I’m looking to extend my program of research to include the A-polynomial. I’ll spare the background of what I want to do with it unless asked off the boards.
Basically, I’ve tracked down the ur-paper of Cooper, Culler, Gillet, Long, and Shalen in Inventiones Mathematicæ, but it’s pretty dense, and my understanding of SL(2,C) representation varieties and character varieties is pretty thin. Also, I know that the A-polynomial has been generalized to multi-component links, but trackbacks haven’t really caught on in journal articles, much less ones published in 1994.
So, if you grok the A-polynomial and its generalizations, or at least know of more lucid references, help a math brother out.
Found these via Google Geek
D. Cooper, M. Culler, H. Gillet, D. D. Long and P. B. Shalen. Plane curves associated to character varieties of 3-manifolds (1994). Inventiones Mathematicae. Publisher: Springer Berlin / Heidelberg. ISSN: 0020-9910 (Paper) 1432-1297 (Online)
DOI: 10.1007/BF01231526. Issue: Volume 118, Number 1. Date: December 1994
Pages: 47 - 84
Gukov, Sergei (2005) Three-Dimensional Quantum Gravity, Chern-Simons Theory, and the A-Polynomial. Communications in Mathematical Physics 255(3)
All roots of unity are detected by the A-polynomial Eric Chesebro
Comments: Published by Algebraic and Geometric Topology at this http URL
Algebr. Geom. Topol. 5 (2005) 207-217
“For an arbitrary positive integer n, we construct infinitely many one-cusped hyperbolic 3-manifolds where each manifold’s A-polynomial detects every n-th root of unity.”
On the relation between the A-polynomial and the Jones polynomial R Gelca - Proc. Amer. Math. Soc, 2002 - ams.org http://scholar.google.com/url?sa=U&q=https://www.ams.org/proc/2002-130-04/S0002-9939-01-06157-3/S0002-9939-01-06157-3.pdf
S-knots and A-Polynomials: http://scholar.google.com/scholar?hl=en&lr=&q=cache:Rjm-VO6BrmIJ:www.emis.ams.org/journals/UW/agt/ftp/main/2004/agt-4l50.pdf+Topological+Polynomials+Cooper,+Culler,+Gillet,+Long
From http://arxiv.org/abs/math.GT/0405353: Non-triviality of the A-polynomial for knots in S^3
Authors: Nathan M. Dunfield, Stavros Garoufalidis
Comments: Published by Algebraic and Geometric Topology at this http URL
Subj-class: Geometric Topology; Quantum Algebra
MSC-class: 57M25, 57M27, 57M50
Journal-ref: Algebr. Geom. Topol. 4 (2004) 1145-1153
“The A-polynomial of a knot in S^3 defines a complex plane curve associated to the set of representations of the fundamental group of the knot exterior into SL(2,C). Here, we show that a non-trivial knot in S^3 has a non-trivial A-polynomial. We deduce this from the gauge-theoretic work of Kronheimer and Mrowka on SU_2-representations of Dehn surgeries on knots in S^3. As a corollary, we show that if a conjecture connecting the colored Jones polynomials to the A-polynomial holds, then the colored Jones polynomials distinguish the unknot “
http://scholar.google.com/scholar?hl=en&lr=&q=Topological+Polynomials+Cooper%2C+Culler%2C+Gillet%2C+Long&btnG=Search
Oh, yaeh …does the “A” stand for “Alexander”? There’s a good bit of literature on both “Alexander” and “Jones” polynomials. There’s also a “C” polynomial out there.
No, you’ve got the right one, though you’d be surprised how many professional mathematicians personally who know I’m a knot theorist think I might be confusing “A” with “Alexander”.
The first one you list is the ur-paper I mentioned – that from which all others flow. The others are basically other properties, but none of them really seem to explain it any better. There’s a point at which you throw out part of the image of the diagonal reps in the character variety, and what’s left is a plane curve (I think), but I don’t really follow what part is thrown out, why it’s thrown out, and why what’s left is a plane curve. Also, none of them really seem to attack how this generalizes to multiple-component links.
Thanks anyhow. I think I’m going to have to tie down Yair Minsky and play “Stuck In the Middle With You” until he talks.