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A further study of quandles and quandle rings

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Pith's one-line read Core quandles of dihedral groups have rank 4, refuting the proposed formula rk(Core(G)) = rk(G)+1.

desk verdict Settles two open questions and gives a valid counterexample to the rank conjecture, but needs a corrected Corollary 7.2 and a filled gap in Proposition 2.2. read the letter →

arxiv 2608.09149 v1 pith:TVQ2OE4C submitted 2026-08-10 math.RA

Gregory Churchill , Indu Rasika Churchill , Neranga Fernando , Bhitali Kousik This is my paper · ORCID
classification math.RA MSC 57K1217A01
keywords QuandleCoreringZero-divisorgraphIdempotentNilpotentNilcleanPrime
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper investigates core quandles of groups and the rings built from quandles. Its headline claim is that the core quandle of the dihedral group D_n of order 2n has rank 4 = rk(D_n)+2 for every n>2, because Core(D_n) splits into two orbits when n is odd and four when n is even; this disproves the conjecture that rk(Core(G)) always equals rk(G)+1. The paper also settles two open idempotent questions, showing that the integer quandle rings Z[R_5] and Z[C_5] have no nontrivial idempotents of augmentation 1, and gives examples showing the quandle ring of the core of an infinite field can carry nontrivial idempotents. It introduces zero-divisor graphs for extended quandle rings, computes them over F_2 for trivial, Joyce, and dihedral quandles, and proves a mirror symmetry in in-degrees and out-degrees. Additional results cover units, nilpotents, the non-nil-clean property, prime and semi-prime quandle rings, and a bivariate Lagrange interpolation polynomial that determines commutative quandles of prime order.

What carries the argument

The load-bearing machinery is the explicit multiplication rule of Core(D_n): with elements written a^i b^j and operation x*y = y $x^{{-1}}$ y, the parity of n decides whether the map j ↦ 2j-i permutes residues modulo n (odd n) or only permutes them modulo n/2 (even n). That decides the number of orbits and lets the rank proof run on subquandle generation. For the ring-theoretic results, the paper works in the augmentation ideal with basis E_i = e_i - e_0, whose multiplication table (Lemma 3.5) converts idempotent equations into polynomial systems solved by Gröbner bases. For units, zero-divisors, and nilpotents it uses the extended quandle ring S = k[Q] ⊕ ke with e central, and characterises units and zero-divisors by solvability of a small linear system. The final construction uses bivariate Lagrange interpolation in (Z_{2n+1}[Q])[X,Y] to write a polynomial f whose values f(a,b) reproduce the commutative quandle operation e_{(n+1)(a+b)}.

What would settle it

For n=4, compute the four orbits of Core(D_4) under the operation x*y = y $x^{{-1}}$ y, choose one representative from each orbit, and check whether the subquandle they generate has all eight elements of D_4; if it does not, the claimed rank 4 is too small.

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Extended reading notes

Core claim

The paper's central discovery is a counterexample to a proposed rank formula. For n>2, rk(Core(D_n)) = 4 = rk(D_n)+2, so the answer to the question posed in [5]—whether rk(Core(G)) always equals rk(G)+1—is no: the core quandle can require two more generators than the group. The orbit structure, two orbits for odd n and four for even n, is what forces the extra generator. Alongside this, the paper proves that the only idempotents of augmentation 1 in Z[R_5] and Z[C_5] are the trivial basis elements, so those rings have no nontrivial idempotents of that augmentation, and it exhibits infinite-field core quandles with nontrivial idempotents, contradicting another conjecture. In the zero-divisor graphs of extended quandle rings over F_2 it finds a mirror symmetry: a vertex with in-degree i and out-degree o is paired with a vertex with in-degree o and out-degree i, and the paired elements add to a non-zero-divisor.

Load-bearing premise

For even n, the rank-4 proof assumes, without a complete derivation, that one element chosen from each of the four orbits of Core(D_n) generates the entire quandle.

