#### Volume 16, issue 4 (2016)

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 The Journal About the Journal Subscriptions Editorial Board Editorial Interests Editorial Procedure Submission Guidelines Submission Page Author Index To Appear ISSN (electronic): 1472-2739 ISSN (print): 1472-2747
Independence of Roseman moves including triple points

### Kengo Kawamura, Kanako Oshiro and Kokoro Tanaka

Algebraic & Geometric Topology 16 (2016) 2443–2458
##### 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.

##### Keywords
surface-link, diagram, Roseman move, $S$–dependence
Primary: 57Q45
Secondary: 57R45