Some generalizations of Satoh's Tube map

TL;DR

This paper extends Satoh's map from virtual knots to virtual m-links, enabling the construction of higher-dimensional welded knots and new geometric representations of virtual links, including 1-links.

Contribution

It generalizes Satoh's map to virtual m-links and introduces higher-dimensional welded knots and novel geometric models for virtual links.

Findings

Generalized Satoh's map to virtual m-links

Constructed higher-dimensional welded knots

Developed new geometric representations of virtual links

Abstract

Satoh has defined a map from virtual knots to ribbon surfaces embedded in $S^4$. Herein, we generalize this map to virtual $m$-links, and use this to construct generalizations of welded and extended welded knots to higher dimensions. This also allows us to construct two new geometric pictures of virtual $m$-links, including 1-links.