n-QUASI-ISOTOPY: III. ENGEL CONDITIONS

In part I it was shown that for each k ≥ 1 the generalized Sato–Levine invariant detects a gap between k-quasi-isotopy of link and peripheral structure preserving isomorphism of the finest quotient Gk of its fundamental group, ‘functorially’ invariant under k-quasi-isotopy. Here we show that Cochran’s derived invariant β, provided k ≥ 3, and a series of μ̄-invariants, starting with μ̄(111112122) for k = 3, also fall in this gap. In fact, all μ̄-invariants where each index occurs at most k + 1 times, except perhaps for one occuring k+2 times, can be extracted from Gk, and if they vanish, Gk is the same as that of the unlink. We also study the equivalence relation on links (called ‘fine k-quasi-isotopy’) generated by ambient isotopy and the operation of interior connected sum with the second component of the (k + 1) Milnor’s link, where the complement to its first component is embedded into the link complement. We show that the finest quotient of the fundamental group, functorially invariant under fine k-quasi-isotopy is obtained from the fundamental group by forcing all meridians to be (k + 2)-Engel elements. We prove that any group generated by two 3-Engel elements has lower central series of length ≤ 5.