Symmetry Groups of Knots

Abstract

The theory of knots, like many other branches of mathematics, had its origin in a physical situation in the real world, namely, the consideration of tangled loops of string. However, due to the tendency towards increasing abstraction, the theory has become more and more divorced from physical reality-and almost unintelligible to anyone working in a different subject, or even in another branch of mathematics. The purpose of this note is to show that interesting, and not entirely trivial, problems exist at the grass-roots level. We shall be considering a strictly geometrical problem in three-dimensional space, a problem which arose from perusal of the knots in the well-known classic of practical knotting, the Ashley Book of Knots [1]. Some of the illustrations (FIGURE 1) suggest that the symmetry groups of knots can be of many kinds, but caution is necessary. For example, the knot in the right half of FIGURE 1 looks, at first sight, as if its symmetry group is [3+, 4] or [39 3]+ (see below for the identification of these groups). As we shall show shortly, this is not so, and in fact the groups [3+,4] and [3,3]+ are not symmetry groups of any knot. The question naturally arises as to which groups can occur in this context; the purpose of our note is to give a complete answer to this problem. It must be emphasized that here we are considering symmetry groups, that is, groups of isometries (distance-preserving transformations) of E3 that map the knot onto itself, and not the various topological groups associated with knots that have been studied extensively over many years. (For a survey of these, and bibliography, see [3]. Since this paper was written, the interesting book [5] has appeared, which contains many illustrations of symmetric knots and links.) To begin with, first we must define what we mean by a knot. From the point of view of this note the reader will not be led astray if he thinks of it as a piece of string that has been tangled and then its ends joined to form a continuous loop. Those who wish to be more formal should regard it as a simple embedding of a closed loop into E3 with some local finiteness and smoothness properties. Various such properties have been proposed, any of which is appropriate in this case: we may insist that the embedding is PL (piecewise linear), or that some two-dimensional projection of the knot has only a finite number of crossings, or that there exists a 8 > 0 such that the intersection of every solid 8-sphere with the loop is either empty or a connected set. By a plane knot we mean any knot which lies entirely in a plane; such knots are, of course, trivial since they are not knotted, in the usual sense of the word.