New York Journal of Mathematics
Volume 17 (2011) 269-279


Stavros Garoufalidis and Thomas W. Mattman

The A-polynomial of the (-2,3,3+2n) pretzel knots

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Published: April 11, 2011
Keywords: Pretzel knots, A-polynomial, Newton polygon, character variety, Culler-Shalen seminorm, holonomic sequences, quasi-polynomials.
Subject: Primary 57N10. Secondary 57M25

We show that the A-polynomial An of the 1-parameter family of pretzel knots Kn=(-2,3,3+2n) satisfies a linear recursion relation of order 4 with explicit constant coefficients and initial conditions. Our proof combines results of Tamura-Yokota and the second author. As a corollary, we show that the A-polynomial of Kn and the mirror of K-n are related by an explicit GL(2,Z) action. We leave open the question of whether or not this action lifts to the quantum level.


The first author was supported in part by the NSF

Author information

Stavros Garoufalidis:
School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332-0160, USA

Thomas W. Mattman:
Department of Mathematics and Statistics, California State University, Chico, Chico, CA 95929-0525, USA