The Noncommutative 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 (noncommutative) 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 noncommutative A-polynomial complements the computation of the A-polynomial of the pretzel knots done by the first author and Mattman, supports the AJ conjecture for knots with reducible A-polynomial, and 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.