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Independence of Roseman moves including triple points

Kengo Kawamura, Kanako Oshiro and Kokoro Tanaka

Algebraic & Geometric Topology 16 (2016) 2443–2458

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 S2–knot such that any sequence of Roseman moves between them must involve at least one tetrahedral move.

surface-link, diagram, Roseman move, $S$–dependence
Mathematical Subject Classification 2010
Primary: 57Q45
Secondary: 57R45
Received: 17 November 2015
Revised: 28 December 2015
Accepted: 9 January 2016
Published: 12 September 2016
Kengo Kawamura
Department of Mathematics
Osaka City University
3-3-138 Sugimoto-cho
Osaka 558-8585
Kanako Oshiro
Department of Information and Communication Sciences
Sophia University
7-1 Kioi-cho
Tokyo 102-8554
Kokoro Tanaka
Department of Mathematics
Tokyo Gakugei University
4-1-1 Nukuikita-machi
Tokyo 184-8501