Doubly sliced knots which are not the double of a disk

In this paper we show that double disk knots can be distinguished from general doubly sliced knots in dimensions 4n + 1. A double disk knot is formed by unioning two identical disk knots along their boundary. J. P. Levine has demonstrated that these knots are all doubly sliced [9], i.e., they can be realized as a slice of the trivial knot. Spun knots, high dimensional ribbon knots [2], and Sumners' knots constructed in [13] are all examples of double disk knots. In [9] Levine also gives an example of a classical knot and a 2-knot that are doubly sliced but not the double of a disk. In this paper we show that doubly sliced knots are distinct from double disk knots in dimensions 4n + 1. Our method of distinguishing double disk knots produces obstructions from the Casson-Gordon invariants. This paper is the main result of the author's Ph.D. thesis and he wishes to thank his advisor, J. P. Levine, for his help and encouragement. An n dimensional knot is a codimension two spherical knot or a smooth oriented pair (Sn+2, K) where K is a submanifold which is homeomorphic to Sn. An n dimensional disk knot is a smooth oriented pair (Bn+2, D) where D is homeomorphic to the n-disk and 9Bn+2 n D = AD. We apply disk knots to the study of knots in two distinct ways. First, the boundary of an n-disk knot (Bn+2, D) is the (n 1)-knot (aBn+2, O9D). Second, we may join two n-disk knots that have diffeomorphic boundaries along their boundaries. Here we obtain an n-knot. If two disk knots (B n+2, D1) and (B n+2, D2) have diffeomorphic boundaries by some orientation preserving diffeomorphism, f: a(B1, D1) a(B2, D2), then we may form the n-knot -(B1, D1) Uf (B2, D2) = (-B1 Uf B2, -D1 Uf D2). Given an n-disk knot (Bn+2, D), one can construct an (n+ 1)-disk knot Z(B, D) called the suspension of D. In the P.L. category the suspension of D may be realized as (Bn+2 x I, D x I) [9]. We use the smooth version, obtained by rounding the corners. The boundary of ED is (-Bn+2, -D) Ui (Bn+2, D), the n-knot formed by doubling the disk (Bn+2, D). Knots formed by doubling a disk are called double disk knots. A knot which is the boundary of a disk knot is called null cobordant. Some knots bound particularly nice disk knots and are slices of the trivial knot. The disk knot (Bn+2, D) = Dn+2'n is invertible if there exists a disk knot (Bn+2, A) = An+2,n and a diffeomorphism, f: 9Dn+2,n _* 09An+2,n such that -Dn+2,n Uf /An+2,n is the Received by the editors September 23, 1985 and, in revised form, December 20, 1985. 1980 Mathematics Subject Classification (1985 Revision). Primary 57Q45, 57Q60. (?)1986 American Mathematical Society 0002-9947/86 $1.00 + $.25 per page