# Yoshikawa moves on marked graphs via Roseman's theorem

## TL;DR

This paper provides an alternative proof that Yoshikawa moves generate isotopy for 2-link surface links in four-dimensional space, utilizing Roseman's theorem and branch-free broken surface diagrams.

## Contribution

It introduces a new proof for the Yoshikawa move conjecture specifically for 2-links, connecting Roseman's theorem with marked graph diagrams.

## Key findings

- Yoshikawa moves generate isotopy for 2-links.
- A new proof using Roseman's theorem is established.
- Construction of marked graphs from branch-free broken surface diagrams is demonstrated.

## 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.