TY - JOUR

T1 - Links and Dynamics

AU - Bardakov, V. g.

AU - Kozlovskaya, T. a.

AU - Pochinka, O. v.

N1 - Bardakov, V. G. Links and Dynamics / V. G. Bardakov, T. A. Kozlovskaya, O. V. Pochinka // Russian Journal of Nonlinear Dynamics. – 2025. – Vol. 21, No. 1. – P. 69-83. – DOI 10.20537/nd241004. The topological part of this work was supported by the Ministry of Science and Higher Education of Russia (agreement No. 075-02-2024-1437). The dynamical part was prepared within the framework of the project “International academic cooperation” HSE University.

PY - 2025/1/1

Y1 - 2025/1/1

N2 - 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 ⩾ 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 S2×S1. It is proved that any essential link can be realized by a diffeomorphism of the class under consideration

AB - 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 ⩾ 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 S2×S1. It is proved that any essential link can be realized by a diffeomorphism of the class under consideration

UR - https://www.scopus.com/record/display.uri?eid=2-s2.0-105002439886&origin=inward&txGid=49fb78b8054a9499b0963071a12628de

UR - https://elibrary.ru/item.asp?id=81021680

U2 - 10.20537/nd241004

DO - 10.20537/nd241004

M3 - Article

VL - 21

SP - 69

EP - 83

JO - Nelineinaya Dinamika

JF - Nelineinaya Dinamika

SN - 2658-5324

IS - 1

ER -