Yoshikawa moves on marked graphs via Roseman's theorem

Abstract

Yoshikawa [Yo] conjectured that a certain set of moves on marked graph diagrams generates the isotopy relation for surface links in ${\mathbb R}^4$, and this was proved by Swenton [S] and Kearton and Kurlin [KK]. In this paper, we find another proof of this fact for the case of 2-links (surface links with spherical components). The proof involves a version of Roseman's theorem [R] for branch-free broken surface diagrams of 2-links and a construction of marked graphs from branch-free broken surface diagrams.