Abstract. For high-dimensional simple knots we give two theorems concerning unique factorization into irreducible knots, and provide examples to show that the hypotheses are necessary in each case. IntroductionThe purpose of this paper is to collate and extend the known results on the factorization of high dimensional knots. By an n-knot we mean an oriented smooth or locally flat PL pair (S n+2 , S m ), where S n is homeomorphic to the n-sphere S n . The sum Jc + l of two n-knots k and I is obtained by excising the interior of a tubular neighbourhood of a point on each S n and identifying the boundaries of the resulting knotted ball pairs so that the orientations match up. A knot k is irreducible if it cannot be written as the sum of two non-trivial knots. It is a result of H. Schubert (16) that for n = I, every knot factorizes into finitely many irreducibles, and that factorization is unique (up to the order of the factors).Given an n-knot k, the exterior K is the closed complement of a tubular neighbourhood of 2™. The knot k is simple if K has the homotopy [(n -l)/2]-type of a circle; that is TJ^K) ~ n 1 (S 1 ) for 1 ^ i < (n-l)/2. For n > 3, this is the most that can be asked without making k trivial (see (ll, 12)). The knot k is fibred if K is fibred over the circle, and we let R denote the infinite cyclic cover of K.In Section 1 we give a short proof that every simple n-knot, n ^ 3, factorizes into finitely many irreducibles. A more general result was published by A. B. Sosinskii in (18), but note the assertion of T. Maeda in (14).Let k be a simple (2q-l)-knot, q ^ 2. There are two ways of classifying such knots in terms of algebraic invariants. The first of these, due to J. Levine, is in terms of the S-equivalence class of the Seifert matrix of k; details may be found in (12). The second method uses the Blanchfield duality pairing, <,>:A o is the field of fractions of A, and H q (K) is regarded as a A-module. Details of this method may be found in (7,8,20,21).Each such knot k has associated with it a quadratic form, as outlined in Section 2. If this form is definite, then k is said to be definite. The knot k is fibred if and only if the leading coefficient of its Alexander polynomial is + 1; this follows easily from the results of R.H.Crowell(3) and W.Browder and J. Levine(2). In Section 2 we show

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Schubert has shown that every classical knot 2:1 c S 3 factorises uniquely into the connected sum of finitely many indecomposable knots (cf. [12]). In particular cancellation holds for these knots. For higher elimensional simple knots factorisation is not always unique (cs [5] and [1]), but in many cases we still have cancellation (see [2], Proposition 6.6).In this note we shall give counter examples to the cancellation of non-singular hermitian and skew-hermitian forms. In order to obtain these examples we shall show that the extension of the 2J-lattice/~4~, n ~= 1, to certain orders is indecomposable.Using the classification of simple (2q--1)-knots 2:2q-lc S2q +1, q ~= 1, in terms of (--1)q+l-hermitian (Blanchfield) forms, we shall then prove that cancellation does not hold for higher odd-dimensional knots.I thank Hans-Joehen Bartels and Larry Gerstein for useful conversations.