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.
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?
  • $\begingroup$ Isn't the Conway polynomial the one which comes more directly from quantum groups? $\endgroup$ Jul 7 '10 at 23:46
  • $\begingroup$ Qiaochu -- in a sense you are right: the coefficients of the Conway polynomial are finite type invariants on the nose; the coefficients of the Alexander polynomial are functions of finite type invariants but are not of finite type themselves. $\endgroup$
    – algori
    Jul 8 '10 at 0:04
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    $\begingroup$ Daniel -- could you explain what you mean by "there is no analogue known for links"? After a change of variable in HOMFLY one gets a polynomial invariant of oriented links that satisfies "positive crossing minus negative crossing equals $t$ times no crossing" $\endgroup$
    – algori
    Jul 8 '10 at 0:37
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    $\begingroup$ I also don't understand the "no analogue known for links" comment. The skein relations require that the Conway polynomial be defined for links, because changing a crossing to two parallel strands changes the number of components of the link. $\endgroup$ Jul 8 '10 at 2:06
  • 1
    $\begingroup$ The "Conway polynomial" for links is a single variable polynomial (of course), and is thus not an analogue to the multivariable Alexander polynomial. You can't recover the multivariable Alexander from the "Conway", certainly not by a change of variables. That might be too much to expect, but a proper (i.e. non-stupid) generalization of the Conway polynomial to some multi-variable polynomial for links seems to be something people are after. I wish I understood this better. There's some sentence that should be in this comment about the relationship to the HOMFLYPT. $\endgroup$ Jul 8 '10 at 16:11

a proper (i.e. non-stupid) generalization of the Conway polynomial to some multi-variable polynomial for links seems to be something people are after

How about this: http://front.math.ucdavis.edu/0312.5007

What is the Conway polynomial supposed to be measuring?

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.

  • $\begingroup$ Thanks- I'll look at this. Was this published anywhere? $\endgroup$ Nov 14 '10 at 19:29
  • $\begingroup$ I have not (yet) submitted it to any journal because I was planning to add more things to it, but then got distracted to other projects. I'm still determined to return to the area some day (if I live up to that day, of course). What I thought was badly missing from the paper was 1) more conceptual proofs instead of miraculous substitutions, 2) some progress on geometry of the coefficients (when $lk\ne 0$ and for 3 components) and 3) on algebra (relating to a shifted Magnus expansion and to Reidemeister torsion), 4) certain applications. In fact I did something related to 2, 4 in the meantime. $\endgroup$ Nov 15 '10 at 20:21

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