Knot as a complete invariant of a Morse-Smale 3-diffeomorphism with four fixed points

Pochinka Olga, Talanova Elena, Shubin Danila

Published 2022-09-11Version 1

Lens spaces are the only 3-manifolds that admit gradient-like flows with four fixed points. This is an immediate corollary of Morse inequality and of the Morse function with four critical points existence. A similar question for gradient-like diffeomorphisms is open. Solution can be approached by describing a complete topological conjugacy invariant of the class of considered diffeomorphisms and constructing of representative diffeomorphism for every conjugacy class by the abstract invariant. Ch. Bonnati and V. Z. Grines proved that the topological conjugacy class of Morse-Smale flows with unique saddle is defined by the equivalence class of the Hopf knot in $\mathbb S^2\times\mathbb S^1$ which is projection of one-dimensional saddle separatrice and used the mentioned approach to prove that the ambient manifold of a diffeomorphism of this class is the three-dimensional sphere. In the present paper similar result is obtained for the gradient-like diffeomorphisms with exactly two saddle points and the unique heteroclinic curve.