Invertible knot cobordisms

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Abstract
In [1], R. H. Fox studies embeddings of S z in S 4 by means of 3-dimensional hyperplanes slicing the embedded S z in 1-complexes. The connected manifold crosssections are slice knots; the non-connected manifold cross-sections are (weakly) slice links. Fox poses the following question [2, # 39] : Which slice knots and weakly slice links can appear as cross-sections of the unknotted S z in $4? Many examples of this phenomenon are known [3, 4, 14, 16, 19], and Hosokawa [3] has given sufficient geometric conditions on the cross-sectional sequence of the S z in S 4 for the S 2 to unknot. This paper considers Fox's question in the setting of higher-dimensional knot theory, but from a different viewpoint - that of inverting a knot cobordism. A slice knot is one which is cobordant to the unknot. If the slice knot bounds a cobordism (to the unknot) which is invertible from the knotted end, then the slice knot is said to be doubly-null-cobordant [19]. The doubly-null-cobordant knots are precisely the cross-sections of the unknot. We supply in this paper detailed proofs of the results announced in [18]. Using techniques of Levine [8], we develop necessary algebraic conditions for an odd-dimensional knot to be doubly-null-cobordant. These conditions are shown to be sufficient in a restricted case. We show that the Stevedore's Knot (61) is not doubly-null-cobordant, and that in fact 946 is the only knot in Reidemeister's table of prime knots 1-10] which is doubly-null-cobordant. We then turn to the problem of geometric realization of doubly-null-cobordant knots. We show that all allowable systems of invariants can be geometrically realized. A development of similar results for higher-dimensional codimension two links will be dealt with in a future paper.

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