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JOURNALS // Russian Journal of Nonlinear Dynamics // Archive

Rus. J. Nonlin. Dyn., 2025 Volume 21, Number 1, Pages 69–83 (Mi nd940)

Mathematical problems of nonlinearity

Links and Dynamics

V. G. Bardakovabcd, T. A. Kozlovskayad, O. V. Pochinkae

a Sobolev Institute of Mathematics, pr. Acad. Koptyuga 4, Novosibirsk, 630090 Russia
b Novosibirsk State Agrarian University, ul. Dobrolyubova 160, Novosibirsk, 630039 Russia
c Novosibirsk State University, ul. Pirogova 2, Novosibirsk, 630090 Russia.
d Regional Scientific and Educational Mathematical Center of Tomsk State University, pr. Lenina 36, Tomsk, 634050 Russia
e National Research University – Higher School of Economics, ul. Bolshaya Pecherskaya 25/12, Nizhny Novgorod, 603155 Russia

Abstract: 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 $\mathbb{S}^2\times \mathbb{S}^1$, 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 $\mathbb{S}^2\times \mathbb{S}^1$.
We prove that, if $M$ is a link complement in $\mathbb{S}^3$, or a handlebody $H_g^{}$ of genus $g\geqslant 0$, 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 $\mathbb{S}^2\times \mathbb{S}^1$. It is proved that any essential link can be realized by a diffeomorphism of the class under consideration.

Keywords: knot, link, equivalence class of links, braid, mixed braid, fundamental quandle, handlebody, $3$-manifold

MSC: 37D15, 37C05, 57M25

Received: 16.05.2024
Accepted: 07.08.2024

Language: English

DOI: 10.20537/nd241004



© Steklov Math. Inst. of RAS, 2025