Abstract
We give a direct proof that if two string links have isotopic closures, then there is a braid-special isomorphism between their n-level group diagrams, for every $n \geqslant 2$. In the case of link-homotopy, we give an alternative proof to our previous result that there is a braid-special isomorphism between the group diagrams for the homotopy classes of two string links if and only if they have link-homotopic closures.
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