Author:
Detcherry Renaud,Garoufalidis Stavros
Abstract
Abstract
The AJ Conjecture relates a quantum invariant, a minimal order recursion for the colored Jones polynomial of a knot (known as the $$\hat{A}$$
A
^
polynomial), with a classical invariant, namely the defining polynomial A of the $${\mathrm {PSL}_2(\mathbb {C})}$$
PSL
2
(
C
)
character variety of a knot. More precisely, the AJ Conjecture asserts that the set of irreducible factors of the $$\hat{A}$$
A
^
-polynomial (after we set $$q=1$$
q
=
1
, and excluding those of L-degree zero) coincides with those of the A-polynomial. In this paper, we introduce a version of the $$\hat{A}$$
A
^
-polynomial that depends on a planar diagram of a knot (that conjecturally agrees with the $$\hat{A}$$
A
^
-polynomial) and we prove that it satisfies one direction of the AJ Conjecture. Our proof uses the octahedral decomposition of a knot complement obtained from a planar projection of a knot, the R-matrix state sum formula for the colored Jones polynomial, and its certificate.
Funder
Max Planck Institute for Mathematics
Publisher
Springer Science and Business Media LLC
Reference40 articles.
1. Andersen, J.E., Malusa, A.: The AJ-conjecture for the Teichmüller TQFT, Preprint (2017)
2. Boyd, D.W., Rodriguez-Villegas, F.: Mahler’s measure and the dilogarithm (II), Preprint (2005)
3. Bostan, A., Lairez, P., Salvy, B: Creative telescoping for rational functions using the Griffiths–Dwork method. In: ISSAC 2013–Proceedings of the 38th International Symposium on Symbolic and Algebraic Computation, pp. 93–100. ACM, New York (2013)
4. Bar-Natan, D., Garoufalidis, S.: On the Melvin–Morton–Rozansky conjecture. Invent. Math. 125(1), 103–133 (1996)
5. Cooper, D., Culler, M., Gillet, H., Long, D., Shalen, P.: Plane curves associated to character varieties of $$3$$-manifolds. Invent. Math. 118(1), 47–84 (1994)
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