Twisting, ladder graphs and A-polynomials

TL;DR

This paper introduces a novel method to compute A-polynomials for hyperbolic knots related by twisting, leveraging cluster algebra structures, and demonstrates its effectiveness on twist and twisted torus knots.

Contribution

It extends existing techniques by connecting deformation equations to cluster algebra exchange relations, providing new formulas for A-polynomials of previously uncomputed knots.

Findings

Derived simplified formulas for A-polynomials of twist knots

Extended the method to twisted torus knots with new results

Established a link between deformation varieties and cluster algebra properties

Abstract

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.