5 n - QUASI - ISOTOPY : II . COMPARISON

Geometric aspects of the filtration on classical links by k-quasi-isotopy are discussed, including the effect of Whitehead doubling and relations with Smythe's n-splitting and Kobayashi's k-contractibility. One observation is: ω-quasi-isotopy is equivalent to PL isotopy for links in a homotopy 3-sphere (resp. contractible open 3-manifold) M if and only if M is homeomorphic to S 3 (resp. R 3). As a byproduct of the proof of the " if " part, we obtain that every compact subset of an acyclic open set in a compact orientable 3-manifold M is contained in a PL homology 3-ball in M. We show that k-quasi-isotopy implies (k + 1)-cobordism of Cochran and Orr. If z m−1 (c 0 + c 1 z 2 + · · · + c n z 2n) denotes the Conway polynomial of an m-component link, it follows that the residue class of c k modulo gcd(c 0 ,. .. , c k−1) is invariant under k-quasi-isotopy. Another corollary is that each Cochran's derived invariant β k is also invariant under k-quasi-isotopy, and therefore assumes the same value on all PL links, sufficiently C 0-close to a given topological link. This overcomes an algebraic obstacle encountered by Kojima and Yamasaki, who " became aware of impossibility to define " for wild links what for PL links is equivalent to the formal power series β n z n by a change of variable.