On the finiteness of higher knot sums

In this paper we show that any higher knot (n≥3) can be decomposed as a sum of irreducible knots and there is a finite upper bound on the number of summands. The case n=1 is due to Schubert 10]. A proof of this present case was published in 12] by Soninskii but subsequently Maeda showed that a crucial lemma was false 7]. The difficulty is to find a bound on decompositions of the knot group π₁. This is achieved here by applying Dunwoody's work in 3]. This results in two theorems, 1.6 and 1.7, which are of some interest in their own right. The rest of the paper follows as in Sosinskii's paper.We are grateful to G.A. Swarup for pointing out the possibility of this line of attack and to A. Bartholomew for an idea used in the proof of Theorem 1.6.