Vol. 28, No. 3, 1969

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Free curves in E3

David Saul Gillman

Vol. 28 (1969), No. 3, 533–542

A 2-sphere in Euclidean 3-dimensional space E3 is called free if it can be pushed into either complementary domain by a map moving no point more than 𝜖, for arbitrary 𝜖. Such 2-spheres have been the object of much recent attention, although the basic problem of whether they must be tame or not remains unsolved. The purpose of this paper is to take a different direction in this study. We introduce a natural generalization of the term free so that it can be used to describe a k-sphere in En, then direct our attention to free 1-spheres and 2-spheres in E3.

Our main tool is Theorem 1, which, roughly speaking, should be viewed as follows: It is well known that if D and E are both polyhedral disks in E3 intersecting only in their interiors (in general position), then E may be altered via a disk replacement process to miss D. Theorem 1 states that even if D were a singular disk in E3, this process would remain valid to an extent.

Mathematical Subject Classification
Primary: 54.78
Received: 3 November 1967
Revised: 30 April 1968
Published: 1 March 1969
David Saul Gillman