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?
Best Answer
How about this:
Cochran, Levine and others gave various good answers - see references in the linked paper. Thinking of what the multi-variable Conway polynomial is measuring seems to give a much clearer picture, at least in the 2-variable case. What is less clear is why it does so - I mean that there should be more conceptual proofs.