On the Euler characteristics for quandles

Ryoya Kai and Hiroshi Tamaru

Abstract

A quandle is an algebraic system whose axioms generalize the algebraic structure of the point symmetries of symmetric spaces. In this paper, we give a definition of Euler characteristics for quandles. In particular, the quandle Euler characteristic of a compact connected Riemannian symmetric space coincides with the topological Euler characteristic. Additionally, we calculate the Euler characteristics of some finite quandles, including generalized Alexander quandles, core quandles, discrete spheres, and discrete tori. Furthermore, we prove several properties of quandle Euler characteristics, which suggest that they share similar properties with topological Euler characteristics.

1 Introduction

The Euler characteristic for a topological space is one of the classical and important topological invariants. In general, the Euler characteristics can be defined for topological spaces using Betti numbers or cell complexes. Hopf and Samelson [Hopf-1941-SatzüberWirkungsräumeGeschlossener] developed a method for calculating the Euler characteristics of homogeneous spaces of compact Lie groups. A symmetric space is a space with a point symmetry at each point. A compact connected symmetric space is a homogeneous space under the action of the compact Lie group generated by these point symmetries. In conclusion, the Euler characteristic of a compact connected Riemannian symmetric space can be computed using the point symmetries.

A quandle is an algebraic system introduced in knot theory [5], that consists of a pair of a non-empty set $X$ and a map $s:X\to\mathrm{Map}(X,X)$ satisfying certain axioms. These axioms align well with the local moves of knot diagrams, known as Reidemeister moves. Recently, quandles have played important roles in many branches of mathematics. For example, symmetric spaces can be viewed as quandles via their point symmetries. In other words, a quandle can be considered as a generalization of an algebraic system that focuses specially on point symmetries in a symmetric space. In particular, quandles can be regarded as a discretization of symmetric spaces, and there have been several recent studies from this view point (for instance, see [1, 6, 4]).

A quandle structure gives rise to certain canonical groups, which act on the quandle in a manner compatible with the point symmetries $s_{x}:=s(x):X\to X$ . The following group, which plays a key role in this study, is defined by Joyce [5, §5] as the transvection group.

Definition 1.1 ([5], [3]).

The group defined by

\mathrm{Dis}(X)=\langle s_{x}\circ s_{y}^{-1}\mid x,y\in X\rangle_{\mathrm{Grp}}

is called the displacement group of a quandle $X$ .

In this paper, we define the quandle Euler characteristic using the action of the displacement group as follows:

Definition 1.2.

Let $X$ be a quandle. Then the quandle Euler characteristic $\chi^{\mathrm{Qdle}}(X)$ is defined by

\chi^{\mathrm{Qdle}}(X):=\inf\{\#\mathrm{Fix}(g,X)\mid g\in\mathrm{Dis}(X)\},

where $\mathrm{Fix}(g,X)$ denotes the set of fixed points of the action of $g$ on $X$ .

By definition, the quandle Euler characteristic satisfies $\chi^{\mathrm{Qdle}}(X)\in\mathbb{Z}_{\geq 0}$ , which aligns with the property of the topological Euler characteristics for compact homogeneous spaces. Furthermore, the following theorem explains why we refer to the number defined above as the quandle Euler characteristic.

Theorem 1.3 (Theorem 3.4).

For a compact connected Riemannian symmetric space, the quandle Euler characteristic is equal to the topological Euler characteristic.

In this paper, we prove several properties of quandle Euler characteristics, which indicate that they share similarities with topological Euler characteristics. In Section 4, we determine the quandle Euler characteristics of some particular examples of quandles.

First examples we consider are generalized Alexander quandles. A generalized Alexander quandle, denoted by $\mathrm{GAlex}(G,\sigma)$ , is defined as a group $G$ whose quandle structure is given by a group automorphism $\sigma$ of $G$ . The class of generalized Alexander quandles is important in quandle theory, as any homogeneous quandle can be realized as a quotient of some generalized Alexander quandle. Note that any homogeneous manifold can be also expressed as a quotient of a Lie group. Therefore, in the theory of quandles, generalized Alexander quandles can be viewed as counterparts of Lie groups. The following theorem determines the quandle Euler characteristics of generalized Alexander quandles.

Theorem 1.4 (Theorem 4.1).

If $\sigma$ is a non-trivial group automorphism of a group $G$ , then the quandle Euler characteristic of the generalized Alexander quandle $\mathrm{GAlex}(G,\sigma)$ is equal to $0$ .

The second examples are core quandles, which are groups with quandle structures derived from the map taking the inverse element. This provides a natural quandle structure for a group from the viewpoint of symmetric spaces. In fact, a symmetric space structure of a Lie group is usually given by the core quandle structure. Similar to the case of generalized Alexander quandles, we determine the quandle Euler characteristics of core quandles, which are analogous to the topological Euler characteristics of non-trivial compact connected Lie groups that are equal to $0$ .

Theorem 1.5 (Theorem 4.3).

The quandle Euler characteristic of a non-trivial core quandle is equal to $0$ .

The third examples consist of discrete subquandles within compact Riemannian symmetric spaces. A typical example is the discrete $n$ -sphere, denoted by $DS^{n}$ , which is defined as a particular finite subquandle of the standard $n$ -sphere $S^{n}$ . The following theorem suggests that the quandle Euler characteristics of certain finite objects approximate the topological Euler characteristics of continuous objects.

Theorem 1.6 (Theorem 4.6).

For any positive integer $n$ , the quandle Euler characteristic of the discrete $n$ -sphere $DS^{n}$ satisfies $\chi^{\mathrm{Qdle}}(DS^{n})=\chi^{\mathrm{Top}}(S^{n})$ , that is,

\chi^{\mathrm{Qdle}}(DS^{n})=\begin{cases*}0&if $n$ is odd,\\ 2&if $n$ is even.\end{cases*}

We also consider the discrete torus $DT^{n}_{u}$ , which is a discrete subquandle of a flat torus $T^{n}=(S^{1})^{n}$ parametrized by $u=(m_{1},\dots,m_{n})\in(\mathbb{Z}_{>0})^{n}$ . Recall that the topological Euler characteristic of a flat torus is equal to $0$ . Similar to discrete spheres, the quandle Euler characteristics of discrete tori approximate the topological Euler characteristics of flat tori.

Theorem 1.7 (Corollary 5.4).

For any positive integer $n$ and any $u=(m_{1},\dots,m_{n})\in(\mathbb{Z}_{>0})^{n}$ with $m_{i}>2$ for all $i$ , the quandle Euler characteristics of the discrete torus $DT^{n}_{u}$ satisfies $\chi^{\mathrm{Qdle}}(DT^{n}_{u})=\chi^{\mathrm{Top}}(T^{n})=0$ .

In Section 5, we study some general properties of quandle Euler characteristics. As a corollary of the Künneth theorem, it is known that the topological Euler characteristic of the product of CW complexes is equal to the product of the topological Euler characteristics of each components (cf. [11]). Note that the direct product of quandles can be defined naturally. We prove a similar statement for the quandle Euler characteristic of the direct product:

Theorem 1.8 (Theorem 5.2).

The quandle Euler characteristic of the direct product of quandles $(X_{1},s^{1})$ and $(X_{2},s^{2})$ satisfies $\chi^{\mathrm{Qdle}}(X_{1}\times X_{2})=\chi^{\mathrm{Qdle}}(X_{1})\cdot\chi^{\mathrm{Qdle}}(X_{2})$ .

For two quandles, one can naturally define a quandle structure on the disjoint union set, which is called the interaction-free union of quandles. In the category of topological spaces, the Euler characteristic of a disjoint union of spaces is equal to the sum of the Euler characteristics of the components, since the homology group of the disjoint union is isomorphic to the direct sum of the homology groups of the individual components. We present an inequality concerning quandle Euler characteristics that mirrors this property:

Theorem 1.9 (Theorem 5.6).

