Independence of Roseman moves including triple points

Abstract

The Roseman moves are seven types of local modifications 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 number of triple points of the two diagrams are the same). Moreover, we find a pair of diagrams of an $S^2$–knot such that any sequence of Roseman moves between them must involve at least one tetrahedral move.