# A diagrammatic approach to the AJ Conjecture

## TL;DR

This paper introduces a diagram-dependent version of the $ ilde{A}$-polynomial for knots, proves it satisfies one direction of the AJ Conjecture using octahedral decompositions and state sum formulas, advancing understanding of knot invariants.

## Contribution

It proposes a new diagram-dependent $ ilde{A}$-polynomial and proves it satisfies one side of the AJ Conjecture using geometric and algebraic methods.

## Key findings

- The diagram-dependent $ ilde{A}$-polynomial aligns with the $ ilde{A}$-polynomial conjecturally.
- The proof employs octahedral decompositions from planar knot diagrams.
- The approach connects geometric decompositions with quantum invariants.

## 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 $\psl$ 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.