In [1], Gordon proved that a certain sequence of preimage knots in S3 is finite. In this paper, we prove that every sequence of preimage knots is finite by using the Gromov invariant for knots. Introduction. Let V be a solid torus and 1 an essential loop in int V. Here 1 is essential in V if aV is incompressible in V 1 and if 1 is not isotopic in V to a core c of V. Let f: V __ S3 (resp. g: V __ S3) be an embedding such that f(c) (resp. g(c)) is knotted (resp. unknotted) in S3, and let T = f(WV). We say f(c) is the companion of f(l) for T, f(l) is the satellite of f(c) for T, and g(l) is a preimage of f(l) for T. Here we do not fix the twistings of embeddings f,g in any manner, so a preimage of f (l) for T is not determined uniquely from f (l) and T. For two knots Ko, K in S3, an inequality Ko K2 > ... > Kn > Kn+l > Let V be a solid torus with a core c and 1 an essential loop in V. Let fi, f2: V S3 be two embeddings. In [6}, Kouno proved that, if fi (1) f2 (1) and if both fi (c) and f2 (c) are knotted, then fi (c) f2 (c). On the other hand, Theorem 1 above shows that, if fi (c) is unknotted and f2 (c) is knotted, then fi (1) J f2 (1). Therefore, by Kouno's theorem [6] together with our theorem, the following corollary is obtained immediately. COROLLARY. With the notation as above if fl (l) f2 (1), then fi (c)-_ f2 (c). 1. Preliminaries. For fundamental notations on 3-manifolds, we refer to Jaco