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eISSN 0454-8124
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### Article

Kyungpook Mathematical Journal 2020; 60(2): 279-288

Published online June 30, 2020

### Torsion in Homology of Dihedral Quandles of Even Order

Seung Yeop Yang

Department of Mathematics, Kyungpook National University, Daegu, 41566, Republic of Korea
e-mail : seungyeop.yang@knu.ac.kr

Received: March 4, 2020; Revised: March 30, 2020; Accepted: March 31, 2020

Niebrzydowski and Przytycki conjectured that the torsion of rack and quandle homology of a dihedral quandle of order 2k is annihilated by k, unless k = 2t for t > 1. We partially prove this conjecture.

Keywords: dihedral quandle, rack and quandle homology, torsion subgroup

### 1. Introduction

Rack [7] and quandle [4] homology are (co)homology theories for self-distributive structures with axioms obtained diagrammatically from Reidemeister moves in classical knot theory. Various homological and homotopical knot invariants have been developed based on these (co)homology theories. More precisely, quandle cocycle invariants [4] are constructed with cocycles of quandle homology and quandle homotopy invariants [13, 17] are obtained using homotopy classes of maps from spheres to the geometric realizations of quandle homology. It is significant to calculate quandle (co)homology and determine explicit quandle cocycles for the study of these invariants.

The free parts of rack and quandle homology groups of finite quandles have been completely computed in [6, 9], but little is known about the torsion parts. The orders of torsion elements of rack and quandle homology were approximated in [9], and their relationship with the inner automorphism group of the quandle was discussed in [6]. Niebrzydowski and Przytycki [11, 12] have developed methods and conjectures about the higher dimensional homology of dihedral quandles. They conjectured that for a dihedral quandle Rp of order odd prime p, $TorHnQ(Rp)=ℤpfn$, where {fn} is a “delayed” Fibonacci sequence, i.e., fn = fn−1 + fn−3 and f1 = f2 = 0, f3 = 1. It was proved independently by Clauwens [5] and Nosaka [14]. In analogy to the result of group homology, it was conjectured [12] that for a finite quasigroup quandle, the torsion subgroups of its rack and quandle homology are annihilated by the order of the quandle. This was proved in [15] and generalized [16, 19] for some connected quandles.

In this paper, we study the torsion subgroups of rack and quandle homology of non-connected quandles. To start with, we partially solve the following conjecture:

### Conjecture 1.1.([12])

The number k annihilates $TorHnW(R2k)$, unless k = 2t, t > 1 and the number 2k is the smallest number annihilating $TorHnW(R2k)$for k = 2t, t > 1, where W = R, Q.

### 1.1. Preliminaries

Definition 1.2.([8, 10])

A quandle (X, *) is an algebraic structure with a set X and a binary operation *: X × XX satisfying the following axioms:

• (Right self-distributivity) (a * b) * c = (a * c) * (b * c) for any a, b, cX;

• (Invertibility) For each bX, the right translation rb: XX given by rb(x) = x * b is invertible;

• (Idempotency) a * a = a for any aX.

If the binary operation satisfies right self-distributivity and invertibility, then (X, *) is call a rack. Note that the three axioms above are motivated by Reidemeister moves in knot theory, and racks and quandles can be used to construct (framed) knot invariants.

Example 1.3

Basic quandles can be obtained from groups and modules as follows:

• An abelian group A equipped with the binary operation *: A × AA defined by g * h = 2hg is called a Takasaki quandle or kei. Specially, when A = ℤn, it is called a dihedral quandle and denoted by Rn. See Table 1 for example.

• A group G with the operation g * h = hg−1h is called a core quandle.

• A group G with the conjugate operation g * h = h−1gh is called a conjugate quandle.

• Let M be a module over the Laurent polynomial ring ℤ[T, T−1]. A quandle M with the operation a * b = Ta + (1 − T)b is called an Alexander quandle.

A trivial quandle, a set X with the binary operation a*b = a (i.e., the operation does not depend on the choice of b), is the most elementary quandle. However, it plays an important role in the proof of Theorem 2.2 because its rack and quandle homology groups do not contain torsion subgroups.

