PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 28, No. 1, April 1971 A SUFFICIENT CONDITION FOR AN ARC TO BE NEARLY POLYHEDRAL JAMES M. MCPHERSON Abstract. We give a simple geometric condition for an arc to be wild at an endpoint, with penetration index at least three. 0. Proving that a given arc is "nearly polyhedral" (see [l]) is a very difficult task and, to the author's knowledge, has only been done twice in the published literature [l], [2]. The method used by Fox and Artin in [2] was algebraic, while the method used by Alford and Ball was purely geometric; no general set of sufficient conditions for an arc to be nearly polyhedral seems to be known. We shall prove that an arc is nearly polyhedral if certain rather mild geometric conditions are satisfied. Since the arcs An of [l] and Example 1.2 of [2] satisfy these conditions, their wildness follows immediately. This result is a generalization of a result in the author's 1970 doctoral thesis at The University of New South Wales, Australia. The author wishes to thank Professor N. F. Smythe for drawing his attention to a serious error in the original draft of this paper, and for suggesting the necessary correction. 1. Let X be a set. We use Bd X, Cl X, and Int X to denote the boundary, closure, and interior respectively of X. N(X) is the number of points of X. If k is an oriented arc in an oriented 3-manifold, and X is an oriented surface, v(kC\X) is the algebraic intersection number of k with X (Schnittzahl— [4, §§69, 70]). Let L =li\Jl2 be a link of two components in Euclidean 3-space E3. Then L is splitlable if there exists a 2-sphere SEE3 such that h and k lie in different components of E3 —S; otherwise L is unsplittable. For L to be unsplittable, it is sufficient that the linking number X(Zi, l2) of L be nonzero [4, p. 278]. Let L=l[[Ul'2 be another link in E3. Then L' arises from L by a simple F-isotopy on the ith component if L — U =L' —l'it and there is a solid torus VEE3 — (L — /,) such that either (i) /,- is a core of V, and l\ has winding number one in V, or (ii) l[ is a core of V, and /, has winding number one in V. Received by the editors April 29, 1970. AMS 1969 subject classifications. Primary 5520, 5705; Secondary 0550. Key words and phrases. Wild arc, nearly polyhedral arc, nice penetration index, "cutting and pasting". Copyright © 1971, American Mathematical Society 229