As far as the ends of a space or of a group are concerned see H. Freudenthal [3]1, [4], H. Hopf [5] especially No. 16, p. 96 and E. Specker [12] p. 320. See also ?3 of this paper. All 3-dimensional manifolds in the present paper are connected and are supposed to be considered with a fixed triangulation. This is not at all a restriction of generality according to E. E. Moise's work [8]. In the present paper S means the 3-sphere and K a polygonal knot in it. In his thesis E. Specker [12] p. 329 proved the following: The conjecture that the knot space S K is aspherical is equivalent to the conjecture that the knot group 7ri(S K) has one or two ends. So naturally the question arises: When has iri(S K) one and when does it have two ends? It seems to be reasonable that the answer should be: 7r,(S K) has two ends if and only if K is algebraically unknotted, i.e., 7r1(S K) is free cyclic. The present paper developed from an attempt to prove asphericity for knots and to clarify when a knot group has one or two ends. Unfortunately a solution of the problem of asphericity of knots escaped us. But on the other hand the basic character for our work of this problem is apparent, for in order to clarify when a knot group has one or two ends it is necessary to assume asphericity. The main result of this paper is: If asphericity of knots holds, then a knot group has one or two ends according as the knot is algebraically knotted or unknotted. This result is an easy consequence of Theorems 1 and 2, and is proved in ?4, where also the different notions of knottedness are explained. As for Theorem 2, which is explained in ?3, we have to notice that its proof is based on two theorems of J. H. C. Whitehead contained in his papers [15] and [16]. However, in proving Theorem 1, which is also explained in ?3, a preparatory work was needed. The proof is based on Corollary 2, explained in ?2, and the main lemma, explained in ?3. Both are based on Lemmas 1 and 2, and Corollary 1, explained in ?2. The author of the present article would like to express his gratitude to the Mathematical Department of Princeton University, which, by giving him the privileges of a Visiting Fellow, enabled him to use the facilities of the Fine Hall Library and to attend the lectures of the Department, especially those of Professors R. H. Fox and N. E. Steenrod. The author is especially indebted to Professor R. H. Fox whose experience on 3-dimensional space was most valuable. In the ??2 and 3 of the present paper some simplifications, suggested by Professor R. H. Fox, have been incorporated.