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A free surface-link is a surface-link whose fundamental group is a free group not necessarily meridian-based. Free ribbon lemma says that every free sphere-link in the 4-sphere is a ribbon sphere-link. Four different proofs of Free ribbon lemma are explained. The first proof is done in an earlier paper. The second proof is done by showing that there is an O2-handle basis of a ribbon surface-link. The third proof is done by removing the commuter relations from a Wirtinger presentation of a free group, which a paper on another proof of Free ribbon lemma complements. The fourth proof is given by the special case of the proof of the result that every free surface-link is a ribbon surface-link which is a stabilization of a free ribbon sphere-link. As a consequence, it is shown that a surface-link is a sublink of a free surface-link if and only if it is a stabilization of a ribbon sphere-link. |
