On tame embeddings of solenoids into 3-space

Volume 214 / 2011

Boju Jiang, Shicheng Wang, Hao Zheng, Qing Zhou Fundamenta Mathematicae 214 (2011), 57-75 MSC: Primary 57N10; Secondary 37E99, 54C25. DOI: 10.4064/fm214-1-4

Abstract

Solenoids are inverse limits of the circle, and the classical knot theory is the theory of tame embeddings of the circle into 3-space. We make a general study, including certain classification results, of tame embeddings of solenoids into 3-space, seen as the “inverse limits” of tame embeddings of the circle.

Some applications in topology and in dynamics are discussed. In particular, there are tamely embedded solenoids $\Sigma\subset \mathbb R^3$ which are strictly achiral. Since solenoids are non-planar, this contrasts sharply with the known fact that if there is a strictly achiral embedding $Y\subset \mathbb R^3$ of a compact polyhedron $Y$, then $Y$ must be planar.

Authors

  • Boju JiangDepartment of Mathematics
    Peking University
    Beijing 100871, China
    e-mail
  • Shicheng WangDepartment of Mathematics
    Peking University
    Beijing 100871, China
    e-mail
  • Hao ZhengDepartment of Mathematics
    Peking University
    Beijing 100871, China
    e-mail
  • Qing ZhouDepartment of Mathematics
    East China Normal University
    Shanghai 200030, China
    e-mail

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