ar X iv : m at h / 03 12 00 7 v 1 [ m at h . G T ] 2 9 N ov 2 00 3 COLORED FINITE TYPE INVARIANTS AND WILD LINKS

We show that no difference between PL isotopy and TOP isotopy (as equivalence relations on PL links in S 3) can be detected by finite type invariants. Next, let L be a link with components K 1 ,. .. , Km, and let Q L denote either the Conway polynomial, or a certain exponential parametrization of the HOMFLY or the Kauffman two-variable polynomial, or any of certain 2 m−1 polynomials, in totality equivalent to the multi-variable Alexander polynomial. Then each coefficient of the power series Q * L := Q L /(Q K 1 · · · Q Km) is invariant under sufficiently close C 0-approximation, and has a unique extension, preserving this property, to all wild links. These are corollaries of the fact that any type k invariant, well-defined up to PL isotopy, assumes same values on k-quasi-isotopic links; more generally, this holds for any Kirk–Livingston type k invariant. Kirk–Livingston finite type invariants can be regarded as finite type invariants of colored links, and include the coefficients of the 2 m−1 polynomials mentioned above. These coefficients C I are generally not of finite type in the usual sense, however, the coefficients of a certain parametrization of the multi-variable Alexander polynomial are finite type invariants, expressible as linear combinations of the C I 's. In the case where Q L is the Conway polynomial, the power series Q * L is of the form z m−1 (α 0 + α 1 z 2 + α 2 z 4 +. . .), and we show that the integer α k is invariant under k-quasi-isotopy, but for k = m = 2 has type > k in either theory.