On tame embeddings of solenoids into 3-space

Abstract

Solenoids are ``inverse limits'' of the circle, and the classical knot theory is the theory of tame embeddings of the circle into the 3-space. We give some general study, including certain classification results, of tame embeddings of solenoids into the 3-space as the ``inverse limits'' of the tame embeddings of the circle. Some applications are discussed. In particular, there are ``tamely'' embedded solenoids $Σ\subset \R^3$ which are strictly achiral. Since solenoids are non-planar, this contrasts sharply with the known fact that if there is a strictly achiral embedding $Y\subset \R^3$ of a compact polyhedron $Y$, then $Y$ must be planar.