Let $(X_{1},s^{1})$ and $(X_{2},s^{2})$ be quandles. Then, the quandle Euler characteristic of the interaction-free union $X_{1}\sqcup^{\mathrm{free}}X_{2}$ satisfies

\chi^{\mathrm{Qdle}}(X_{1}\sqcup^{\mathrm{free}}X_{2})\leq\chi^{\mathrm{Qdle}}(X_{1})+\chi^{\mathrm{Qdle}}(X_{2}).

Unlike the case of topological Euler characteristics, the equality in the above theorem does not hold in general. We will provide an example that does not achieve this equality (see Example 5.7).

2 Preliminary

In this section, we review some notions of quandles and basic facts on symmetric spaces. In particular, the displacement groups play an important role. First, let us recall the definition of quandles.

Definition 2.1 ([5]).

Let $X$ be a non-empty set and let $\mathrm{Map}(X,X)$ be the set of all maps from $X$ to $X$ . For a map $s:X\to\mathrm{Map}(X,X)$ , the pair $(X,s)$ is called a quandle if the following three conditions hold:

1.

$s_{x}(x)=x$ for any $x\in X$ ,
2.

$s_{x}:X\to X$ is a bijection,
3.

$s_{x}\circ s_{y}=s_{s_{x}(y)}\circ s_{x}$ for any $x,y\in X$ .

For quandles $(X,s^{X})$ and $(Y,s^{Y})$ , a map $f:X\to Y$ is called a quandle homomorphism if $f\circ s_{x}^{X}=s_{f(x)}^{Y}\circ f$ holds for any $x\in X$ . A bijective quandle homomorphism is called a quandle isomorphism.

Definition 2.2.

The quandle automorphism group $\mathrm{Aut}(X)$ is the group consisting of all quandle isomorphisms from $X$ to $X$ . The quandle automorphism group is a group under composition, and acts on $X$ . A quandle $X$ is called homogeneous if its automorphism group acts transitively on $X$ .

Note that the maps $s_{x}:X\to X$ are quandle automorphisms.

Definition 2.3.

The subgroup of $\mathrm{Aut}(X)$ generated by the set $\{s_{x}\mid x\in X\}$ is called the inner automorphism group and is denoted by $\mathrm{Inn}(X)$ . A quandle $X$ is called connected if the inner automorphism group acts transitively on $X$ .

The following group is defined by Joyce [5, §5] as the transvection group.

Definition 2.4 ([5], [3]).

The subgroup of $\mathrm{Inn}(X)$ defined by

\mathrm{Dis}(X)=\langle s_{x}\circ s_{y}^{-1}\mid x,y\in X\rangle_{\mathrm{Grp}}

is called the displacement group of $X$ .

These groups and their actions are studied in detail in [3]. In particular, the actions of the displacement group and the inner automorphism group on a quandle have the same orbits.

According to the definition of a symmetric space given by Loos [7], its point symmetries provide a quandle structure. For this reason, the map $s_{x}:X\to X$ in the definition of a quandle is called the point symmetry at $x$ of $X$ . In the case of symmetric spaces, Loos [7] gave the following properties about the displacement group. These properties will be used in the latter argument.

Proposition 2.5 ([7, Chapter 2, Theorems 2.8 and 3.1]).

Let $X$ be a symmetric space, and suppose that $X$ is connected as a topological space. Then the followings hold:

1.

The displacement group $\mathrm{Dis}(X)$ is a connected Lie group.
2.

$X$ is connected as a quandle, and in particular $X$ is a homogeneous space of $\mathrm{Dis}(X)$ .

At the end of this section, we see the $n$ -dimensional unit sphere as a typical example of symmetric spaces.

Example 2.6.

The $n$ -dimensional unit sphere $S^{n}$ is a compact connected Riemannian symmetric space, and hence the point symmetries give rise to a quandle structure on $S^{n}$ . The point symmetry $s_{p}:S^{n}\to S^{n}$ at a point $p\in S^{n}$ is defined by

s_{p}(x)=2\langle x,p\rangle p-x,

where $\langle\cdot,\cdot\rangle$ is the standard inner product of $\mathbb{R}^{n+1}$ . In this case, the inner automorphism group $\mathrm{Inn}(S^{n})$ and the displacement group $\mathrm{Dis}(S^{n})$ are

	$\displaystyle\mathrm{Inn}(S^{n})$	$\displaystyle=\begin{cases}\mathrm{O}(n+1)&if $n$ is odd,\\ \mathrm{SO}(n+1)&if $n$ is even,\end{cases}$
	$\displaystyle\mathrm{Dis}(S^{n})$	$\displaystyle=\mathrm{SO}(n+1).$

Then, the quandle $S^{n}$ is connected, since the orthogonal group $\mathrm{O}(n+1)$ and the special orthogonal group $\mathrm{SO}(n+1)$ acts transitively on $S^{n}$ . In particular, the displacement group is a connected Lie group for any dimension $n$ .

3 The quandle Euler characteristics

In this section, we define the quandle Euler characteristics by using the displacement group. We also show that this notion is a natural generalization of the topological Euler characteristics of compact connected symmetric spaces.

When a group $G$ acts on a set $X$ , we denote the set of fixed points of this action by $\mathrm{Fix}(G,X)$ . Similarly, we denote the set of fixed points of the action of an element $g\in G$ by $\mathrm{Fix}(g,X)$ . The following is the definition of the quandle Euler characteristics.

Definition 3.1.

Let $X$ be a quandle. Then the quandle Euler characteristic $\chi^{\mathrm{Qdle}}(X)$ is defined by

\chi^{\mathrm{Qdle}}(X):=\inf\{\#\mathrm{Fix}(g,X)\mid g\in\mathrm{Dis}(X)\}.

In order to calculate the quandle Euler characteristic of a given quandle $X$ , one needs to know the displacement group $\mathrm{Dis}(X)$ . Since the displacement group of a trivial quandle is the trivial group, we have the following result.

Example 3.2.

The Euler characteristic of a trivial quandle is equal to its cardinality.

The definition of the quandle Euler characteristic involves a group action. This concept is inspired by a classical result from Hopf and Samelson [Hopf-1941-SatzüberWirkungsräumeGeschlossener]. Let $G$ be a compact connected Lie group and $T$ a maximal torus of $G$ . Then, there exists an element $g_{0}\in T$ such that the topological closure in $G$ of the group generated by $g_{0}$ is equal to $T$ . Such an element $g_{0}\in T$ is called a generator of $T$ . Hopf and Samelson provided a formula for the topological Euler characteristic of a compact connected homogeneous space in terms of a maximal torus and its generator.

Proposition 3.3 ([Hopf-1941-SatzüberWirkungsräumeGeschlossener], see also [Püttmann-2002-HomogeneityRankAtomsActions]).

Let $M$ be a homogeneous space of a compact connected Lie group $G$ . Let us consider a maximal torus $T$ of $G$ , and let $g_{0}\in T$ be a generator of $T$ . Then, the topological Euler characteristic $\chi^{\mathrm{Top}}(M)$ satisfies

\chi^{\mathrm{Top}}(M)=\#\mathrm{Fix}(T,M)=\#\mathrm{Fix}(g_{0},M).

Using this proposition, one can show that the notion of quandle Euler characteristics is a generalization of the notion of topological Euler characteristics of compact connected Riemannian symmetric spaces. Recall that a Riemannian symmetric space is a quandle by the point symmetries, and each point symmetry is an isometry.

Theorem 3.4.

For a compact connected Riemannian symmetric space, the quandle Euler characteristic is equal to the topological Euler characteristic.

Proof.

