Partial conjugations suffice

In their 1990 classification of links up to link homotopy, Habegger and Lin prove a Markov-type theorem saying that two string links have link homotopic closures if and only if they are related by a sequence of conjugations and partial conjugations. In this paper we show that one can dispense with conjugations, in the sense that any change produced by conjugation can also be produced by an appropriately-chosen sequence of partial conjugations. We also give an interpretation of this result in terms of the action of 2n-component string links on n-component string links used by Habegger and Lin to prove their theorem. This interpretation suggests a canonical form for sequences of partial conjugations. These results simplify the Habegger–Lin classification scheme, as well as the still-open problem of classifying links up to link homotopy by means of a complete set of invariants.