A Geometric Filtration of Links modulo Knots: I. Questions of Nilpotence

For each k = 0, 1, 2, . . . we define an equivalence relation called k-quasi-isotopy on the set of classical links in R3 up to isotopy in the sense of Milnor (1957), such that all sufficiently close approximations of a topological link are k-quasi-isotopic. Whereas 0-quasi-isotopy coincides with link homotopy, 1-quasi-isotopy is not implied by concordance, with aid of the generalized (lk 6= 0) Sato–Levine invariant. Thus kquasi-isotopy is not completely described by the lower central series quotients of the fundamental group. More intriguingly, it is almost certainly not determined by the derived series quotients, since the natural quotient with respect to k-quasi-isotopy (for k = 0 isomorphic to Milnor’s link group) is an Engel-type group whose derived subgroups are interposed between the lower central subgroups. A special case of the Isotopic Realization Problem motivates the question, whether this quotient is always nilpotent; we could only show that all of its finite epimorphic images are nilpotent. Nevertheless, we prove k-quasi-isotopy invariance of Milnor’s μ̄-invariants of length ≤ 2k + 3 (leading to invariance of certain coefficients of Conway’s polynomial) and sharpness of this restriction. We also discuss relations with n-splitting of Smythe and the effect of Whitehead doubling on our filtration.