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A further study of quandles and quandle rings

This paper investigates core quandles and idempotents in quandle rings, resolves specific questions regarding nontrivial idempotents and units in extended quandle rings, analyzes nilpotent elements and prime ring properties, and introduces zero-divisor graphs to reveal mirror symmetry and a polynomial characterization for commutative quandles of prime order.

Original authors: Gregory Churchill, Indu Rasika Churchill, Neranga Fernando, Bhitali Kousik

Published 2026-08-11

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CC BY 4.0

Original authors: Gregory Churchill, Indu Rasika Churchill, Neranga Fernando, Bhitali Kousik

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a world where the rules of algebra don't always play nice. In the standard math you learn in school, if you multiply numbers, the order doesn't matter ( $3 \times 4$ is the same as $4 \times 3$ ), and if you multiply something by itself enough times, you usually get a predictable result. But there is a wilder branch of mathematics called "knot theory," which studies how strings can get tangled in space. To solve the puzzle of whether two knots are actually the same (just twisted differently), mathematicians invented a special kind of algebraic structure called a quandle. Think of a quandle as a set of magical buttons. When you press one button ( $x$ ) using another button ( $y$ ), it changes $x$ into a new state ( $x * y$ ). These buttons follow three strict rules that mimic the physical moves you can make on a knotted string without cutting it. They are non-associative, meaning the order in which you press them changes the outcome, making them tricky but incredibly powerful for untangling mathematical knots.

Now, imagine taking these magical buttons and building a whole new kind of number system out of them, called a quandle ring. It's like taking a deck of cards (the quandle) and allowing yourself to mix them, add them up, and multiply them in complex ways. Mathematicians are obsessed with finding "idempotents" in these rings. An idempotent is a special number that, when you multiply it by itself, stays exactly the same ( $x^2 = x$ ). In normal number systems, the only idempotents are usually 0 and 1. But in these quandle rings, the question is: are there any other hidden numbers that act like this? Finding them is like discovering secret codes in a video game; they could help mathematicians build even better tools to tell knots apart. This paper dives deep into these rings to see where these secret numbers hide, where they don't, and how the "zero-divisors" (numbers that multiply to zero) are arranged in a beautiful, symmetrical pattern.

The authors of this paper, Gregory Churchill, Indu Rasika Churchill, Neranga Fernando, and Bhitali Kousik, set out to investigate several specific types of these quandle rings. They started by looking at "core quandles," which are built from groups of symmetries (like the rotations and flips of a regular polygon). They tackled a big question: does the complexity of the underlying group determine the complexity of the quandle? They found that for the symmetries of a polygon (the dihedral group), the answer depends on whether the number of sides is odd or even. If the number of sides is odd, the quandle splits into two distinct "orbits" (groups of buttons that can reach each other); if it's even, it splits into four. This discovery helped them prove that a previous guess about how these quandles are built was wrong.

Next, they hunted for those elusive "non-trivial idempotents." They showed that while some infinite quandle rings definitely have these secret numbers, others might not. They specifically solved two long-standing puzzles about the rings built from a 5-sided polygon (the dihedral quandle of order 5) and a 5-element commutative quandle. Using a powerful computer algebra technique called Gröbner bases (think of it as a super-organized way to solve a massive system of equations), they proved that for the ring of integers, the only idempotents in these specific cases are the "trivial" ones (the basic building blocks). In other words, there are no hidden codes in these specific rings; the only numbers that stay the same when squared are the ones you started with.

The paper then moves on to "extended quandle rings," which are like adding a special "universal" button to the mix. Here, they investigated "units" (numbers that have an inverse, like how 2 and 0.5 are inverses in normal math) and "nilpotent" elements (numbers that vanish to zero if you multiply them by themselves enough times). They found clear rules for when these extended rings have units and proved that the rings built from "trivial quandles" (where the buttons don't really change each other) are not "nil clean." This is a fancy way of saying you can't break every number in these rings down into a simple sum of a "staying-the-same" number and a "vanishing" number.

One of the most visually striking parts of their work involves "zero-divisor graphs." Imagine a map where every number that can multiply with another to make zero is a city, and a road connects two cities if they cancel each other out. The authors drew these maps for various quandle rings and found a stunning "mirror symmetry." If there is a city with 3 roads going out and 1 road coming in, there is another city with 1 road going out and 3 coming in. It's like a perfect reflection in a funhouse mirror. They proved that if you add these two "mirror" numbers together, the result is never a zero-divisor, which is a neat and surprising property.

Finally, the authors looked at prime and semi-prime rings (rings that don't have "hidden" zero-square ideals) and found that if the size of the quandle is divisible by a prime number, the ring fails to be prime. They also showed how to write a specific polynomial (a fancy algebraic formula with two variables) that can recreate the entire multiplication table of a commutative quandle of a certain size, essentially turning the whole structure into a single equation.

In summary, this paper doesn't just ask questions; it answers them with rigorous proofs. It rules out the existence of non-trivial idempotents in specific integer-based quandle rings, corrects misconceptions about the rank of core quandles, and reveals a beautiful, symmetrical structure in the way zero-divisors interact. While they don't claim to have solved every mystery in knot theory, they have provided a clearer, more detailed map of the algebraic landscape, showing us exactly where the secrets lie and where the paths are empty.