Editorial extensions

If this is right

  • The proposed formula rk(Core(G)) = rk(G)+1 is false for dihedral groups, so a correct general statement about core-quandle ranks must depend on more than the group's rank.
  • The quandle rings Z[R_5] and Z[C_5] have no nontrivial idempotents with augmentation 1, so idempotent-based knot invariants built from these rings must use augmentation-0 elements or other coefficient rings.
  • Since latin core quandles of odd-order groups have nontrivial idempotents without containing a trivial subquandle of order greater than 1, subquandle criteria cannot characterise nontrivial idempotents in quandle rings.
  • Over F_2, the zero-divisor graphs of extended quandle rings of trivial and Joyce quandles obey a mirror symmetry between in-degrees and out-degrees, and mirror pairs sum to a non-zero-divisor.
  • If p divides |Q| then Z_p[Q] is not semi-prime and not prime; moreover the bivariate polynomial f(X,Y) evaluates to exactly the elements of the commutative quandle on Z_{2n+1}.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The mirror pairs in the paper all add to 1 in the extended ring, suggesting the general symmetry is the map u -> u+1; proving that for all finite fields would extend the F_2 computation to arbitrary characteristic.
  • The even-n rank proof's gap suggests a direct computer test of Core(D_4) and Core(D_6) would confirm or refute the orbit-generation step, and similar orbit methods could be tried on other two-generator groups to see how widely the rank jump rk(G)+2 occurs.
  • Because the polynomial systems for Z[R_5] and Z[C_5] were solved by Gröbner bases, the same pipeline can be applied to R_p and C_p for other primes to classify nontrivial idempotents systematically.
  • The Lagrange interpolation result connects quandle-ring-valued polynomials with local permutation polynomials and latin squares, a direction the paper mentions only in motivation; one testable extension is whether every finite latin quandle admits such a polynomial over its quandle ring.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper studies core quandles of dihedral groups and algebraic properties of quandle rings. It claims a counterexample to the conjecture rk(Core(G)) = rk(G)+1 by proving rk(Core(D_n)) = rk(D_n)+2 for n>2, determines idempotents in several quandle rings (including solving two systems left open in earlier work), analyzes units and nilpotents in extended quandle rings, introduces zero-divisor graphs for extended quandle rings and observes a mirror symmetry in their in- and out-degrees, discusses prime/semi-prime quandle rings, and constructs a bivariate polynomial over Z_{2n+1}[Q] that realizes a commutative quandle of prime order.

Significance. If the main results stand, the paper makes a useful contribution: Proposition 2.2 gives a clean counterexample to a published conjecture, and Section 4 settles two previously open systems of equations on idempotents. The zero-divisor graph computations are concrete and checkable, and the mirror symmetry observation is a genuinely new phenomenon that could spur further work. The paper is also commendably explicit: it reports Gröbner bases, multiplication tables, and degree tables rather than only existence statements. However, the rank proof has a load-bearing gap in the even case, and Section 7 contains a false corollary and an unjustified ideal computation, so the paper is not yet acceptable in its current form.

