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On tame embeddings of solenoids into 3-space

Boju Jiang, Shicheng Wang, Hao Zheng, Qing Zhou

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.

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 which are strictly achiral. Since solenoids are non-planar, this contrasts sharply with the known fact that if there is a strictly achiral embedding of a compact polyhedron , then must be planar.

Paper Structure

This paper contains 15 sections, 16 theorems, 5 equations.

Table of Contents

  1. Introduction and motivations
  2. Tame embeddings of solenoids, Preliminaries
  3. Definitions of solenoids and their tame embeddings
  4. Convergence of homeomorphisms
  5. Tame solenoids as mapping tori
  6. Non-tame embeddings of solenoids
  7. Classification and applications
  8. Maximal defining sequences
  9. Classification of tame solenoids
  10. Knotting, linking and invariants
  11. Unknotted 2-adic tame solenoids
  12. Smale solenoids
  13. Chirality of tame solenoids
  14. Criterions
  15. Examples

Key Result

Theorem 2.3

(1) The above two definitions of solenoids are equivalent; each solenoid is determined by its type $\varpi$. (2) Two solenoids $\Sigma$ and $\Sigma'$ of types $\varpi$ and $\varpi'$ respectively are homeomorphic if deleting finitely many terms from $\varpi$ and $\varpi'$ can make them identical. Mor

Theorems & Definitions (49)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • Definition 2.7
  • Remark 2.8
  • Lemma 2.9
  • proof
  • ...and 39 more