Technical Summary: A Further Study of Quandles and Quandle Rings

Problem Statement
This paper investigates the algebraic structure of quandle rings and their extensions, focusing on several open questions regarding idempotents, units, nilpotent elements, and zero-divisor graphs. Specifically, the authors address:

The relationship between the rank of a group $G$ and the rank of its core quandle $\text{Core}(G)$ , challenging the conjecture that $\text{rk}(\text{Core}(G)) = \text{rk}(G) + 1$ .
The existence of non-trivial idempotents in quandle rings $k[\text{Core}(F)]$ for infinite fields $F$ , and specifically solving open systems of equations regarding non-trivial idempotents in $\mathbb{Z}[R_5]$ and $\mathbb{Z}[C_5]$ (where $R_5$ and $C_5$ are the dihedral and commutative quandles of order 5, respectively).
The characterization of units and nilpotent elements in extended quandle rings, particularly for trivial and Joyce quandles.
The structural properties of zero-divisor graphs for extended quandle rings, including the observation of "mirror symmetry" in vertex degrees.
The conditions under which quandle rings are prime or semi-prime, and the construction of polynomials that define commutative quandles.

Methodology
The authors employ a combination of algebraic manipulation, combinatorial analysis, and computational algebra techniques:

Orbit Analysis: In Section 2, the authors analyze the orbits of the core quandle of the dihedral group $D_n$ under the action of inner automorphisms to determine the rank of $\text{Core}(D_n)$ .
Gröbner Basis Techniques: To resolve open problems concerning idempotents in $\mathbb{Z}[R_5]$ and $\mathbb{Z}[C_5]$ (Section 4), the authors translate the conditions for idempotents into systems of non-linear homogeneous multivariate equations. They utilize Gröbner bases (computed via Mathematica) to solve these systems over the integers.
Augmentation Ideal Analysis: The study of units and nilpotent elements (Section 5) relies on decomposing elements of the extended quandle ring $S = k[Q] \oplus ke$ relative to the augmentation ideal $\Delta(Q)$ . The authors derive necessary and sufficient conditions for solvability of specific systems of equations to determine invertibility and nilpotency.
Graph Theory: Section 6 defines zero-divisor graphs for extended quandle rings (directed for non-commutative cases, undirected for commutative ones). The authors compute in-degrees and out-degrees for vertices in these graphs over finite fields ( $F_2$ ) and identify structural symmetries.
Polynomial Interpolation: In Section 8, the authors apply bivariate Lagrange interpolation to construct a polynomial in $(\mathbb{Z}_{2n+1}[Q])[X, Y]$ that reproduces the operation of a commutative quandle of prime order $2n+1$ .

Key Contributions and Results

Rank of Core Quandles: The authors prove that for the dihedral group $D_n$ ( $n > 2$ ), the number of orbits in $\text{Core}(D_n)$ is 2 if $n$ is odd and 4 if $n$ is even. Consequently, they establish that $\text{rk}(\text{Core}(D_n)) = \text{rk}(D_n) + 2 = 4$ , providing a counterexample to the conjecture that $\text{rk}(\text{Core}(G)) = \text{rk}(G) + 1$ .
Idempotents in Quandle Rings:
- The paper provides examples of infinite fields $F$ where $k[\text{Core}(F)]$ possesses non-trivial idempotents, answering Question 1.7 negatively.
- It is shown that the existence of a trivial subquandle of order $>1$ is sufficient but not necessary for the existence of non-trivial idempotents.
- Using Gröbner bases, the authors demonstrate that the quandle rings $\mathbb{Z}[R_5]$ and $\mathbb{Z}[C_5]$ possess no non-trivial idempotents $u$ with augmentation $\epsilon(u) = 1$ . The only integer solutions to the relevant systems are the trivial ones.
Units and Nilpotents:
- Necessary and sufficient conditions are derived for an element to be a unit in the extended quandle ring of a trivial quandle.
- It is proven that for a trivial quandle $Q$ , the quandle ring $k[Q]$ and its extension $S$ are not nil clean. Specifically, the sum of an idempotent and a nilpotent element in $k[Q]$ is always an idempotent, preventing the representation of all elements as such sums.
Zero-Divisor Graphs:
- The authors introduce zero-divisor graphs for extended quandle rings.
- A "mirror symmetry" is identified: for every vertex $x$ with in-degree $i$ and out-degree $o$ , there exists a vertex $y$ with in-degree $o$ and out-degree $i$ . Furthermore, the sum of such symmetric elements is never a zero divisor. This symmetry is proven for extended quandle rings of trivial quandles over $F_2$ .
- The paper presents specific graphs and degree tables for trivial, Joyce, and dihedral quandles over $F_2$ .
Prime and Semi-Prime Rings: A sufficient condition is provided: if $p$ is a prime dividing the order $n$ of a quandle $Q$ , then the quandle ring $\mathbb{Z}_p[Q]$ is not semi-prime (and thus not prime).
Polynomial Representation: The paper constructs a bivariate polynomial in $(\mathbb{Z}_{2n+1}[Q])[X, Y]$ that determines the operation of a commutative quandle of order $2n+1$ (where $2n+1$ is prime), drawing an analogy to local permutation polynomials used in constructing Latin squares.

Significance
The paper claims to advance the algebraic theory of quandle rings by resolving specific open questions regarding idempotents in small-order cases ( $R_5, C_5$ ) and extending the study of ring-theoretic properties (units, nilpotents, zero-divisors) to the non-associative setting of extended quandle rings. The identification of mirror symmetry in zero-divisor graphs offers a new structural perspective on these non-associative rings. Additionally, the construction of polynomials defining commutative quandles bridges the gap between quandle theory and the theory of local permutation polynomials. The authors note that these results, particularly the zero-divisor graphs, are intended to motivate further investigation into the interplay between quandle rings and graph theory.

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