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.

中文翻译：

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AJ 猜想的图解方法

AJ 猜想将量子不变量、一个结的有色琼斯多项式的最小阶递归（称为 $$\hat{A}$$ A ^ 多项式）与经典不变量相关联，即定义多项式 A $${\mathrm {PSL}_2(\mathbb {C})}$$ PSL 2 ( C ) 结的字符变体。更准确地说，AJ 猜想断言 $$\hat{A}$$ A ^ -多项式的不可约因数集（在我们设置 $$q=1$$ q = 1 之后，并排除 L -degree零）与 A 多项式的那些一致。在本文中，我们介绍了 $$\hat{A}$$ A ^ -多项式的一个版本，它依赖于一个结的平面图（推测与 $$\hat{A}$$ A ^ -多项式），我们证明它满足 AJ 猜想的一个方向。