The A-Polynomial


For more information about the A-poynomial we have posted a pdf version of a seminar presentation given by Marc Culler. The original source for the A-polynomial is the paper by Cooper, Culler, Gillet, Long, and P. Shalen, referenced below. We thank Abhijit Champanerkar for helping with the exposition on this page.

There is a map of the SL2(C) representation space of a knot complement to C* x C*, given by evaluating the trace of the representation on the meridian and longitude. The closure of the image is a variety defined by a single polynomial, called the A-Polynomial. Jim Hoste gave us information on 2-bridge knots and Marc Culler provided us with further tables, based on glueing equations. These have not been proved to equal the A-polynomial; the issue is described next.

The set of isometry classes of ideal hyperbolic tetrahedra is paramaterized by the upper half complex plane. Thus, if the complement of a knot is decomposed into tetrahedra, the set of glueings that yield hyperbolic structures on the knot complement is determined by the solutions to glueing equations. The set of glueing equations defines an algebraic variety that maps to the PSL2( C) character variety of the knot. To the image variety there is associated an "A-polynomial", which is the PSL2( C) version of the classical A-polynomial. In many cases the PSL2( C) A-polynomial can be computed directly from the glueing and completeness equations by eliminating the tetrahedral parameters to get a 2-variable polynomial. However, the resulting polynomial depends on the choice of the triangulation and in general only divides the PSL2( C ) A-polynomial. For an exposition of this alternative viewpoint of A-polynomials, see the appendix by N. Dunfield to Mahler's Measure and the Dilogarithm by Boyd, Rodrigues-Villegas, and Dunfield or "A-polynomial and Bloch invariants of hyperbolic 3-manifolds" by A. Champanerkar.

We have provided three tables of A-polynomials, all linked in Table of A-Polynomials: two-bridge knots. Jim Hoste has provided us with this table of values for 2-bridge knots of 9 crossings or less.

Table of A-Polynomials (Glueing equations approach). This data, based on glueing equaitons, was provided by Marc Culler.

Table of A-Polynomial: tetrahedral census (Glueing equations approach). This table, also provided by Marc Culler, lists the A-polynomials of knots in the tetrahedral enumeration. There is an overlap in the two tables. Warning: in the overlap, orientations changed for some knots, so one polynomial is related to the other by a change of variable (something like L -> L-1).


Warning: a change of orientation, from a knot to its mirror image, changes the A-polynomial. The data in our tables has not be checked for its match to the choice of orientation in our diagrams. Also, the A-polynomial can be defined so that repeated factors are significant. In our table repeated factors have been removed.

References

There are now many papers written about the A-Polynomial. The original source is:

D. Boyd, F. Rodrigues-Villegas, and N. Dunfield, "Mahler's Measure and the Dilogarithm", http://arxiv.org/abs/math/0308041

A. Champanerkar, "A-polynomial and Bloch invariants of hyperbolic 3-manifolds", PhD thesis, Columbia University, May 2003.

D. Cooper, M. Culler, H. Gillet, D. D. Long, and P. B. Shalen, Plane curves associated to character varieties of $3$-manifolds, Invent. Math. 118 (1994), no. 1, 47--84.