Let n>1 be odd, let R_n=Z/nZ with dihedral quandle operation ab=2b-a, let Delta(R_n) be the augmentation ideal of its integral quandle ring, and let Delta^k(R_n) denote its kth power. The conjecture asserts that Delta^k(R_n)/Delta^k+1(R_n)congZ_n for every k≥ 1, and that the right Peirce spectrum of the complex quandle algebra C[R_n] is 0,1,-1 for every odd n>1.

Odd-order dihedral quandle spectrum conjecture

The conjecture asserts a uniform spectrum for odd-order dihedral quandle rings, including the quotient pattern $\Delta^k(R_n)/\Delta^{k+1}(R_n)\cong\mathbb{Z}_n$ for odd $n>1$ . Earlier literature attributes this odd-order result to Elhamdadi, Fernando, and Tsvelikhovskiy.

Known results

Elhamdadi, Fernando, and Tsvelikhovskiy (2019): for odd $n>1$ , $\Delta^k(R_n)/\Delta^{k+1}(R_n)\cong\mathbb{Z}_n$ for every $k\ge1$ .
Panja and Prasad (2022): the analogous even-order prediction fails, with $\Delta^2(R_8)/\Delta^3(R_8)\cong\mathbb{Z}_4\oplus\mathbb{Z}_4$ ; they proposed a revised even-order pattern.

October 2026 claimed proof

Zhi-Lin Zhang's preprint Group-ring methods for Alexander quandle rings claims to prove the conjectured odd-order spectrum and supplies a quotient theorem correcting the earlier false order- $m$ prediction. It also treats augmentation filtrations, idempotents, and finite medial commutative quandles; the claim has not been independently verified.

Current status (as of October 2026): The odd-order spectrum is claimed proved for the specified finite and medial Alexander-quandle classes, but the claim is unverified and does not settle arbitrary quandle rings.

Odd-order dihedral quandle spectrum conjecture

Primary source

Additional references

Progress summary

Solutions 0