New York Journal of Mathematics
Volume 29 (2023), 739-770


Em K. Thompson

Twisting, ladder graphs, and A-polynomials

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Published: June 22, 2023.
Keywords: Dehn filling, layered solid torus, A-polynomial, cluster algebras, perfect matchings, ladder graphs.
Subject [2020]: 57K10 (primary), 57K31, 57K32, 13F60.

We extend recent work by Howie, Mathews and Purcell to simplify the calculation of A-polynomials for any family of hyperbolic knots related by twisting. The main result follows from the observation that equations defining the deformation variety that correspond to the twisting are reminiscent of exchange relations in a cluster algebra. We prove two additional results with analogues in the context of cluster algebras: the Laurent phenomenon, and intersection numbers appearing as exponents in the denominator. We demonstrate our results on the twist knots, and on a family of twisted torus knots for which A-polynomials have not previously been calculated.


The author is supported by an Australian Government Research Training Program (RTP) Scholarship.

Author information

Em K. Thompson
School of Mathematics
Monash University
VIC 3800, Australia