n - QUASI - ISOTOPY : II . COMPARISON

SERGEY A. MELIKHOV and DUŠAN REPOVŠ ABSTRACT We prove that k-quasi-isotopy implies (k + 1)-cobordism of Cochran–Orr, leading to k-quasi-isotopy invariance of Cochran's derived invariants β i , i ≤ k, and Milnor's ¯ µ-invariants of length ≤ 2k + 3. Secondly, k-quasi-isotopic links cannot be distinguished by any Vassiliev invariant of type ≤ k which is well-defined up to PL isotopy, where type ≤ k invariants are to be understood either in the regular sense or, more generally, in the sense of Kirk–Livingston (1997). In particular, any linear combination of the coefficients of the Conway polynomial of an m-component link at the powers ≤ k + m − 1, invariant under PL isotopy, is invariant under k-quasi-isotopy. We also notice that the strong version of k-quasi-isotopy, if extended for k an infinite ordinal number, coincides with PL isotopy, and observe a relation between weak 1-quasi-isotopy of semi-contractible links and link homotopy of their Jin suspensions.