Let $X$ be a compact connected Riemannian symmetric space. By Proposition 2.5, the displacement group $G:=\mathrm{Dis}(X)$ is a connected Lie group, and acts transitively on $X$ . Since each point symmetry is an isometry, the displacement group $G:=\mathrm{Dis}(X)$ is a subgroup of the isometry group $\mathrm{Isom}(X)$ of $X$ . Since $X$ is a compact Riemannian symmetric space, the isometry group is compact [8, §5], and in particular, $G$ is compact. Let $T$ be a maximal torus in $G$ and let $g_{0}$ be a generator of $T$ . Since $\chi^{\mathrm{Top}}(X)=\#\mathrm{Fix}(g_{0},X)$ by Proposition 3.3, we have

\chi^{\mathrm{Top}}(X)\geq\inf_{g\in G}\mathrm{Fix}(g,X)=\chi^{\mathrm{Qdle}}(X).

It remains to prove the converse inequality. Let us take any $g\in G$ , and we will show

\chi^{\mathrm{Top}}(X)\leq\#\mathrm{Fix}(g,X).

Since $G$ is a compact Lie group, all maximal tori are conjugate. Hence there exists $h\in G$ such that $hgh^{-1}\in T$ . Therefore, it satisfies

\#\mathrm{Fix}(g,X)=\#\mathrm{Fix}(hgh^{-1},X)\geq\#\mathrm{Fix}(T,X)=\chi^{\mathrm{Top}}(X),

which completes the proof. ∎

4 Examples of the quandle Euler characteristics

In this section, we calculate the quandle Euler characteristics of specific examples of quandles.

4.1 The generalized Alexander quandles

As the first example, we consider the generalized Alexander quandles. For a group $G$ and a group automorphism $\sigma\in\mathrm{Aut}(G)$ , the generalized Alexander quandle $\mathrm{GAlex}(G,\sigma)$ is a quandle $(G,s)$ , where the map $s:G\to\mathrm{Map}(G,G)$ is defined by $s_{h}(g)=h\sigma(h^{-1}g)$ for $g,h\in G$ . The inverse of a point symmetry $s_{h}$ is given by $s_{h}^{-1}(g)=h\sigma^{-1}(h^{-1}g)$ . Note that a generalized Alexander quandle $\mathrm{GAlex}(G,\sigma)$ is trivial if and only if the automorphism $\sigma$ is trivial. The next theorem determines the quandle Euler characteristics of generalized Alexander quandles.

Theorem 4.1.

If $\sigma$ is a non-trivial group automorphism of a group $G$ , then the quandle Euler characteristic of the generalized Alexander quandle $\mathrm{GAlex}(G,\sigma)$ is equal to $0$ .

Proof.

Since $\sigma$ is non-trivial, there exists $g\in G$ such that $\sigma(g)\neq g$ . Consider $s_{g}\circ s_{1}^{-1}\in\mathrm{Dis}(\mathrm{GAlex}(G,\sigma))$ for $g\in G$ , where $1$ is the identity of $G$ . It is enough to show that $s_{g}\circ s_{1}$ has no fixed points. For any $x\in G$ , one has

\displaystyle s_{g}\circ s_{1}^{-1}(x)=s_{g}(\sigma^{-1}(x))=g\sigma(g^{-1}\sigma^{-1}(x))=g\sigma(g)^{-1}x.

Since $\sigma(g)\neq g$ , we have $g\sigma(g)^{-1}\neq 1$ , and thus $s_{g}\circ s_{1}^{-1}(x)\neq x$ . This yields that $s_{g}\circ s_{1}^{-1}$ has no fixed points, which completes the proof. ∎

A generalized Alexander quandle is called an Alexander quandle (or an affine quandle) if it is defined by an abelian group $G$ . Recall that the dihedral quandle $R_{n}$ is a typical example of Alexander quandles.

Example 4.2.

The dihedral quandle $R_{n}$ is a quandle isomorphic to $\mathrm{GAlex}(\mathbb{Z}/n\mathbb{Z},\sigma)$ , where $\sigma:\mathbb{Z}/n\mathbb{Z}\to\mathbb{Z}/n\mathbb{Z}$ is the group automorphism defined by $\sigma(a)=-a$ . The dihedral quandles can be realized as discrete subquandles of the circle $S^{1}$ (see Figure 1). Note that $R_{n}$ is non-trivial if $n>2$ . Hence, for any integer $n>2$ , it follows from Theorem 4.1 that

\chi^{\mathrm{Qdle}}(R_{n})=\chi^{\mathrm{Top}}(S^{1})=0.

Refer to caption — Figure 1: The dihedral quandle $R_{5}$ realized as a discrete subquandle of $S^{1}$ .

4.2 The core quandles

In this subsection, we consider the core quandles. For a group $G$ , the core quandle $\mathrm{Core}(G)$ is a quandle $(G,s)$ , where the map $s:G\to\mathrm{Map}(G,G)$ is defined by $s_{h}(g)=hg^{-1}h$ for $g,h\in G$ . In particular, each point symmetry is an involution. Note that the core quandle $\mathrm{Core}(G)$ is trivial if and only if the order of any element in $G$ is less than or equal to $2$ ; in other words, the group $G$ is isomorphic to a direct product of some copies of $\mathbb{Z}/2\mathbb{Z}$ . Recall that any Lie group has a symmetric space structure given by the core quandle structure. The next theorem is an analogy of the fact that the Euler characteristic of a non-trivial compact connected Lie group is 0.

Theorem 4.3.

The quandle Euler characteristic of a non-trivial core quandle is equal to $0$ .

Proof.

Let $G$ be a group. If $G$ is abelian, the quandle $\mathrm{Core}(G)$ is isomorphic to the Alexander quandle $\mathrm{GAlex}(G,\sigma)$ , where the group automorphism $\sigma:G\to G$ is given by $\sigma(g)=g^{-1}$ for $g\in G$ . Since the quandle is non-trivial, the quandle Euler characteristic is equal to $0$ by Theorem 4.1.

Let us assume that $G$ is not abelian. Then, there exist $g,h\in G$ such that $[g,h]:=ghg^{-1}h^{-1}\neq 1$ , where $1$ denotes the identity element. Consider the elements $g_{1}:=s_{g^{-1}h^{-1}}\circ s_{1}^{-1},g_{2}:=s_{h}\circ s_{g^{-1}}^{-1}\in\mathrm{Dis}(\mathrm{Core}(G))$ . For any $x\in\mathrm{Core}(G)$ , we have

	$\displaystyle g_{1}(x)=s_{g^{-1}h^{-1}}\circ s_{1}^{-1}(x)=s_{g^{-1}h^{-1}}(x^{-1})=g^{-1}h^{-1}xg^{-1}h^{-1},$
	$\displaystyle g_{2}(x)=s_{h}\circ s_{g^{-1}}^{-1}(x)=s_{h}(g^{-1}x^{-1}g^{-1})=hgxgh.$

Hence, the map $\varphi:=g_{1}\circ g_{2}\in\mathrm{Dis}(X)$ satisfies

\displaystyle\varphi(x)=g_{1}\circ g_{2}(x)=g_{1}(hgxgh)=g^{-1}h^{-1}(hgxgh)g^{-1}h^{-1}=x[g,h].

Since $[g,h]\neq 1$ , we have $\varphi(x)\neq x$ . Therefore, the map $\varphi$ has no fixed points, which completes the proof. ∎

4.3 Discrete spheres

In this subsection, we determine the Euler characteristics of discrete spheres, which are defined as particular finite subquandles in spheres. Recall that the $n$ -dimensional unit sphere $S^{n}$ is a Riemannian symmetric space. See Example 2.6.

Definition 4.4.

Let $\{e_{i}\}$ be the standard basis of $\mathbb{R}^{n+1}$ . Then the subset $DS^{n}:=\{\pm e_{1},\dots,\pm e_{n+1}\}$ is called the discrete $n$ -sphere.