major comments (3)
  1. [§2, Proposition 2.2] The even-n case of the proof is incomplete at the point where the upper bound on rk(Core(D_n)) is established. After Theorem 2.1 gives four orbits, the text asserts, without derivation, that fewer than four elements cannot generate Core(D_n) and that any choice of one element from each orbit generates the whole quandle. Neither assertion is immediate from the displayed orbit analysis. A correct proof needs the invariance observation that the type (rotation/reflection) and the parity of the exponent of the left operand are preserved by the quandle operation, so no orbit can be omitted, and then a doubling argument (or an equivalent) showing that, for example, a^0 and a^t with t odd generate all rotations of both parities. The odd-n case also silently assumes that two suitably chosen rotations generate the full rotation orbit; this is true since 2 is invertible modulo odd n, but it should be stated. As written, the central rank theorem is not fully proved.
  2. [§7, Corollary 7.2] Corollary 7.2 is false as stated. Let Q be the trivial quandle of order 3 and take p=5. Writing S=x0+x1+x2, the trivial-quandle multiplication gives S^2=3S and more generally S^k=3^{k-1}S. Then S^3=9S≡4S, but (S^3)^2=S^6=3^5S≡3S, which is not 4S in Z_5. Hence S^3 is not an idempotent. The intended statement becomes true only with an additional hypothesis such as p|n, under which S^2=0 and the nth power is 0. The attribution 'which also appeared in [3]' should be checked against the exact statement in that reference.
  3. [§7, Proposition 7.1] The proof of Proposition 7.1 does not justify the assertion that the two-sided ideal generated by S=x0+...+x_{n-1} is exactly {αS : α∈Z_p}. The displayed identity verifies only products of the form S·(Σα_i x_i). For arbitrary A, the product A·S involves the row sums of the quandle multiplication table, which are not shown to be scalar multiples of S unless the quandle is Latin or has some additional balanced-row property. Consequently the square of a general element of the ideal is never computed, and the conclusion I^2=0 is not established for arbitrary quandles. The proposition may be salvageable by a different argument or by restricting the class of quandles, but the present proof is incomplete.
minor comments (4)
  1. [§4, C5 system after (4.4)] The rational root candidates for the displayed one-variable polynomial in β4 are the integer divisors of 6, namely ±1, ±2, ±3, ±6, not ±1, ±2, ±3, ±4. The proof should either replace the list or explicitly check the missing candidates ±6.
  2. [§6.1, Case 3] The line 'in-degree = 2.2^{n-1}-1' should read 2^n-1, and the notation 'x0+x1 x2' in Remark 6.2 is missing commas and likely a missing plus sign.
  3. [§8, Lagrange basis] In the displayed definition of L_{(X,2n)}(X), the product is written with 'i≠n−1'; it should be 'i≠2n'.
  4. [Throughout] There are numerous typos and inconsistent notations, including 'commutaitve', 'whcih', 'zero-divsor', 'Proporition', 'ε' versus 'ϵ', and the duplicated Propositions 1.8 and 3.3. A careful copyedit is needed.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the main results are derived by explicit computation or from external cited results; the only self-citation is background and not load-bearing.

full rationale

The derivation chain is not circular. Proposition 2.2 computes rk(Core(D_n)) from the orbit count in Theorem 2.1; the even-n generation claim is asserted rather than fully expanded, but it is a sufficiency argument that does not presuppose the rank being proved. Section 3's idempotent conclusions rest on [4, Propositions 12.11 and 12.13], authored by Bardakov, Elhamdadi, and Singh, not by the present authors; Section 4 solves systems imported from [2] by explicit Grobner basis computations. Section 5 proves unit and nilpotent characterizations from the definitions, with [7, Theorem 3.5] as an external, non-self input, and Section 6 computes zero-divisor degrees from those characterizations. Proposition 7.1 is proved directly, and Section 8 constructs its bivariate polynomial by Lagrange interpolation from the known operation, so the 'determines' claim is a construction rather than an assumed input. The sole self-citation is [10], where N. Fernando is a coauthor, cited in the introduction for background facts on power-associativity, integral domains, and Noetherianity; none of these facts is used in the later proofs, so the self-citation is not load-bearing. The omitted proof of Lemma 5.1 and the unexpanded generation step in Proposition 2.2 are expositional gaps, not circularities.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no fitted parameters and no new postulated entities. Its central claims rest on standard quandle and ring theory, plus two cited external results: [4, Proposition 12.11] (existence of nontrivial idempotents from subquandles of order n invertible in k) used in Examples 3.1 and 3.4, and [7, Theorem 3.5] (square-zero augmentation ideal for trivial quandles) used in Theorem 5.2. The Gröbner basis computations are presented as outputs without code or certificates.

assumptions (4)
  • standard math Quandle axioms: x*x=x, right multiplication by each element is invertible, and the operation distributes over itself.
    Foundation of the paper; used throughout without proof.
  • domain assumption [4, Proposition 12.11]: If a quandle Q contains a subquandle of order n and n is a unit in k, then k[Q] has a nontrivial idempotent.
    Used in Examples 3.1 and 3.4 to answer Question 1.7 and show the converse of Proposition 3.3 is false. The paper cites this proposition rather than proving it.
  • domain assumption [7, Theorem 3.5]: The augmentation ideal of the quandle ring of a trivial quandle squares to zero.
    Used in the proof of Theorem 5.2 to simplify products in the extended quandle ring.
  • standard math Lagrange interpolation over the finite field Z_{2n+1} yields a unique bivariate polynomial taking prescribed values on F_q^2.
    Basis for Section 8's interpolating polynomial.

