# Why is the Alexander polynomial a quantum invariant?

When we think of quantum invariants, we usually think of the Jones polynomial or of the coloured HOMFLYPT. But (arguably) the simplest example of a quantum invariant of a knot or link is its Alexander polynomial. From the beginning, the central problem in the study of quantum invariants has been what do they mean topologically? The Alexander polynomial has clear algebraic topological meaning as the order of the Alexander module (first homology of the infinite cyclic cover as a module over the group of deck transformations). Can people conceptually explain (in terms of both physics and mathematics) why the representation theory of certain small quantum groups naturally gives rise to this quantity? Computationally I can understand it, but not conceptually.
Update: I posted on this question here and here. See also this question.

• This is a very good question! Dec 27 '09 at 2:07
• What does it mean for something to be a "quantum" knot invariant? Dec 27 '09 at 2:27
• I interpret this as saying that it arises from Chern-Simons theory on some suitable Lie (super)group. Hence the question, the way I interpret it, is whether one can reconcile the original topological definition of the Alexander-Conway invariant with the Chern-Simons point of view and, in particular, explain why one takes the group that one does. Dec 27 '09 at 8:06
• A quantum invariant is an operator invariant of a link which comes from a representation of a ribbon Hopf algebra such as a quantum group. You first slice a diagram of your link into basic tangles. Then you assign certain elements in the image of the representation to each basic tangle, and compose in a certain way. The most important of these is the R-matrix, which gets assigned to a crossing, An equivalent definition comes from Chern-Simons theory. The R-matrix for the Alexander polynomial comes classically from the Burau representation. Dec 27 '09 at 10:22