n-QUASI-ISOTOPY II: COMPARISON
Abstract
Geometric aspects of the filtration on classical links by k-quasi-isotopy are discussed, including the effect of Whitehead doubling, 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 (respectively, contractible open 3-manifold) M if and only if M is homeomorphic to S3 (respectively, ℝ3). 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 zm-1 (c0+c1z2+⋯+cnz2n) denotes the Conway polynomial of an m-component link, it follows that the residue class of ck modulo gcd(c0,…,ck-1) is invariant under k-quasi-isotopy. Another corollary is that each Cochran's derived invariant βk is also invariant under k-quasi-isotopy, and therefore assumes the same value on all PL links, sufficiently C0-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 ∑βnzn by a change of variable.
