On twins in the four-sphere. I.

Abstract

E. C. Zeeman [Trans. Amer. Math. Soc. 115 (1965), 471–495; MR0195085 (33 #3290)] introduced the process of twist spinning a 1-knot to obtain a 2-knot (in S4), and proved that a twist-spun knot is fibered with finite cyclic structure group. R. A. Litherland [ibid. 250 (1979), 311–331; MR0530058 (80i:57015)] generalized twist-spinning by performing during the spinning process rolling operations and other motions of the knot in three-space. The first paper generalizes those results by introducing the concept of a twin. A twin W is a subset of S4 made up of two 2-knots R and S that intersect transversally in two points. The prototype of a twin is the n-twist spun of K (that is, the union of the n-twist spun knot of K and the boundary of the 3-ball in which the original knot lies). The exterior of a twin, X(W), is the closure of S4−N(W), where N(W) is a regular neighborhood of W in S4. The first paper considers properties of X(W), and uses these to characterize the automorphisms of a 2-torus standardly embedded in S4, which extend to S4, and also to prove that any homotopy sphere obtained by Dehn surgery on such a 2-torus is the real S4. The second paper is devoted to the fibration problem, i.e. given a twin in S4, try to understand what surgeries in W give a twin W′ which has a component that is a fibered knot (as in the Zeeman theorem). This approach yields alternative proofs of the twist-spinning theorem of Zeeman, and of the roll-twist spinning results of Litherland. New fibered 2-knots are produced through these methods.