As in [1, Example 2.4], it is easy to show that $DS^{n}$ is a subquandle in $S^{n}$ . In fact, the point symmetries satisfy

\displaystyle s_{e_{i}}=s_{-e_{i}},\qquad s_{e_{i}}(\pm e_{i})=\pm e_{i},\qquad s_{e_{i}}(\pm e_{j})=\mp e_{j}\quad(\text{for }i\neq j).

Note that these maps are the restrictions of some orthogonal transformations of $\mathbb{R}^{n+1}$ . Hence, we obtain an injective orthogonal representation $\rho:\mathrm{Inn}(DS^{n})\to O(n+1)$ given by

\rho(s_{e_{i}})=\mathrm{diag}(\varepsilon_{1},\dots,\varepsilon_{n+1}),\quad\varepsilon_{j}=\begin{cases*}1&if $i=j$,\\ -1&if $i\neq j$,\end{cases*}

where $\mathrm{diag}(\varepsilon_{1},\dots,\varepsilon_{n+1})$ denotes the diagonal matrix whose $(i,i)$ -entry is $\varepsilon_{i}$ .

Lemma 4.5.

The image $\rho(\mathrm{Dis}(DS^{n}))$ satisfies

\rho(\mathrm{Dis}(DS^{n}))=\{A=\mathrm{diag}(\varepsilon_{1},\dots,\varepsilon_{n+1})\mid\varepsilon_{i}\in\{\pm 1\},\det(A)=1\}.

(1)

Proof.

In this case, it is easy to see that

\displaystyle\rho(\mathrm{Inn}(DS^{n}))\subset\left\{\mathrm{diag}(\varepsilon_{1},\dots,\varepsilon_{n+1})\in O(n+1)\mid\varepsilon_{i}\in\{\pm 1\}\right\}.

Recall that $S=\{s_{e_{i}}\circ s_{e_{j}}^{-1}\}$ is a generating set of $\mathrm{Dis}(DS^{n})$ . For $i\neq j$ , one can obtain the matrix expression

\rho(s_{e_{i}}\circ s_{e_{j}}^{-1})=\mathrm{diag}(\varepsilon_{1},\dots,\varepsilon_{n+1}),\qquad\text{where }\varepsilon_{k}=\begin{cases*}1&$k\not\in\{i,j\}$,\\ -1&$k\in\{i,j\}$.\end{cases*}

Hence, the determinant of any element in $\rho(S)$ is equal to $1$ . Since $\rho(\mathrm{Dis}(DS^{n}))$ is generated by $\rho(S)$ , this shows the inclusion $(\subset)$ in the assertion. In order to prove the inverse inclusion, let us take $B=\mathrm{diag}(\varepsilon_{1},\dots,\varepsilon_{n+1})$ in the right-hand side of the assertion. Then the number of indices $i$ with $\varepsilon_{i}=1$ is even. Therefore the matrix $B$ can be written as a product of elements in $\rho(S)$ . This completes the proof. ∎

In the following, we identify $\mathrm{Inn}(DS^{n})$ with $\rho(\mathrm{Inn}(DS^{n}))$ , and also $\mathrm{Dis}(DS^{n})$ with $\rho(\mathrm{Dis}(DS^{n}))$ . The next theorem determines the quandle Euler characteristics of the discrete spheres.

Theorem 4.6.

For any positive integer $n$ , it satisfies $\chi^{\mathrm{Qdle}}(DS^{n})=\chi^{\mathrm{Top}}(S^{n})$ , that is,

\chi^{\mathrm{Qdle}}(DS^{n})=\begin{cases*}0&if $n$ is odd,\\ 2&if $n$ is even.\end{cases*}

Proof.

Suppose that $n$ is odd. Then the displacement group $\mathrm{Dis}(DS^{n})$ contains the identity matrix $-I_{n+1}$ . The matrix $-I_{n+1}$ has no fixed points in $DS^{n}$ . Therefore we have $\chi^{\mathrm{Qdle}}(DS^{n})=0$ .

Suppose that $n$ is even. Then we have

A_{0}=\left(\begin{matrix}1&0\\ 0&-I_{n}\end{matrix}\right)\in\mathrm{Dis}(DS^{n}).

It is clear that $A_{0}$ fixes two elements $\pm e_{1}$ in $DS^{n}$ , which shows $\chi^{\mathrm{Qdle}}(DS^{n})\leq 2$ . Recall that any element $g\in\mathrm{Dis}(DS^{n})$ can be expressed as

\displaystyle g=\mathrm{diag}(\varepsilon_{1},\dots,\varepsilon_{n+1}),

where $\varepsilon_{i}\in\{\pm 1\}$ and $\det(g)=1$ . Since $n+1$ is odd and hence $\#\{i\mid\varepsilon_{i}=1\}$ is odd, we have $\#\{i\mid\varepsilon_{i}=1\}\geq 1$ . Therefore, $g$ has at least two fixed points in $DS^{n}$ . This shows $\chi^{\mathrm{Qdle}}(DS^{n})\geq 2$ , which completes the proof. ∎

4.4 Quandles obtained from weighted graphs

In this subsection, we study the quandle Euler characteristics of the quandles obtained from weighted graphs, as introduced in [10]. See also [1].

Proposition 4.7 ([10]).

Let $A$ be an abelian group, $V$ a finite set and $d:V\times V\to A$ a map with $d(v,v)=0$ for all $v\in V$ . Then the following map $s:V\times A\to\mathrm{Map}(V\times A)$ is a quandle structure on $V\times A$ :

s_{(v,a)}(w,b)=(w,d(v,w)+b).

The constructed quandle is called the quandle obtained from $(V,A,d)$ , and denoted by $V\times_{d}A$ . Note that $(V,A,d)$ is called an $A$ -weighted graph, whose edges are weighted by elements in $A$ . More precisely, $(V,A,d)$ is an oriented graph with the vertex set $V$ , the edge set $E=\{(v,w)\mid v,w\in V,d(v,w)\neq 0\}$ , and the weight of the edge $(v,w)\in E$ is given by $d(v,w)$ . We express a weighted-edge $(v,w)\in E$ by an arrow from $v$ to $w$ that is labeled by $d(v,w)$ .

Let us denote an $n$ -point set by $V=V_{n}=\{v_{1},\dots,v_{n}\}$ for a positive integer $n$ . For an $A$ -weighted graph $(V,A,d)$ , let us define the adjacency matrix $D$ with respect to $(V,A,d)$ by $D:=(d(v_{i},v_{j}))_{i,j\in\{1,\dots,n\}}$ , and denote the $i$ -th row vector of $D$ by $d_{i}$ .

Remark 4.8.

For any $a,b\in A$ and any $v\in V_{n}$ , the point symmetries satisfy $s_{(v,a)}=s_{(v,b)}$ . In the following, we denote the point symmetry at $(v,a)\in V_{n}\times_{d}A$ by $s_{v}$ . Note that $s_{v_{i}}$ is described by $d_{i}$ , which is the $i$ -th row vector of the adjacency matrix $D$ . Hence, $s_{v_{i}}=s_{v_{j}}$ as inner automorphisms of $V_{n}\times_{d}A$ if and only if $d_{i}=d_{j}$ . We can regard $\mathrm{Inn}(V_{n}\times_{d}A)$ as a subgroup of $A^{n}$ by the map $\iota:\mathrm{Inn}(V_{n}\times_{d}A)\to A^{n}$ defined by $\iota(s_{v_{i}})=d_{i}$ . In the following we identify $\mathrm{Inn}(V_{n}\times_{d}A)$ with its image of $\iota$ .

Remark 4.9.

