Fourier Analysis and Idempotents in Quandle Algebras

Mohamed Elhamdadi; Luc Ta; Bryce Virgin

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arXiv · 2609.03799

Fourier Analysis and Idempotents in Quandle Algebras

Mohamed Elhamdadi·Luc Ta·Bryce Virgin

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Abstract

In 2023, the first author, Nunez, Singh, and Swain [10] formulated an analogue of Kaplansky's idempotent conjecture for integral quandle rings. In this paper, we develop a new Fourier-analytic approach to quandle rings and use it to establish the conjecture for two major classes of quandles: Takasaki quandles, including dihedral quandles, and medial commutative quandles. A central contribution of the paper is the introduction of Fourier analysis on Alexander quandles, which provides a new framework for studying quandle rings and, in particular, their idempotents. Using this Fourier-analytic framework, we also prove that a previously known sufficient condition for the existence of counterexamples to the conjecture is in fact necessary, thereby resolving a problem of Jablonowski [12].

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BibTeX RIS

Mohamed Elhamdadi, Luc Ta, Bryce Virgin. 2026-09-03. Fourier Analysis and Idempotents in Quandle Algebras. https://arxiv.org/abs/2609.03799

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