Some closed incompressible surfaces in knot complements which survive surgery  pp. 179-194

Some closed incompressible surfaces in knot complements which survive surgery

By Hamish Short

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I wish to present here two theorems, the first of which is mainly due to Menasco (and is not unrelated to the little known work of Connor). We show that certain closed incompressible surfaces in the complement of a knot remain incompressible after all (p,q)-surgeries (“survive” surgery).

The surfaces which interest us are generalizations of the incompressible, non-boundary parallel torus in the complement of a composite knot, and later, generalizations of the boundary of the solid torus used in the construction of doubled knots. The first class are also present in the complements of Connor's splittable knots (see §2, ii and Connor, A.C., 1969); Connor showed that these knots have property P (amongst other things), and in fact it is usually a straightforward task to generalise the proof of a result for composite knots to splittable knots.

Menasco shows (Menasco, W., 1984) that what I call 2m-surfaces (Definition 1) survive surgery, giving a simple geometric proof. It was in the course of studying Menasco's proof (case (b) of §3) that the generalizations to m-surfaces (Theorem 1) and d-surfaces (Theorem 2) were found. I have since learnt that Menasco was aware of the generalization to m-surfaces. Unfortunately here the result is not as strong, as we cannot deal with the case of integer (q = ± 1) surgery.

As corollaries we obtain properties P and R for knots with 2m- or d-surfaces, and the proof of the properties for knots with m-surfaces is reduced to the case of ± 1 and (0,1)-surgeries.