It was proved in [10] that the inner automorphism group $\mathrm{Inn}(V\times_{d}A)$ is abelian, and every finite homogeneous quandle with abelian inner automorphism group can be constructed in this way. An $A$ -weighted graph $(V,A,d)$ is said to be homogeneous if there exists a graph isomorphism $f:(V,E)\to(V,E)$ such that $d(f(v),f(w))=d(v,w)$ for any $v,w\in V$ . If an $A$ -weighted graph $(V,A,d)$ is homogeneous, then $V_{n}\times_{d}A$ is a homogeneous quandle [10, Theorem A].

Example 4.10.

The discrete sphere $DS^{n}$ is isomorphic to the quandle $V_{n+1}\times_{d}(\mathbb{Z}/2\mathbb{Z})$ , where the map $d:V_{n+1}\times V_{n+1}\to(\mathbb{Z}/2\mathbb{Z})$ is defined by

d(v,w)=\begin{cases*}1&if $v\neq w$,\\ 0&if $v=w$.\end{cases*}

The corresponding $A$ -weighted graph is a perfect graph with $n+1$ vertexes, all of whose edges are weighted by $1\in\mathbb{Z}/2\mathbb{Z}$ . In the case $n=3$ , the corresponding $A$ -weighted graph is described in Figure 2, and the corresponding adjacency matrix $D$ is

D=\begin{pmatrix}0&1&1&1\\ 1&0&1&1\\ 1&1&0&1\\ 1&1&1&0\\ \end{pmatrix}.

The next proposition is useful for calculating the displacement group and the quandle Euler characteristic of the quandle $V\times_{d}A$ .

Proposition 4.11.

Let $V_{n}\times_{d}A$ be the quandle obtained from an $A$ -weighted graph $(V,A,d)$ , and $d_{i}$ be the $i$ -th row vector of the adjacency matrix $D$ . Then we have the following:

$(1)$: $\mathrm{Dis}(V_{n}\times_{d}A)=\left\{\sum_{i=0}^{n-1}k_{i}(d_{i}-d_{i+1})\in A^{n}\mid k_{i}\in\mathbb{Z}\right\}$ ,
$(2)$: $\chi^{\mathrm{Qdle}}(V_{n}\times_{d}A)=\#A\cdot\inf\left\{\#\{i\mid a_{i}=0\}\mid(a_{1},\dots,a_{n})\in\mathrm{Dis}(V_{n}\times_{d}A)\right\}$ .

Proof.

We show $(1)$ . Recall that the displacement group $G:=\mathrm{Dis}(V_{n}\times_{d}A)$ is generated by

\{s_{v_{i}}\circ s_{v_{j}}^{-1}\mid v_{i},v_{j}\in V_{n}\}=\{d_{i}-d_{j}\mid i,j\in\{1,\dots n\}\}.

Since $(d_{i}-d_{j})=-(d_{j}-d_{i})$ for any $i,j$ , and $d_{i}-d_{j}=\sum_{k=i}^{j}(d_{k}-d_{k+1})$ for $j>i$ , we can replace the generating set to $\{d_{i}-d_{i+1}\mid i\in\{1,\dots,n-1\}\}$ . Therefore we obtain the assertion.

We show $(2)$ . An element $a=(a_{1},\dots,a_{n})\in\mathrm{Dis}(V_{n}\times_{d}A)$ acts on $V_{n}\times_{d}A$ by $a\cdot(v_{i},b)=(v_{i},b+a_{i})$ . Hence the element $(v_{i},b)\in V_{n}\times_{d}A$ is fixed by $a$ if and only if $a_{i}=0$ . Note that if there exists an element $(v_{i},b_{0})\in V_{n}\times_{d}A$ fixed by $a$ , then for any $b\in A$ , every element $(v_{i},b)\in V_{n}\times_{d}A$ is fixed by $a$ . Hence the number $\#A\cdot\#\{i\mid a_{i}=0\}$ is equal to the number of elements fixed by $a$ . This completes the proof. ∎

Now we calculate quandle Euler characteristics of a concrete example of $(V,A,d)$ . This example will be used to construct a quandle satisfying particular inequality (see Theorem 5.6 and Example 5.7).

Proposition 4.12.

Let $n$ be a positive integer with $n>1$ , and consider $V_{n}=\{v_{1},\dots,v_{n}\}$ . Let us define a map $d:V_{n}\times V_{n}\to\mathbb{Z}/2\mathbb{Z}$ by

d(v_{i},v_{j})=\begin{cases*}1&if $i-j=1$ in $\mathbb{Z}/n\mathbb{Z}$,\\ 0&otherwise.\end{cases*}

Then $C_{n}:=V\times_{d}(\mathbb{Z}/2\mathbb{Z})$ is a non-trivial homogeneous quandle and the quandle Euler characteristic is given by

\chi^{\mathrm{Qdle}}(C_{n})=\begin{cases*}2&if $n$ is odd,\\ 0&if $n$ is even.\end{cases*}

Proof.

Since $d\neq 0$ and the weighted graph $(V_{n},\mathbb{Z}/2\mathbb{Z},d)$ is homogeneous, the quandle $C_{n}$ is non-trivial and homogeneous. In this case, the $i$ -th row vector of the adjacency matrix is given by

d_{i}=e_{i-1}^{T},

where ^T denotes the transpose. By Proposition 4.11 $(1)$ , we have

\mathrm{Dis}(C_{n})=\left\{(a_{1},\dots,a_{n})\in(\mathbb{Z}/2\mathbb{Z})^{n}\mid a_{k}\in\mathbb{Z}/2\mathbb{Z},\,\textstyle{\sum_{k=1}^{n}}a_{k}=0\right\}.

If $n$ is even, then $(1,\dots,1)\in\mathrm{Dis}(C_{n})$ has no fixed points, and hence the quandle Euler characteristic is equal to $0$ . If $n$ is odd, then $(1,\dots,1,0)\in\mathrm{Dis}(C_{n})$ and any element in $\mathrm{Dis}(C_{n})$ has at least one entry that is $0$ . Hence the Euler characteristic is equal to $\#(\mathbb{Z}/2\mathbb{Z})=2$ by Proposition 4.11 $(2)$ . This completes the proof. ∎

The next proposition shows that, given positive integer $n>1$ , there exists a non-trivial quandle whose Euler characteristic is equal to $n$ .

Proposition 4.13.

Let $n>1$ be a positive integer, and consider $V_{2}=\{v_{1},v_{2}\}$ . Let us define a map $d:V_{2}\times V_{2}\to\mathbb{Z}/n\mathbb{Z}$ by

d(v_{i},v_{j})=\begin{cases*}1&if $(i,j)=(1,2)$,\\ 0&otherwise.\end{cases*}

Then $B_{n}:=V\times_{d}\mathbb{Z}/n\mathbb{Z}$ is a non-trivial quandle and the quandle Euler characteristic is given by

\chi^{\mathrm{Qdle}}(B_{n})=n.

Proof.

Since $d$ is not the zero-map, $B_{n}$ is non-trivial. In this case, the displacement group satisfies

\mathrm{Dis}(B_{n})=\left\{(0,k)\in(\mathbb{Z}/n\mathbb{Z})^{2}\mid k\in\mathbb{Z}/n\mathbb{Z}\right\}

by Proposition 4.11. Since the first entry of any element in $W$ is $0$ , the Euler characteristic of $B_{n}$ is equal to $\#(\mathbb{Z}/n\mathbb{Z})=n$ . This completes the proof. ∎

5 Properties of the quandle Euler characteristics

In this section, we prove several properties of the quandle Euler characteristics. These properties can be regarded as analogous to certain properties of topological Euler characteristics.

5.1 The direct product of quandles

Recall that the Euler characteristic of a product of CW complexes is equal to the direct product of the Euler characteristics of the individual components. In this subsection, we show that the same equality holds for the quandle Euler characteristic of the product of quandles.

