Links and Dynamics

Russian Journal of Nonlinear Dynamics. 2025. Vol. 21. No. 1. P. 69–83.

Bardakov V. G., Kozlovskaya T. A., O. V. Pochinka

Knots naturally appear in continuous dynamical systems as flow periodic trajectories. However,
discrete dynamical systems are also closely connected with the theory of knots and links.
For example, for Pixton diffeomorphisms, the equivalence class of the Hopf knot, which is the
orbit space of the unstable saddle separatrix in the manifold S2 × S1, is a complete invariant of
the topological conjugacy of the system. In this paper we distinguish a class of three-dimensional
Morse – Smale diffeomorphisms for which the complete invariant of topological conjugacy is the
equivalence class of a link in S2 × S1.
We prove that, if M is a link complement in S3, or a handlebody Hg of genus g, or
a closed, connected, orientable 3-manifold, then the set of equivalence classes of tame links in M
is countable. As a corollary, we prove that there exists a countable number of equivalence classes
of tame links in S2 ×S1. It is proved that any essential link can be realized by a diffeomorphism
of the class under consideration.

Keywords: зацепление, link

Publication based on the results of: Качественная теория систем обыкновенных уравнений и уравнений в частных производных (2026)