Seifert surfaces of knots inS4

Abstract

This paper uses some ideas from 3-dimensional topology to study knots in S4 . We show that the Poincare conjecture implies the existence of a non-fibered knot whose complement fibers homotopically. In a different direction, we show that is an obstruction to a knot having a Seifert surface made out of Seifert fibered spaces, and hence to being ribbon. We also prove that any 3-manifold is invertibly homology cobordant to a hyperbolic 3-manifold, so that every knot in S4 has a hyperbolic Seifert surface. One of the reasons that the study of knots in the 4-sphere has a special character is that the Seifert surfaces that such knots bound are 3-dimensional. Hence the peculiar nature of the topology of 3manifolds can lead to interesting behavior of 2-knots. In this paper we give several examples of this principle. The first example is to show that the 3-dimensional Poincare conjecture implies the existence of non-fibered (topological) knots in *S 4 whose exteriors are homotopy equivalent to the exterior of a fibered knot. (Similar phenomena have been noticed by J. Hillman and C. B. Thomas [12, 13] and S. Weinberger [32].) The second instance is to see how the existence of a structure on a Seifert surface influences topological properties of the knot. Restrictions on the possible geometric structures are obtained via Gromov's norm of a 2-knot, defined below. We show that a knot with non-zero cannot have a Seifert surface which is a connected sum of Seifert-fiber ed 3-manifolds. In particular, the is seen to be an obstruction to a knot in S4 being ribbon. A similar obstruction has been found by Bruce Trace [30]. In contrast, we will show that any knot has a Seifert surface which is a hyperbolic manifold. This follows from Theorem 2.6, which states that any 3-manifold has an invertible homology cobordism to a hyperbolic manifold. A cobordism W from M to N is called invertible if there is another cobordism W, so that W U N W = M x I. Without the requirement that the homology cobordism be invertible, this theorem is due to R. Myers [23].