Satellites of an oriented surface link and their local moves

TL;DR

This paper introduces a new way to represent satellite surface links in four-dimensional space using $m$-charts on surface diagrams, and shows how to define Roseman moves for these charts to understand their equivalences.

Contribution

It develops the concept of $m$-charts on surface diagrams for satellite surface links and establishes the framework for Roseman moves in this context.

Findings

$m$-charts effectively represent 2-dimensional braids over surface links.

Roseman moves can be extended to diagrams with $m$-charts.

The framework aids in understanding surface link equivalences.

Abstract

For an oriented surface link $F$ in $\mathbb{R}^4$, we consider a satellite construction of a surface link, called a 2-dimensional braid over $F$, which is in the form of a covering over $F$. We introduce the notion of an $m$-chart on a surface diagram $\pi(F)\subset \mathbb{R}^3$ of $F$, which is a finite graph on $\pi(F)$ satisfying certain conditions and is an extended notion of an $m$-chart on a 2-disk presenting a surface braid. A 2-dimensional braid over $F$ is presented by an $m$-chart on $\pi(F)$. It is known that two surface links are equivalent if and only if their surface diagrams are related by a finite sequence of ambient isotopies of $\mathbb{R}^3$ and local moves called Roseman moves. We show that Roseman moves for surface diagrams with $m$-charts can be well-defined.