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Cite this review

Pith. "Pith review of A further study of quandles and quandle rings." pith.science (2026). https://pith.science/paper/TVQ2OE4C

@misc{pith2026260809149,
  author       = {Pith},
  title        = {Pith review of: A further study of quandles and quandle rings},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TVQ2OE4C}},
  note         = {Machine review of arXiv:2608.09149}
}
abstract

We investigate core quandles and idempotents in quandle rings of core quandles. We answer several questions on the rank of core quandles and nontrivial idempotents in quandle rings. We also present solutions to two questions raised in a recent paper about non-trivial idempotents in quandle rings $\mathbb{Z}[R_5]$ and $\mathbb{Z}[C_5]$, where $\mathbb{Z}$, $R_5$ and $C_5$ are the ring of integers, the dihedral quandle of order 5, and the commutative quandle of order 5, respectively. We then study units in extended quandle rings of a trivial quandle and the Joyce quandle, and nilpotent elements in extended quandles rings of a trivial quandle, where the ground ring is an integral domain. As a consequence, we show that the quandle ring and the extended quandle ring of a trivial quandle are not nil clean rings. We also explore prime rings and semi-prime rings among quandle rings. We introduce zero-divisor graphs of quandle rings and find an intriguing mirror symmetry among the in-degree and out-degree of the vertices. Moreover, we find a bivariate polynomial in the ring $(\mathbb{Z}_{2n+1}[Q])[X,Y]$ that determines the commutative quandle of order $2n+1$, where $2n+1$ is prime.

Figures

Figures reproduced from arXiv: 2608.09149 by the authors.

Figure 1
Figure 1. Zero-divisor graph of the extended quandle ring of the trivial quandle of order 2 [PITH_FULL_IMAGE:figures/full_fig_p022_1.png] view at source ↗
Figure 2
Figure 2. Zero-divisor graph of the extended quandle ring of the trivial quandle of order 3 Vertex In-degree Out-degree x0 1 2 3 − 1 x1 1 2 3 − 1 x2 1 2 3 − 1 x0 + 1 2 3 − 1 1 x1 + 1 2 3 − 1 1 x2 + 1 2 3 − 1 1 x0 + x1 2 3 − 1 2 3 − 1 x0 + x2 2 3 − 1 2 3 − 1 x1 + x2 2 3 − 1 2 3 − 1 x0 + x1 + x2 1 2 3 − 1 x0 + x1 + x2 + 1 2 3 − 1 1 [PITH_FULL_IMAGE:figures/full_fig_p024_2.png] view at source ↗
Figure 3
Figure 3. Zero-divisor graph of the extended quandle ring of the Joyce quandle over F2 Vertex In-degree Out-degree e0 1 2 3 − 1 ej , j ̸= 0 1 2 2 − 1 e0 + 1 2 2 − 1 1 ej + 1, j ̸= 0 2 3 − 1 1 e0 + ej , j ̸= 0 2 2 − 1 2 2 − 1 e1 + e2 2 3 − 1 2 3 − 1 e0 + e1 + e2 1 2 3 − 1 e0 + e1 + e2 + 1 2 2 − 1 1 [PITH_FULL_IMAGE:figures/full_fig_p029_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Zero-divisor graph of the extended quandle ring of the dihe￾dral quandle over F2 From the graph, it is evident that every element of the extended dihedral quandle ring over F2, other than 0 and the unity, is a zero divisor. It is also observed that any vertex x with ϵ …

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Works this paper leans on

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