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} polynomial), with a classical invariant, namely the defining polynomial A of the {mathrm {PSL}_2(mathbb {C})} character variety of a knot. More precisely, the AJ Conjecture asserts that the set of irreducible factors of the hat{A}-polynomial (after we set 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}-polynomial that depends on a planar diagram of a knot (that conjecturally agrees with the hat{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.
Highlights
1.1 The colored Jones polynomial and the AJ ConjectureThe Jones polynomial of a knot [20] is a powerful knot invariant with deep connections with quantum field theory, discovered by Witten [36]
Since the dependence of the colored Jones polynomial JK (n) on the variable q plays no role in our paper, we omit it from the notation
The starting point of the AJ Conjecture [9] is the fact that the colored Jones polynomial JK (n) of a knot K is q-holonomic [15], that is, it satisfies a nontrivial linear recursion relation d c j (q, qn)JK (n + j ) = 0, for all n ∈ N, (1)
Summary
The Jones polynomial of a knot [20] is a powerful knot invariant with deep connections with quantum field theory, discovered by Witten [36]. The starting point of the AJ Conjecture [9] is the fact that the colored Jones polynomial JK (n) of a knot K is q-holonomic [15], that is, it satisfies a nontrivial linear recursion relation d c j (q, qn)JK (n + j ) = 0, for all n ∈ N,. The aim of the paper is to highlight the fact that formulas of the form (4) lead to further knot invariants which are natural from the point of view of holonomic modules and form a rephrasing of the AJ Conjecture that connects well with the results of [22] and [24]. Where F is allowed to change by for instance, consequences of the q-binomial identity, one can obtain an operator Acf (q, Q, E) which is independent of the chosen presentation
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