The colored Jones polynomials J_n(K) of a knot K are known to satisfy some recurrence relations, which are described by a non commutative two variable polynomial Â(q,L,M). The AJ conjecture states that the specialization Â(q=1,L,M) is equal to the A-polynomial of K, whose zero set is the SL_2(C) character variety of K.
Fixing a diagram of the knot K, we will compare the state sum formula for colored Jones polynomials of K with the Thurston gluing equations for the decomposition of S^3\K into ideal octahedra arising from the diagram. As a result, we prove a weak form of the AJ conjecture: For any knot K, Â(q=1,L,M) divides a power of the A-polynomial.
Joint with Stravos Garoufalidis.
Séminaire dans le cadre de la 2nd rencontre de l'ANR AlMaRe (29-30 Avril 2021)