For quandles $(X_{1},s^{1})$ and $(X_{2},s^{2})$ , we can define a quandle structure $s$ on the direct product $X_{1}\times X_{2}$ by setting $s_{(x_{1},x_{2})}:=s^{1}_{x_{1}}\times s^{2}_{x_{2}}$ , that is

s_{(x_{1},x_{2})}(y_{1},y_{2})=(s^{1}_{x_{1}}(y_{1}),s^{2}_{x_{2}}(y_{2})).

The quandle $(X_{1}\times X_{2},s)$ is called the direct product of quandles $(X_{1},s^{1})$ and $(X_{2},s^{2})$ , and for simplicity, we denote it as $X_{1}\times X_{2}$ .

Lemma 5.1.

Let $(X_{1},s^{1})$ and $(X_{2},s^{2})$ be quandles. Then

$(1)$: the group homomorphism $f:\mathrm{Inn}(X_{1}\times X_{2})\to\mathrm{Inn}(X_{1})\times\mathrm{Inn}(X_{2})$ defined by $f(s_{(x_{1},x_{2})})=(s^{1}_{x_{1}},s^{2}_{x_{2}})$ is injective,
$(2)$: $f(\mathrm{Dis}(X_{1}\times X_{2}))=\mathrm{Dis}(X_{1})\times\mathrm{Dis}(X_{2})$ .

Proof.

We show $(1)$ , that is $\ker(f)=\{1\}$ . Let us take $g\in\ker(f)$ . Then there exist $x_{1},\dots,x_{n}\in X_{1}\times X_{2}$ and $\varepsilon_{i}\in\{\pm 1\}$ such that $g=s_{x_{1}}^{\varepsilon_{1}}\circ\cdots\circ s_{x_{n}}^{\varepsilon_{n}}$ . Here, it follows $s_{x_{i}}=s_{x_{i}^{1}}\times s_{x_{i}^{2}}$ for each $x_{i}=(x_{i}^{1},x_{i}^{2})\in X_{1}\times X_{2}$ by the definition of the point symmetries of the product quandle. By the definition of $f$ , one has

f(g)=((s^{1}_{x_{1}^{1}})^{\varepsilon_{1}}\circ\cdots\circ(s^{1}_{x_{n}^{1}})^{\varepsilon_{n}},(s^{2}_{x_{1}^{2}})^{\varepsilon_{1}}\circ\cdots\circ(s^{2}_{x_{n}^{2}})^{\varepsilon_{n}})=:(g_{1},g_{2})\in\mathrm{Inn}(X_{1})\times\mathrm{Inn}(X_{2}).

Since $g\in\ker(f)$ , we have $g_{i}$ is trivial for $i\in\{1,2\}$ . On the other hand, it satisfies

	$\displaystyle g_{1}\times g_{2}$	$\displaystyle=\left((s^{1}_{x_{1}^{1}})^{\varepsilon_{1}}\circ\cdots\circ(s^{1}_{x_{n}^{1}})^{\varepsilon_{n}}\right)\times\left((s^{2}_{x_{1}^{2}})^{\varepsilon_{1}}\circ\cdots\circ(s^{2}_{x_{n}^{2}})^{\varepsilon_{n}}\right)$
		$\displaystyle=\left((s^{1}_{x_{1}^{1}})^{\varepsilon_{1}}\times(s^{2}_{x_{1}^{2}})^{\varepsilon_{1}}\right)\circ\cdots\circ\left((s^{1}_{x_{n}^{1}})^{\varepsilon_{n}}\times(s^{2}_{x_{n}^{2}})^{\varepsilon_{n}}\right)$
		$\displaystyle=s_{x_{1}}^{\varepsilon_{1}}\circ\cdots\circ s_{x_{n}}^{\varepsilon_{n}}=g.$

This yields that $g=1$ . Therefore the homomorphism $f$ is injective as desired.

We show $(2)$ . First, we demonstrate that $f(\mathrm{Dis}(X_{1}\times X_{2}))\subset\mathrm{Dis}(X_{1})\times\mathrm{Dis}(X_{2})$ . The left-hand side is generated by

D=\{(s^{1}_{x_{1}}\circ(s^{1}_{y_{1}})^{-1},s^{2}_{x_{2}}\circ(s^{2}_{y_{2}})^{-1})\mid(x_{1},x_{2}),(y_{1},y_{2})\in X_{1}\times X_{2}\}.

Since $s_{x_{k}}^{i}\circ(s_{y_{k}}^{i})^{-1}\in\mathrm{Dis}(X_{i})$ , the right-hand side contains $D$ , which shows $(\subset)$ . Next, we prove the converse inclusion $(\supset)$ . The right-hand side $\mathrm{Dis}(X_{1})\times\mathrm{Dis}(X_{2})$ is generated by

D^{\prime}=\{(s^{1}_{x_{1}}\circ(s^{1}_{y_{1}})^{-1},1)\mid x_{1},y_{1}\in X_{1}\}\cup\{(1,s^{2}_{x_{2}}\circ(s^{2}_{y_{2}})^{-1})\mid x_{2},y_{2}\in X_{2}\}.

The left-hand side contains $D^{\prime}$ . In fact, we have

	$\displaystyle(s^{1}_{x_{1}},s^{2}_{x_{2}})\cdot(s^{1}_{y_{1}},s^{2}_{x_{2}})^{-1}$	$\displaystyle=(s^{1}_{x_{1}}\circ(s^{1}_{y_{1}})^{-1},1),$
	$\displaystyle(s^{1}_{x_{1}},s^{2}_{x_{2}})\cdot(s^{1}_{x_{1}},s^{2}_{y_{2}})^{-1}$	$\displaystyle=(1,s^{2}_{x_{2}}\circ(s^{2}_{y_{2}})^{-1}).$

Since both belong to the left-hand side, it follows $(\supset)$ , which completes the proof. ∎

By this lemma, the displacement group of the direct product quandle is isomorphic to the product group of the displacement groups of the individual components. Using this property, we obtain the following result.

Theorem 5.2.

Proof.

We identify the group $\mathrm{Dis}(X_{1}\times X_{2})$ with $\mathrm{Dis}(X_{1})\times\mathrm{Dis}(X_{2})$ by Lemma 5.1. Hence any element $g\in\mathrm{Dis}(X_{1})$ can be written as $g=(g_{1},g_{2})$ with $g_{1}\in\mathrm{Dis}(X_{1})$ and $g_{2}\in\mathrm{Dis}(X_{2})$ . Since the action is given by $(g_{1},g_{2}).(x_{1},x_{2})=(g_{1}(x_{1}),g_{2}(x_{2}))$ , it follows that the set of fixed points satisfies

\mathrm{Fix}((g_{1},g_{2}),X_{1}\times X_{2})=\mathrm{Fix}(g_{1},X_{1})\times\mathrm{Fix}(g_{2},X_{2}).

Therefore, $(g_{1},g_{2})$ attains the infimum of $\#\mathrm{Fix}((g_{1},g_{2}),X_{1}\times X_{2})$ if and only if both of $g_{1}$ and $g_{2}$ attain the infimums of $\#\mathrm{Fix}(g_{1},X_{1})$ and $\#\mathrm{Fix}(g_{2},X_{2})$ , respectively. This completes the proof of the desired equality. ∎

As an application of Theorem 5.2, one can calculate the quandle Euler characteristics of the discrete tori. As seen in Example 4.2, the dihedral quandles can be regarded as discrete subquandles of the circle $S^{1}$ . Thus, we define the discrete tori as follows:

Definition 5.3.

For a positive integer $n$ and an integer vector $u=(m_{1},\dots,m_{n})\in(\mathbb{Z}_{>0})^{n}$ , the discrete torus $DT^{n}_{u}$ is defined by

DT^{n}_{u}:=\prod_{k=1}^{n}R_{m_{k}}.

