**Background story**:

I have just come out from a talk by Misha Polyak on generalizations of an arrow-diagram formula for coefficients of the Conway polynomial by Chmutov, Khoury, and Rossi. In it, he described how to obtain formulae for a Conway polynomial of string links, string links in open-closed surfaces etc. These formulae do **not** give generalizations of the Alexander polynomial. This is funny, because I always thought that the Alexander polynomial and the Conway polynomial were basically the same, but that turns out not to be the case at all, philosophically.

**Background**:

The Alexander polynomial is a topological invariant for knots. It is a palindromic polynomial in t and t^{-1}, which can be viewed as representing deck transformations of the infinite cyclic covering of the complement. Knot Floer Homology categorifies it. There is a multivariable version for links.

The Conway polynomial of a knot is obtained from the Alexander polynomial by a change of variables (see the wikipedia page for details). It's an honest polynomial, satisfying a particularly satisfying skein relation. There is no analogue known for links, although Misha mentioned some recent thesis which gives partial results in this direction. It's categorification seems to be unknown, and Misha suggested that a solution to this problem would lead to a categorification of the linking number.

**My question**:

How are these two knot invariants different, beyond what I said above? Why is one "better" than the other? What is the Conway polynomial supposed to be measuring?

requirethat the Conway polynomial be defined for links, because changing a crossing to two parallel strands changes the number of components of the link. $\endgroup$ – Theo Johnson-Freyd Jul 8 '10 at 2:06