A Remarkable Simple Closed Curve
Ralph H. Fox · Annals of Mathematics · 1949
Two simple closed curves in ordinary three dimensional space R are called equivalent' if there is an orientation-preserving homeomorphism of R on itself which transforms one curve into the other. A curve is called unknotted if it is equivalent to the circle X2 + y2 = 1, Z = 0, otherwise knotted. In a recent paper2 we have introduced the terms tame, for a curve equivalent to a polygon, and wild, for one not equivalent to a polygon 3. It is clear that an unknotted curve is tame and has an abelian (hence infinite cyclic) group. Whether conversely a curve which is tame and has an abelian group is necessarily unknotted is still an open question 4. On the other hand a wild (and hence knotted) curve whose group is abelian has been constructed 5. It is known that a simple closed curve in R is unknotted if and only if it is equivalent to a curve F which is a subset of a plane in R. Let us call a simple closed curve almost unknotted if it is equivalent to a curve which has the following property: (*) For any neighborhood U of a given p there is a neighborhood V C U of p and a homeomorphism so of R on itself such that (i) jp(q) = q for every q e V, and (ii) p(F V) is a subset of a plane in R. There exists a simple closed curve whose group is non-abelian although it is almost unknotted 6. A regular normed projection of this simple closed curve I is shown in figure 1. (Using the approach of F A one could easily construct a precise description). It has one singular point p; by applying the necessary condition developed in F A example 1.2 to a subarc of F of which p is the left endpoint it can be shown that F is wild 7. The group 8 of F is generated by elements an , bn X c, (n > 0) indicated in the usual way in figure 1. A set of defining relations is