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

Short, Hamish

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

MSC:

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

Keywords:

Dehn surgery; Haken-3-manifolds; incompressible surface in the; complement of a knot; m-surface; composite knots; incompressible tangle; prime tangle; d-surface

Citations:

Zbl 0561.00016; Zbl 0525.57003