THE NON-COMMUTATIVE A-POLYNOMIAL OF (−2, 3, n) PRETZEL KNOTS

Abstract. We study q-holonomic sequences that arise as the colored Jones polynomial of knots in 3-space. The minimal-order recurrence for such a sequence is called the (non-commutative) A-polynomial of a knot. Using the method of guessing, we obtain this polynomial explicitly for the Kp = (−2, 3, 3+2p) pretzel knots for p = −5,..., 5. This is a particularly interesting family since the pairs (Kp,−K−p) are geometrically similar (in particular, scissors congruent) with similar character varieties. Our computation of the non-commutative A-polynomial (a) complements the computation of the A-polynomial of the pretzel knots done by the first author and Mattman, (b) supports the AJ Conjecture for knots with reducible A-polynomial and (c) numerically computes the Kashaev invariant of pretzel knots in linear time. In a later publication, we will use the numerical computation of the Kashaev invariant to numerically verify the Volume Conjecture for the above mentioned pretzel knots. 1. The colored Jones polynomial: a q-holonomic sequence of natural origin 1.1. Introduction. The colored Jones polynomial of a knot K in 3-space is a q-holonomic sequence of Laurent polynomials of natural origin in Quantum Topology [GL05]. As a canonical recursion relation for this sequence we choose the one with minimal order; this is the so-called non-commutative A-polynomial of a knot [Gar04]. Using the computational method of guessing with undetermined coefficients [Kau09a, Kau09b