According to the classification given in [4], a flat connected finite quandle is a discrete torus. Recall that a compact connected flat Riemannian symmetric space is just a flat torus $T^{n}=(S^{1})^{n}$ , and its topological Euler characteristic is equal to $0$ . The Euler characteristic of a discrete torus has the same property:

Corollary 5.4.

For any positive integer $n$ and any $u=(m_{1},\dots,m_{n})\in(\mathbb{Z}_{>0})^{n}$ with $m_{i}>2$ , the quandle Euler characteristics of the discrete torus $DT^{n}_{u}$ satisfies $\chi^{\mathrm{Qdle}}(DT^{n}_{u})=\chi^{\mathrm{Top}}(T^{n})=0$ .

Proof.

Since $m_{k}>2$ , any component $R_{m_{k}}$ is non-trivial. As seen in Example 4.2, the quandle Euler characteristics of $R_{m_{k}}$ is equal to $0$ . By applying Theorem 5.2, we have the quandle Euler characteristics of discrete torus is equal to $0$ , which completes of proof. ∎

5.2 The interaction-free union of quandles

Recall that the Euler characteristic of a disjoint union of topological spaces is equal to the sum of the Euler characteristics of the individual components. In this subsection, we provide a comparable inequality for the Euler characteristics of the interaction-free union of quandles.

For two quandles $(X_{1},s^{1})$ and $(X_{2},s^{2})$ , we can define a quandle structure $s$ on the disjoint union $X_{1}\sqcup X_{2}$ by setting

s_{x}(y)=\begin{cases*}y&if $\{x,y\}\not\subset X_{0},X_{1}$,\\ s_{x}^{i}(y)&if $\{x,y\}\subset X_{i}$.\end{cases*}

The quandle $(X_{1}\sqcup X_{2},s)$ is called the interaction-free union of quandles $(X_{1},s^{1})$ and $(X_{2},s^{2})$ , and for simplicity, we denote it by $X_{1}\sqcup^{\mathrm{free}}X_{2}$ . Note that the natural inclusion map $\iota_{i}:X_{i}\to X_{1}\sqcup^{\mathrm{free}}X_{2}$ is an injective quandle homomorphism and induces a group homomorphism $\iota_{i}:\mathrm{Inn}(X_{i})\to\mathrm{Inn}(X_{1}\sqcup^{\mathrm{free}}X_{2})$ .

Lemma 5.5.

Let $(X_{1},s^{1})$ and $(X_{2},s^{2})$ be quandles. Let us define the map $\iota:\mathrm{Inn}(X_{1})\times\mathrm{Inn}(X_{2})\to\mathrm{Inn}(X_{1}\sqcup^{\mathrm{free}}X_{2})$ by $\iota(g_{1},g_{2})=\iota_{1}(g_{1})\iota_{2}(g_{2})$ . Then,

$(1)$

the map $\iota:\mathrm{Inn}(X_{1})\times\mathrm{Inn}(X_{2})\to\mathrm{Inn}(X_{1}\sqcup^{\mathrm{free}}X_{2})$ is a group isomorphism,
$(2)$

$\iota(\mathrm{Dis}(X_{1})\times\mathrm{Dis}(X_{2}))\subset\mathrm{Dis}(X_{1}\sqcup^{\mathrm{free}}X_{2})$ .

Proof.

We show $(1)$ , that is, $\iota$ is a group homomorphism and is a bijection. First, we show that $\iota$ is a group homomorphism. For $x_{1}\in X_{1}$ and $x_{2}\in X_{2}$ , the maps $s_{x_{1}}$ and $s_{x_{2}}$ are commutative in $\mathrm{Inn}(X_{1}\sqcup^{\mathrm{free}}X_{2})$ by the definition of the interaction-free union. Thus, an element in $\iota_{1}(X_{1})$ and an element in $\iota_{2}(X_{2})$ are commutative. Hence for $(g_{1},g_{2}),(h_{1},h_{2})\in\mathrm{Inn}(X_{1})\times\mathrm{Inn}(X_{2})$ , we have

	$\displaystyle\iota((g_{1},g_{2})(h_{1},h_{2}))$	$\displaystyle=\iota(g_{1}h_{1},g_{2}h_{2})$
		$\displaystyle=\iota_{1}(g_{1}h_{1})\iota_{2}(g_{2}h_{2})$
		$\displaystyle=\iota_{1}(g_{1})\iota_{1}(h_{1})\iota_{2}(g_{2})\iota_{2}(h_{2})$
		$\displaystyle=\iota_{1}(g_{1})\iota_{2}(g_{2})\iota_{1}(h_{1})\iota_{2}(h_{2})$
		$\displaystyle=\iota(g_{1},g_{2})\iota(h_{1},h_{2}).$

Therefore the map $\iota$ is a group homomorphism. Next, we show that $\iota$ is surjective. To prove this, let us consider

S:=\{(s^{1}_{x_{1}},1)\mid x_{1}\in X_{1}\}\sqcup\{(1,s^{2}_{x_{2}})\mid x_{2}\in X_{2}\},

which is a generating set of $\mathrm{Inn}(X_{1})\times\mathrm{Inn}(X_{2})$ . Note that $\iota(s_{x_{1}}^{1},1)=s_{x_{1}}$ and $\iota(1,s_{x_{2}}^{2})=s_{x_{2}}$ . Then the image $\iota(S)$ satisfies

\iota(S)=\{s_{x}\in\mathrm{Inn}(X_{1}\sqcup^{\mathrm{free}}X_{2})\mid x\in X_{1}\sqcup X_{2}\},

which is a generating set of $\mathrm{Inn}(X_{1}\sqcup^{\mathrm{free}}X_{2})$ . Thus, the group homomorphism $\iota$ is surjective. Lastly, we show that $\iota$ is injective, that is $\ker\iota=\{1\}$ . Let us take $(g_{1},g_{2})\in\ker(\iota)$ , where $g_{1}\in\mathrm{Inn}(X_{1})$ and $g_{2}\in\mathrm{Inn}(X_{2})$ . Then the map $\iota(g_{1},g_{2})=\iota_{1}(g_{1})\iota_{2}(g_{2})$ acts trivially on $X_{1}\sqcup^{\mathrm{free}}X_{2}$ . In particular, the element $\iota_{1}(g_{1})=\iota(g_{1},g_{2})\iota_{2}(g_{2})^{-1}$ acts trivially on $X_{1}$ . Thus we conclude that $g_{1}$ is trivial, and similarly we conclude that $g_{2}$ is trivial. Therefore, we have $(g_{1},g_{2})=(1,1)$ as desired.

We show $(2)$ . Let us consider

S^{\prime}:=\{(s^{1}_{x_{1}}\circ(s^{1}_{y_{1}})^{-1},1)\mid x_{1},y_{1}\in X_{1}\}\sqcup\{(1,s^{2}_{x_{2}}\circ(s^{2}_{y_{2}})^{-1})\mid x_{2},y_{2}\in X_{2}\},

which is a generating set of $\mathrm{Dis}(X_{1})\times\mathrm{Dis}(X_{2})$ . Then the image $\iota(S^{\prime})$ is given by

\iota(S^{\prime})=\{s_{x_{1}}\circ(s_{y_{1}})^{-1}\mid x_{1},y_{1}\in X_{1}\}\sqcup\{s_{x_{2}}\circ(s_{y_{2}})^{-1}\mid x_{2},y_{2}\in X_{2}\},

which is a subset of the group $\mathrm{Dis}(X_{1}\sqcup^{\mathrm{free}}X_{2})$ . Since the group $\iota(\mathrm{Dis}(X_{1})\times\mathrm{Dis}(X_{2}))$ is generated by $\iota(S^{\prime})$ , we have $(2)$ , which completes the proof. ∎

By this lemma, we can regard the product group of the displacement groups of the individual components as a subgroup of the displacement group of the interaction-free union quandle. Using this property, we obtain the following result.

