Our goal is to compute the minimal-order recurrence of the colored Jones polynomial
of the
knot, as well as for the first four double twist knots. As a
corollary, we verify the AJ Conjecture for the simplest knot
with reducible
nonabelian
character variety. To achieve our goal, we use symbolic summation techniques of
Zeilberger’s holonomic systems approach and an irreducibility criterion for
–difference
operators. For the latter we use an improved version of the qHyper
algorithm of Abramov–Paule–Petkovšek to show that a given
–difference
operator has no linear right factors. En route, we introduce exterior power Adams
operations on the ring of bivariate polynomials and on the corresponding affine
curves.
Keywords
$q$–holonomic module, $q$–holonomic sequence, creative
telescoping, irreducibility of $q$–difference operators,
factorization of $q$–difference operators, qHyper, Adams
operations, quantum topology, knot theory, colored Jones
polynomial, AJ conjecture, double twist knot, $7_4$
