Fig. 1. The Hopf index in the cylindrical dots as a function of the dot radius for the fixed dot aspect ratio . The dot radius is in units of the hopfion radius . The hopfion vorticities : a) – solid green line, b) – dotted red line, and c) dashed blue line. Fig. 2. The Hopf index in the spherical particles as a function of the particle radius ( is in units of the hopfion radius ). The hopfion vorticities : a) – solid green line, b) – dotted red line, and c) dashed blue line.

Acknowledgements The authors are grateful to B.A. Ivanov and M. Krawczyk for fruitful discussions. The research of K.G. and E.T was supported by the National Science Centre of Poland, project no UMO-2023/49/B/ST3/02920