Theorem 5.6.

Let $(X_{1},s^{1})$ and $(X_{2},s^{2})$ be quandles. Then, the quandle Euler characteristic of the interaction-free union $X_{1}\sqcup^{\mathrm{free}}X_{2}$ satisfies

\chi^{\mathrm{Qdle}}(X_{1}\sqcup^{\mathrm{free}}X_{2})\leq\chi^{\mathrm{Qdle}}(X_{1})+\chi^{\mathrm{Qdle}}(X_{2}).

Proof.

We regard $\mathrm{Dis}(X_{1})\times\mathrm{Dis}(X_{2})$ as a subgroup of $\mathrm{Dis}(X_{1}\sqcup^{\mathrm{free}}X_{2})$ by the map $\iota$ in Lemma 5.5. For each $i\in\{1,2\}$ , there exists $g_{i}\in\mathrm{Dis}(X_{i})$ such that

\#\mathrm{Fix}(g_{i},X_{i})=\chi^{\mathrm{Qdle}}(X_{i}).

Then $g:=g_{1}g_{2}$ is regarded as an element in $\mathrm{Dis}(X_{1}\sqcup^{\mathrm{free}}X_{2})$ . Since $g_{i}$ acts trivially on the other component, it follows that the set of fixed points satisfies

\mathrm{Fix}(g,X_{1}\sqcup^{\mathrm{free}}X_{2})=\mathrm{Fix}(g_{1},X_{1})\sqcup\mathrm{Fix}(g_{2},X_{2}).

This concludes that

\chi^{\mathrm{Qdle}}(X_{1}\sqcup^{\mathrm{free}}X_{2})\leq\#\mathrm{Fix}(g,X_{1}\sqcup^{\mathrm{free}}X_{2})=\#\mathrm{Fix}(g_{1},X_{1})+\#\mathrm{Fix}(g_{2},X_{2})=\chi^{\mathrm{Qdle}}(X_{1})+\chi^{\mathrm{Qdle}}(X_{2}),

which completes the proof. ∎

The following provides an example that does not satisfy the equality in Theorem 5.6. We will use a quandle obtained from a weighted graph. Note that this quandle is homogeneous, and therefore, the equality in Theorem 5.6 does not hold in general, even for homogeneous quandles.

Example 5.7.

Let $C_{3}$ be the quandle in Proposition 4.12. Then we have

\chi^{\mathrm{Qdle}}(C_{3}\sqcup^{\mathrm{free}}C_{3})=0<4=\chi^{\mathrm{Qdle}}(C_{3})+\chi^{\mathrm{Qdle}}(C_{3}).

Proof.

Since we proved $\chi^{\mathrm{Qdle}}(C_{3})=2$ in Proposition 4.12, we have only to show that the Euler characteristic of $C_{3}\sqcup^{\mathrm{free}}C_{3}$ is equal to $0$ . Recall that $C_{3}$ is obtained from the $\mathbb{Z}/2\mathbb{Z}$ -weighted graph in Figure 3. We have that $C_{3}\sqcup^{\mathrm{free}}C_{3}$ coincides with the quandle obtained from the $\mathbb{Z}/2\mathbb{Z}$ -weighted graph in Figure 5, which is the disjoint union of two copies of the graph of $C_{3}$ . With respect to the labeling in Figure 5, the corresponding adjacency matrix $D$ is given by

D=\begin{pmatrix}0&1&0&0&0&0\\ 0&0&1&0&0&0\\ 1&0&0&0&0&0\\ 0&0&0&0&1&0\\ 0&0&0&0&0&1\\ 0&0&0&1&0&0\end{pmatrix}.

It follows from Proposition 4.11 that

(1,1,1,1,1,1)=(d_{1}-d_{2})+(d_{3}-d_{4})+(d_{5}-d_{6})\in\mathrm{Dis}(C_{3}\sqcup^{\mathrm{free}}C_{3}).

Since this element has no fixed points, we have $\chi^{\mathrm{Qdle}}(C_{3}\sqcup^{\mathrm{free}}C_{3})=0$ by Proposition 4.11 $(2)$ . ∎

Acknowledgement

The authors would like to thank Hirotaka Akiyoshi, Katsunori Arai, Seiichi Kamada, Akira Kubo, Fumika Mizoguchi, Takayuki Okuda, Makoto Sakuma, and Yuta Taniguchi for helpful comments and useful discussions. The first author was supported by JST SPRING, Grant Number JPMJSP2139. The second author was supported by JSPS KAKENHI Grant Numbers JP22H01124 and JP24K21193. The authors were partly supported by MEXT Promotion of Distinctive Joint Research Center Program JPMXP00723833165.

References

[1] Furuki, K., Tamaru, H.: Homogeneous Quandles with Abelian Inner Automorphism Groups and Vertex-Transitive Graphs. International Electronic Journal of Geometry 17(1), 184–198 (2024). DOI 10.36890/iejg.1438745
[2] Hopf, H., Samelson, H.: Ein Satz über die Wirkungsräume geschlossener Liescher Gruppen. Commentarii Mathematici Helvetici 13, 240–251 (1941). DOI 10.1007/BF01378063
[3] Hulpke, A., Stanovský, D., Vojtěchovský, P.: Connected quandles and transitive groups. Journal of Pure and Applied Algebra 220(2), 735–758 (2016). DOI 10.1016/j.jpaa.2015.07.014
[4] Ishihara, Y., Tamaru, H.: Flat connected finite quandles. Proceedings of the American Mathematical Society 144(11), 4959–4971 (2016). DOI 10.1090/proc/13095
[5] Joyce, D.: A classifying invariant of knots, the knot quandle. Journal of Pure and Applied Algebra 23(1), 37–65 (1982). DOI 10.1016/0022-4049(82)90077-9
[6] Kubo, A., Nagashiki, M., Okuda, T., Tamaru, H.: A commutativity condition for subsets in quandles — a generalization of antipodal subsets. In: B.Y. Chen, N. Brubaker, T. Sakai, B. Suceavă, M. Tanaka, H. Tamaru, M. Vajiac (eds.) Contemporary Mathematics, vol. 777, pp. 103–125. American Mathematical Society, Providence, Rhode Island (2022). DOI 10.1090/conm/777/15631
[7] Loos, O.: Symmetric spaces. I: General theory. W. A. Benjamin, Inc., New York-Amsterdam (1969)
[8] Myers, S.B., Steenrod, N.E.: The Group of Isometries of a Riemannian Manifold. The Annals of Mathematics 40(2), 400 (1939). DOI 10.2307/1968928
[9] Püttmann, T.: Homogeneity rank and atoms of actions. Annals of Global Analysis and Geometry 22(4), 375–399 (2002). DOI 10.1023/A:1020540622377
[10] Saito, T., Sugawara, S.: Homogeneous quandles with abelian inner automorphism groups. Journal of Algebra 663, 150–170 (2025). DOI 10.1016/j.jalgebra.2024.09.004
[11] Weintraub, S.H.: Fundamentals of Algebraic Topology, Graduate Texts in Mathematics, vol. 270. Springer New York, New York, NY (2014). DOI 10.1007/978-1-4939-1844-7

(R. Kai) Department of Mathematics, Graduate School of Science, Osaka Metropolitan University, 3-3-138, Sugimoto, Sumiyoshi-ku, Osaka, 558-8585, Japan

Email address: [email protected]

(H. Tamaru) Department of Mathematics, Graduate School of Science, Osaka Metropolitan University, 3-3-138, Sugimoto, Sumiyoshi-ku, Osaka, 558-8585, Japan

Email address: [email protected]