A quandle homomorphism is a map h: XY between two quandles (X, *) and (Y, *′) such that h(a * b) = h(a) *′ h(b) for all a, bX. A bijective quandle homomorphism is called a quandle isomorphism, and a quandle isomorphism from a quandle X onto itself is called a quandle automorphism. Let X be a quandle. The quandle automorphism group Aut(X) of X is the group consisting of quandle automorphisms of X. Note that every right translation of a quandle is a quandle automorphism. The subgroup of Aut(X) generated by all the right translations of X is called the quandle inner automorphism group, denoted by Inn(X).

A quandle X is said to be quasigroup or Latin if every left translation, lb: XX defined by lb(x) = b * x is invertible. A quandle X is connected if the canonical action of Inn(X) on X is transitive. Otherwise, X is said to be non-connected. Note that every quasigroup quandle is connected, but the converse does not hold in general.

Example 1.4
• (1) The dihedral quandle Rn of order n is quasigroup (i.e., connected) if n is odd. Otherwise, it is non-connected.

• (2) The set of all 4-cycles in the symmetric group S4 with the conjugate operation is a connected quandle, but it is not quasigroup.

• (3) An Alexander quandle is quasigroup if and only if 1 − T is invertible.

We next review the rack and quandle homology theories.

Definition 1.5.([4, 7])
• (1) For a given rack X, let $CnR(X)$ be the free abelian group generated by n-tuples x = (x1, …, xn) of elements of X. We define the boundary homomorphism $∂n:CnR(X)→Cn-1R(X)$ by for n ≥ 2,

$∂n(x)=∑i=2n(-1)i{(x1,…,xi-1,xi+1,…,xn)-(x1*xi,…,xi-1*xi,xi+1,…,xn)}$

and for n < 2, ∂n = 0. ($CnR(X)$, ∂n) is called the rack chain complex of X.

• (2) For a quandle X, define the subgroup $CnD(X)$ of $CnR(X)$ for n ≥ 2 generated by n-tuples (x1, …, xn) with xi = xi+1 for some i. We let $CnD(X)=0$ if n < 2. Then ($CnD(X)$, ∂n) forms a sub-chain complex of ($CnR(X)$, ∂n), called the degenerate chain complex of X.

The quotient chain complex ($CnQ(X)=CnR(X)/CnD(X),∂n′$), where $∂n′$ is the induced homomorphism, is called the quandle chain complex of X. Hereafter, we denote all boundary maps by ∂n.

• (3) Let A be an abelian group. Define the chain and cochain complexes

$C*W(X;A)=C*W(X)⊗A,∂=∂⊗Id,CW*(X;A)=Hom(C*W(X),A),δ=Hom(∂,Id)$

for W=R, D, and Q. The yielded homology groups

$HnW(X;A)=Hn(C*W(X;A)) and HWn(X;A)=Hn(CW*(X;A))$

for W=R, D, and Q are called the nth rack, degenerate, and quandle homology groups and the nth rack, degenerate, and quandle cohomology groups of a rack/quandle X with coefficient group A.

The free parts of the rack and quandle homology groups of a finite quandle X, denoted by $FreeHnR(X)$ and $FreeHnQ(X)$, respectively, were completely computed [6, 9]. In particular, for the dihedral quandle Rm of order m we have:

$FreeHnR(Rm)={ℤ,if m is odd;ℤ2n,if m is even,FreeHnQ(Rm)={ℤ,if m is odd and n=1;0,if m is odd and n>1;ℤ2,if m is even.$

However, it is a bit difficult to compute the torsion parts because there are fewer methods to calculate them than the group homology theory. As for the torsion parts of the rack and quandle homology of dihedral quandles, if m is odd prime, then

$TorHnQ(Rm)=ℤmfn,$

where fn = fn−1 + fn−3 and f1 = f2 = 0, f3 = 1 [5, 14]. Moreover,

$∣Rm∣TorHnR(Rm)=0 and ∣Rm∣TorHnQ(Rm)=0$

if m is odd [14, 15]. However, little is known when m is even.

