Colored Finite Type Invariants and a Multi-variable Analogue of the Conway Polynomial

The multi-variable Alexander polynomial (in the form of Conway’s potential function), when stripped of a redundant summand, is shown to be of the form ▽L(x1−x −1 1 , . . . , xm−x −1 m ) for some polynomial ▽L over Z; the Conway polynomial ∇L(z) coincides with z▽L(z, . . . , z). The coefficients of ▽L and of the power series ▽ L := ▽L/(▽K1 · · · ▽Km), where Ki denote the components of L, are finite type invariants in the sense of Kirk–Livingston. When m = 2, they are integral liftings of Milnor’s invariants μ̄(1 . . . 12 . . . 2) of even length, including, in the case of ▽ L , Cochran’s derived invariants β. The coefficients of ▽L and ▽ ∗ L are closely related to certain Q-valued invariants of (genuine) finite type, among which we find alternative extensions β̂ of β to the case lk 6= 0, such that 2β̂/ lk is the Casson–Walker invariant of the Q-homology sphere obtained by 0-surgery on the link components. Each coefficient of ▽ L (hence of ∇ ∗ L := ∇L/(∇K1 · · ·∇Km )) is invariant under TOP isotopy and under sufficiently close C-approximation, and can be extended, preserving these properties, to all topological links. The same holds for H L and F ∗ L , where HL and FL are certain exponential parameterizations of the two-variable HOMFLY and Kauffman polynomials. Next, we show that no difference between PL isotopy and TOP isotopy (as equivalence relations on PL links in S) can be detected by finite type invariants. These are corollaries of the fact that any type k invariant (genuine or Kirk–Livingston), well-defined up to PL isotopy, assumes same values on k-quasi-isotopic links. We prove that ck, where ∇ ∗ L (z) = z(c0+c1z+c2z+. . . ), is invariant under k-quasi-isotopy, but c2 fails to have type 2 in either theory.