Independence of Roseman moves including triple points

TL;DR

This paper investigates the independence of Roseman moves involving triple points in surface-link diagrams, demonstrating that certain moves are essential and cannot be derived from others, with implications for understanding surface-link isotopies.

Contribution

It constructs diagrams of surface-links where moves involving triple points are necessary, proving their independence within Roseman move sets.

Findings

Certain Roseman moves involving triple points are proven to be independent.

Diagrams are constructed requiring moves with triple points for isotopy.

At least one tetrahedral move is necessary for some surface-link diagrams.

Abstract

Roseman moves are seven types of local modification for surface-link diagrams in $3$-space which generate ambient isotopies of surface-links in $4$-space. In this paper, we focus on Roseman moves involving triple points, one of which is the famous tetrahedral move, and discuss their independence. For each diagram of any surface-link, we construct a new diagram of the same surface-link such that any sequence of Roseman moves between them must contain moves involving triple points (and the numbers of triple points of the two diagrams are the same). Moreover, we can find a pair of two diagrams of an $S^2$-knot such that any sequence of Roseman moves between them must involve at least one tetrahedral move.