### 2. Annihilation Theorems for Quandle Extensions

Quandle cocycles can be used to construct extensions of quandles in a similar way to obtain extensions of groups using group cocycles. An abelian extension theory for quandles was introduced by Carter, Elhamdadi, Nikiforou, and Saito [2], and a generalization to extensions with a dynamical cocycle was defined by Andruskiewitsch and Graña [1].

### Definition 2.1.([1, 3])

Let X be a quandle and S be a non-empty set. Let α: X × X → Fun(S × S, S) = SS×S be a function, so that for a, bX and s, tS we have αa,b(s, t) ∈ S. Then S × X is a quandle by the operation (s, a) * (t, b) = (αa,b(s, t), a * b), where a * b denotes the quandle operation in X, if and only if α satisfies the following conditions:

• (1) αa,a(s, s) = s for all aX and sS,

• (2) αa,b(−, t): SS is a bijection for each a, bX and for each tS,

• (3) αa*b,c(αa,b(s, t), u) = αa*c,b*c(αa,c(s, u), αb,c(t, u)) for all a, b, cX and s, t, uS.

Such a function α is called a dynamical quandle cocycle. The quandle constructed above is denoted by S×α X, and is called the extension of X by a dynamical cocycle α.

We first discuss annihilation of rack and quandle homology groups of quandle extensions using certain dynamic cocycles.

### Theorem 2.2

Suppose that X is a finite quasigroup quandle and S is a non-empty set. Let α be the dynamical cocycle defined by αa,b(−, t) = IdSfor all a, bX and for all tS. Then the torsion subgroups of $HnR(S×αX)$and $HnQ(S×αX)$are annihilated by |X|.

Proof

Denote an element (s, x) of S×αX by xs. Let $x=(x1s1,⋯,xnsn)∈CnR(S×αX)$. We define two chain maps $frj,fsj:CnR(S×αX)→CnR(S×αX)$ by

$frj(x)=∣X∣ (xjs1,…,xjsj,xj+1sj+1,…,xnsn) for 1≤j≤n,fsj(x)=∑y∈X(ys1,…,ysj,xj+1sj+1,…,xnsn) for 1≤j≤n.$

Using the following chain homotopies $Dnj$ and $Fnj$, we show that $Dnj:frj≃fsj$ for each 1 ≤ jn and $Fnj:fsj-1≃frj$ for each 2 ≤ jn:

$Dnj(x)=∑y∈X(xjs1,…,xjsj,ysj,xj+1sj+1,…,xnsn) for 1≤j≤n,Fnj(x)=∑y∈X(xjs1,…,xjsj-1,ysj,xjsj,xj+1sj+1,…,xnsn) for 2≤j≤n.$

Note that (s, a) * (t, b) = (s, a * b), i.e., as * bt = (a * b)s since αa,b(−, t) = IdS for any a, bX and for any s, tS.

Define face maps $di(*0),di(*):CnR(X)→Cn-1R(X)$ of the boundary homomorphism ∂n by

$di(*0)(x)=(x1,…,xi-1,xi+1,…,xn) anddi(*)(x)=(x1*xi,…,xi-1*xi,xi+1,…,xn),$

i.e., $∂n=∑i=2n(-1)i(di(*0)-di(*))$.

Let us first consider the chain homotopy $Dnj:CnR(S×αX)→Cn+1R(S×αX)$.

(1) Assume that ij. The idempotent condition of a quandle implies that

$di(*)Dnj(x)∑y∈X(xjs1,…,xjsi-1,xjsi+1,…,xjsj,ysj,xj+1sj+1,…,xnsn).$

Note that the formula above does not depend on *, in particular $(di(*0)-di(*))Dnj=0$. Moreover,

$Dn-1jdi(*)(x)=∑y∈X(xj+1s1,…,xj+1si-1,xj+1si+1,…,xj+1sj+1,ysj+1,xj+2sj+2,…,xnsn)$

which is the same as $Dn-1jdi(*0)(x)$, hence $Dn-1j(di(*0)-di(*))=0$.

