^ 2\times \{i\}\) \((i=1,2)\). For example, complements of doubles of non-trivial knots contain d-surfaces.
Generalizing results and methods of W. Menasco [Topology 23, 37-44 (1984; Zbl 0525.57003)] the author shows that m-surfaces (for \(| q| >1)\), 2m-surfaces, and d-surfaces remain incompressible after all (p,q)-surgeries on k. In particular, the surgered manifold is Haken.
Reviewer: W.Heil


57M25 Knots and links in the \(3\)-sphere (MSC2010)
57N10 Topology of general \(3\)-manifolds (MSC2010)