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.