(2) When j + 2 ≤ in + 1, $∑y∈X(ysj*xi-1si-1)=∑y∈X(y*xi-1)sj=∑y∈X(ysj)$ by the invertibility condition of a quandle, and therefore

$di(*)Dnj(x)=∑y∈X((xj*xi-1)s1,…,(xj*xi-1)sj,ysj,(xj+1*xi-1)sj+1,…,(xi-2*xi-1)si-2,xisi,⋯,xnsn).$

On the other hand, if j + 1 ≤ i, then we have

$Dn-1jdi(*)(x)=∑y∈X((xj*xi)s1,…,(xj*xi)sj,ysj,(xj+1*xi)sj+1,…,(xi-1*xi)si-1,xi+1si+1,⋯,xnsn),.$

i.e., $di+1(*0)Dnj=Dn-1jdi(*0)$ and $di+1(*0)Dnj=Dn-1jdi(*)$ for j + 1 ≤ in.

(3) Suppose that i = j + 1. Then we have

$di(*0)Dnj(x)=∣X∣ (xjs1,…xjsj,xj+1sj+1,…,xnsn)=frj(x).$

Note that $∑y∈X(xjsk*ysj)=∑y∈X(xj*y)sk=∑y∈X(ysk)$ as X is a quasigroup quandle.

Therefore, we obtain the following equality:

$di(*)Dnj(x)=∑y∈X(ys1,…,ysj,xj+1sj+1,…,xnsn)=fsj(x).$

By (1), (2), and (3),

$∂n+1Dnj(x)+Dn-1j∂n(x)=(-1)j+1(frj(x)-fsj(x)),$

hence, $Dnj:frj≃fsj$ for each 1 ≤ jn.

We next consider the chain homotopy $Fnj:CnR(S×αX)→Cn+1R(S×αX)$.

(4) If ij − 1, then

$di(*)Fnj(x)=∑y∈X(xjs1,…,xjsi-1,xjsi+1,…,xjsj-1,ysj,xjsj,xj+1sj+1,…,xnsn),$

so this formula does not depend on *, in particular $(di(*0)-di(*))Fnj=0$.

Moreover, if ij, then

$Fn-1jdi(*)(x)=∑y∈X(xj+1s1,…,xj+1si-1,xj+1si+1,…,xj+1sj,ysj+1,xj+1sj+1,xj+2sj+2,…,xnsn)$

which is the same as $Fn-1jdi(*0)(x)$, hence $Fn-1j(di(*0)-di(*))=0.$.

(5) Note that $∑y∈X(ysj*xi-1si-1)=∑y∈X(y*xi-1)sj=∑y∈X(ysj)$ by the invertibility condition of a quandle.

If i = j + 1, then $(di(*0)-di(*))Fnj=0$. Assume that j + 2 ≤ in + 1.

Then

$di(*)Fnj(x)=∑y∈X((xj*xi-1)s1,…,(xj*xi-1)sj-1,ysj,(xj*xi-1)sj,…,(xi-2*xi-1)si-2,xisi,⋯,xnsn).$

On the other hand, if j + 1 ≤ i, then

$Fn-1jdi(*)(x)=∑y∈X((xj*xi)s1,…,(xj*xi)sj-1,ysj,(xj*xi)sj,…,(xi-1*xi)si-1,xi+1si+1,⋯,xnsn).$

Thus, $di+1(*0)Fnj=Fn-1jdi(*0)$ and $di+1(*)Fnj=Fn-1jdi(*)$ for j + 1 ≤ in.

(6) Assume that i = j. Then we have

$di(*0)Fnj(x)=∣X∣ (xjs1,…,xjsj,xj+1sj+1,…,xnsn)=frj(x).$

Since X is a quasigroup quandle, $∑y∈X(xjsk*ysj)=∑y∈X(xj*y)sk=∑y∈X(ysk)$. Hence, we have

$di(*)Fnj(x)=∑y∈X(ys1,…,ysj-1,xjsj,…,xnsn)=fsj-1(x).$

By (4), (5), and (6),

$∂n+1Fnj(x)+Fn-1j∂n(x)=(-1)j(frj(x)-fsj-1(x)),$

thus, $Fnj:fsj-1≃frj$for each 2 ≤ jn.

