Certification of linkedness by curves on handles

I. S. Alekseev, A. A. Malova

Saint Petersburg State University

Abstract: This work develops geometric certificates for detecting non-splittability of links in the three-sphere. We establish sufficient conditions for the non-splittability of two-component links in which one component bounds a handle (a punctured torus) or a cross-handle (a punctured Klein bottle) in the complement of the other. These conditions are expressed in terms of the linking behavior of curves on these surfaces with the other component. This lays the foundation for a geometric framework to certify non-splittability, an approach conceptually akin to Cochran's calculus and Milnor invariants.

Key words and phrases: knot theory, non-splittability, geometric certificate, Seifert surface, punctured torus, handle, punctured Klein bottle, non-orientable handle, cross-handle.

UDC: 515.162.8

Received: 29.12.2025