Classification of string links up to $2n$-moves and link-homotopy

Abstract

Two string links are equivalent up to 2n-moves and link-homotopy if and only if their all Milnor link-homotopy invariants are congruent modulo n. Moreover, the set of the equivalence classes forms a finite group generated by elements of order n. The classification induces that if two string links are equivalent up to 2n-moves for every n > 0, then they are link-homotopic.

Milnor invariants, $2n$-moves and $V^{n}$-moves for welded string links

“…By arguments similar to those in the proof of Theorem 3.1, l ′ i is obtained from l i by replacing a kl with a ′ kl for all k, l and inserting the pth powers of elements in the free group A ′ . The following claim, which was proved in [22], completes the proof. (2) For any 1 ≤ i ≤ m and 1 ≤ j ≤ r(i), we have…”

“…m r [12,Section 3]. We remark that the quotient SL(m)/(2n+lh) of SL(m) under (2n+lh)equivalence forms a finite group generated by elements of order n, and that the order of the group is n sm [22,Corollary 1.2].…”

“…In particular, the connections between 2n-moves and classical link invariants are increasingly well-understood; see for example [16,23,8,9]. Recently, the authors [22] established the relation between Milnor link-homotopy invariants and 2n-moves as follows. Here, the (2n+lh)-equivalence is the equivalence relation generated by the 2n-move, self-crossing change and ambient isotopy.…”