Finally, we obtain the following sequence of chain homotopic chain maps:

$∣X∣IdCnR(S×αX)=fr1≃fs1≃fr2≃⋯≃frn-1≃fsn-1≃frn≃fsn.$

Let τ(S) denote the trivial quandle with the set S, i.e., s * t = s for any s, tτ(S). Consider the chain maps $p:CnR(S×αX)→CnR(τ(S))$ and $ϕ:CnR(τ(S))→CnR(S×αX)$ given by

$p(x1s1,⋯,xnsn)=(s1,⋯,sn) and ϕ(s1,⋯,sn)=∑y∈X(ys1,⋯,ysn).$

Clearly $ϕ∘p=fsn$, so that we have the same induced homomorphisms

$∣X∣IdHnR(S×αX)=(fr1)*=(fsn)*=ϕ*∘p*,$

where $p*:HnR(S×αX)→HnR(τ(S))$ and $ϕ*:HnR(τ(S))→HnR(S×αX)$. Since τ(S) is a trivial quandle, $HnR(τ(S))$ has no torsion. Therefore,

$∣X∣z=∣X∣IdHnR(S×αX)(z)=ϕ*(p*(z))=ϕ*(0)=0$

for every $z∈Tor(HnR(S×αX))$ as desired.

Furthermore, since the rack homology of a quandle splits into the quandle homology and the degenerate homology [9], i.e., $HnR(S×αX)=HnQ(S×αX)⊕HnD(S×αX)$, the torsion of $HnQ(S×αX)$ is also annihilated by |X|.

Using Theorem 2.2, we partially prove Conjecture 1.1.

### Corollary 2.3

Let R2k be the dihedral quandle of order 2k. The number k annihilates both $TorHnR(R2k)$and $TorHnQ(R2k)$, if k is odd.

Proof

If k = 1, we are done because R2 is a trivial quandle and therefore $HnR(R2)$ and $HnQ(R2)$ have no torsion by definition.

Suppose that k > 1. Let S = {e, o} be the set with two elements. We define the dynamical cocycle α: Rk × RkSS×S by α[a],[b](−, t) = IdS for all [a], [b] ∈ Rk and for all tS. Note that if k is odd, then S ×α RkR2k via the quandle isomorphism h: S×αRkR2k defined by h(e, [m]) = [2m] and h(o, [m]) = [2m + k] for each [m] ∈ Rk. Furthermore, since k is odd, the dihedral quandle Rk is a quasigroup quandle. Therefore, Theorem 2.2 implies that the torsion subgroups of $HnR(R2k)=HnR(S×αRk)$ and $HnQ(R2k)=HnQ(S×αRk)$ are annihilated by k.

Table 2 contains some computational results on homology groups of dihedral quandles of small even order.

### Remark 2.4

Corollary 2.3 does not hold when we replace the condition “k is odd” with “k is even” in the corollary. For example, $H3Q(R8)=ℤ2⊕ℤ22⊕ℤ82$.

### 3. Future Research

Corollary 2.3 can be used to compute rack and quandle homology groups of dihedral quandles of even order. In 2016, Takefumi Nosaka suggested the following open problem during the Knots in Hellas conference:

### Problem 3.1

$H3Q(R2p)=ℤ⊕ℤ⊕ℤp⊕ℤp$if p is odd prime.

An open problem of whether |Inn(X)| of a finite quandle X annihilates $TorHnR(X)$ and $TorHnQ(X)$ for every dimension n was suggested in [16]. It is known that Inn(Rm) is isomorphic to the dihedral group of order 2m if m is odd and the dihedral group of order m if m is even. One can prove the following open problem by generalizing Corollary 2.3 in case of even k:

### Problem 3.2

$TorHnR(Rm)$and $TorHnQ(Rm)$are annihilated by |Inn(Rm)| for all n.

Dihedral quandle of order 6

*024351
0042042
2420420
4204204
3315315
5153153
1531531

Homology of dihedral quandles of order 2k when k is odd

n123
$HnR(R2)$248
$HnQ(R2)$222
$HnD(R2)$026
$HnR(R6)$24$ℤ8⊕ℤ32$
$HnQ(R6)$22$ℤ2⊕ℤ32$
$HnD(R6)$026
$HnR(R10)$24$ℤ8⊕ℤ52$
$HnQ(R10)$22$ℤ2⊕ℤ52$
$HnD(R10)$026

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