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 $K_p = (−2, 3, 3 + 2p)$ pretzel knots for $p= −5, \dots , 5$. This is a particularly interesting family, since the pairs $(K_p,−K_{−p})$ 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.