More on complements of minimal spanning surfaces

Abstract

W. R. Alford in volume 91 of the Annals of Mathematics has shown the existence of a knot which has two minimal spanning surfaces whose complements in S are not homeomorphic. The trefoil knot is a companion to the knot. This paper shows that any nontrivial knot k is a companion to a knot K which has at least two minimal spanning surfaces. Introduction. In [1], W. R. Alford exhibited a knot k and two minimal spanning surfaces Sx and S2 for k such that S 3 — S{ are not homeomorphic. The knot was formed by sending the torus T containing the knot I in Fig. 1 faithfully to a regular neighborhood of the trefoil knot. In a later paper [2], Alford and C. B. Schaufele constructed knots with 2 really distinct minimal spanning surfaces; the surfaces do not have homeomorphic complements. The examples were constructed by sending the torus T containing the knot I in Fig. 1 faithfully to a regular neighborhood of the sum of m nice knots. The selection of the knots was strongly influenced by their algebraic properties. The purpose of this paper is to show that any nontrivial knot is a companion to a knot K which has at least two minimal surfaces. The knot K is the image of the knot I in T in Fig. 1 under a faithful homeomorphism of the solid torus T to a regular neighborhood V of the knot I. The Alexander polynomial of K is (2 — t) • (2t — 1) [4] for any nontrivial k used. Thus K had genus at least one. The spanning surfaces for K have genus one, so K has genus one. The surfaces. The surfaces for K are constructed as in [2]. The knot I is spanned by a singular disk in T as shown in Fig. 2. Only one side is shown; the singularities are in heavy lines. Received by the editors June 4,1971 and, in revised form, November 8,1971. AMS (MOS) subject classifications (1970). Primary 55A25; Secondary 57A10.