A Geometric Filtration of Links modulo Knots: Ii. Comparison

We continue the study of the equivalence relations introduced in the first part. For finite k, it is shown that k-quasi-isotopy implies (k + 1)-cobordism of Cochran–Orr, leading to invariance of Cochran’s derived invariants β, i ≤ k. Furthermore, if two links are k-quasi-isotopic then they 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 usual sense or in the sense of Kirk–Livingston (1997). In particular the coefficients of the Conway polynomial of an m-component link at the powers ≤ k + m − 1 are invariant under k-quasi-isotopy. Next we show that the strong version of k-quasi-isotopy, if extended for k an infinite ordinal number, coincides with PL isotopy. We also observe a relation between weak 1-quasi-isotopy of semi-contractible links and link homotopy of their Jin suspensions.