Determining incompressibility of surfaces in alternating knot and link complements (Q1066521)

Let L be a non-split, prime, alternating (relative to a fixed projection \({\mathbb{R}}^ 3\to {\mathbb{R}}^ 2)\) link in \({\mathbb{R}}^ 3\). In a previous paper [Topology 23, 37-44 (1984; Zbl 0525.57003)], the author introduced the notion of a standard position embedding (relative to the projection) for an incompressible surface in \({\mathbb{R}}^ 3-L\) and proved that for \(n>0\) there are only finitely many such n-punctured surfaces in \({\mathbb{R}}^ 3-L\). In the present paper a method is given for determining when a given standard position embedding \(S\subset {\mathbb{R}}^ 3\) is incompressible. The method is applied to understanding when ''peripheral tubing'' preserves incompressibility.