Some Non‐Fibred 3‐Knots

It is well known that a simple n-knot, n > 3, is fibred if and only if the leading coefficient of its Alexander polynomial is + 1 . This can be proved directly by doing ambient surgery on a Seifert surface of the knot, in the manner of J. Levine [6], or by using the fibration theorem of W. Browder and J. Levine [2, Corollary 1.6]. Using a recent result of S. K. Donaldson [3], concerning the intersection form on a smooth closed simply-connected 4-manifold, we are able to construct examples of simple 3-knots which satisfy the condition above but are not fibred. As a corollary, we see at once that the Browder-Levine fibration theorem does not extend to dimension 5. Furthermore, it follows easily that the quasi-fibring conjecture for 5-manifolds (with infinite-cyclic fundamental group), and the smoothing conjecture for simplyconnected finite Poincare complexes in dimension 4 also fail. For details of both these conjectures the reader is referred to [7]. An n-knot k is a smooth pair (S" + , Z") where Z" is homeomorphic to the n-sphere S". Let K denote the exterior of k; that is, the complement of an open tubular neighbourhood of I " in S" + . The knot k is said to be fibred if K fibres smoothly over S. The {2q— l)-knot k is simple if n^K) ^ ^.(S) for 1 ^ i < q\ for q ^ 2, such knots have been classified in terms of the S-equivalence class of the Seifert matrix (see [6] for details). There exists a 16 x 16 integer matrix A with the following properties (i) det(A + A') = 1 (ii) signature {A + A') = 16 (iii) detA = 1.