Satellites of an oriented surface link and their local moves

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.

16 pages, 12 figures. Theorem 6.4 is changed to Remark 6.4. Proofs of Theorem 5.5 and Lemma 8.3 are rewritten

References in corpus (1)

• Surface links which are coverings over the standard torus

Cited by in corpus (5)

• Simplifying branched covering surface-knots by an addition of 1-handles with chart loops
• On addition of 1-handles with chart loops to 2-dimensional braids
• Simplifying branched covering surface-knots by chart moves involving black vertices
• Branched covering surface-knots with degree three have the simplifying numbers less than three
• Showing distinctness of surface links by taking 2-dimensional braids