The Alexander's Theorem states that every link can be represented as a closed braid. How about if only the pure braid are allowed to be used in the representation? Obviously the closure of every pure $n-$braid is an $n-$link. But the question is:
What $n$-links can be represented as closed pure braids?
One answer would be "if and only if the $n$-link has braid index $n$". But how can one characterise $n-$links with braid index $x$?