Ribbon-clasp surface-links and normal forms of singular surface-links

TL;DR

This paper introduces ribbon-clasp surface-links, generalizes normal forms for immersed surface-links, and establishes their characterizations via symmetric normal forms and ribbon-clasp normal forms.

Contribution

It defines ribbon-clasp surface-links, extends normal form theory to immersed links, and characterizes these links using symmetric and ribbon-clasp normal forms.

Findings

Any immersed surface-link can be described in a normal form.

Embedded surface-links are ribbon if and only if they have a symmetric normal form.

Ribbon-clasp surface-links are characterized by symmetric normal forms.

Abstract

We introduce the notion of a ribbon-clasp surface-link, which is a generalization of a ribbon surface-link. We generalize the notion of a normal form on embedded surface-links to the case of immersed surface-links and prove that any (immersed) surface-link can be described in a normal form. It is known that an embedded surface-link is a ribbon surface-link if and only if it can be described in a symmetric normal form. We prove that an (immersed) surface-link is a ribbon-clasp surface-link if and only if it can be described in a symmetric normal form. We also introduce the notion of a ribbon-clasp normal form, which is a simpler version of a symmetric normal form.