Integral quandle-ring idempotent conjecture for semi-latin quandles

Establish whether the integral quandle ring \(\mathbb{Z}[X]\) of every semi-latin quandle \(X\) has only trivial idempotents, and in particular whether the integral quandle ring of every finite latin quandle has only trivial idempotents.

Background

The paper formulates a quandle-theoretic analogue of Kaplansky’s idempotent conjecture for integral quandle rings. A trivial idempotent is a basis element exe_x associated with an element xx of the quandle; the conjecture asserts that no other idempotents occur in the integral quandle ring of a semi-latin quandle.

The paper proves the conjecture for all semi-latin Takasaki quandles and all medial commutative quandles, including finite latin dihedral quandles, but does not settle the conjecture for arbitrary semi-latin or finite latin quandles. Thus the general assertion remains an unresolved problem beyond the classes treated in the paper.

References

In 2023, the first author, Nunez, Singh, and Swain posed the following quandle-theoretic analogue of Kaplansky's idempotent conjecture. The conjecture is related to certain knot invariants and has become one of the most widely studied conjectures in quandle theory in recent years; see \cites{bardakov-elhamdadi-2026, BES, BPS-2, MR4579329, churchill, ta-2} for previous work on the conjecture.

The integral quandle ring $Z[X]$ of a semi-latin quandle $X$ has only trivial idempotents. In particular, the integral quandle ring of a finite latin quandle has only trivial idempotents.

— Fourier Analysis and Idempotents in Quandle Algebras  (2609.03799 - Elhamdadi et al., 3 Sep 2026) in Introduction, Conjecture 1 (labeled “Main conjecture”)