Article
Kyungpook Mathematical Journal 2020; 60(2): 279288
Published online June 30, 2020
Copyright © Kyungpook Mathematical Journal.
Torsion in Homology of Dihedral Quandles of Even Order
Seung Yeop Yang
Department of Mathematics, Kyungpook National University, Daegu, 41566, Republic of Korea
email : seungyeop.yang@knu.ac.kr
Received: March 4, 2020; Revised: March 30, 2020; Accepted: March 31, 2020
Abstract
Niebrzydowski and Przytycki conjectured that the torsion of rack and quandle homology of a dihedral quandle of order 2
Keywords: dihedral quandle, rack and quandle homology, torsion subgroup
1. Introduction
Rack [7] and quandle [4] homology are (co)homology theories for selfdistributive 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
In this paper, we study the torsion subgroups of rack and quandle homology of nonconnected quandles. To start with, we partially solve the following conjecture:
Conjecture 1.1.([12])
1.1. Preliminaries
A

(Right selfdistributivity) (
a *b ) *c = (a *c ) * (b *c ) for anya ,b ,c ∈X ; 
(Invertibility) For each
b ∈X , the right translationr _{b} :X →X given byr _{b} (x ) =x *b is invertible; 
(Idempotency)
a *a =a for anya ∈X .
If the binary operation satisfies right selfdistributivity and invertibility, then (
Basic quandles can be obtained from groups and modules as follows:

An abelian group
A equipped with the binary operation *:A ×A →A defined byg *h = 2h −g is called aTakasaki quandle orkei . Specially, whenA = ℤ_{n} , it is called adihedral quandle and denoted byR _{n} . See Table 1 for example. 
A group
G with the operationg *h =hg ^{−1}h is called acore quandle . 
A group
G with the conjugate operationg *h =h ^{−1}gh is called aconjugate quandle . 
Let
M be a module over the Laurent polynomial ring ℤ[T ,T ^{−1}]. A quandleM with the operationa *b =Ta + (1 −T )b is called anAlexander quandle .
A
A
A quandle

(1) The dihedral quandle
R _{n} of ordern is quasigroup (i.e., connected) ifn is odd. Otherwise, it is nonconnected. 
(2) The set of all 4cycles in the symmetric group
S _{4} 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.

(1) For a given rack
X , let${C}_{n}^{R}(X)$ be the free abelian group generated byn tuplesx = (x _{1}, …,x _{n} ) of elements ofX . We define the boundary homomorphism${\partial}_{n}:{C}_{n}^{R}(X)\to {C}_{n1}^{R}(X)$ by forn ≥ 2,$${\partial}_{n}(\mathbf{x})=\sum _{i=2}^{n}{(1)}^{i}\{({x}_{1},\dots ,{x}_{i1},{x}_{i+1},\dots ,{x}_{n})({x}_{1}*{x}_{i},\dots ,{x}_{i1}*{x}_{i},{x}_{i+1},\dots ,{x}_{n})\}$$ and for
n < 2, ∂_{n} = 0. (${C}_{n}^{R}(X)$ , ∂_{n} ) is called therack chain complex ofX . 
(2) For a quandle
X , define the subgroup${C}_{n}^{D}(X)$ of${C}_{n}^{R}(X)$ forn ≥ 2 generated byn tuples (x _{1}, …,x _{n} ) withx _{i} =x _{i} _{+1} for somei . We let${C}_{n}^{D}(X)=0$ ifn < 2. Then (${C}_{n}^{D}(X)$ , ∂_{n} ) forms a subchain complex of (${C}_{n}^{R}(X)$ , ∂_{n} ), called thedegenerate chain complex ofX .The quotient chain complex (
${C}_{n}^{Q}(X)={C}_{n}^{R}(X)/{C}_{n}^{D}(X),{\partial}_{n}^{\prime}$ ), where${\partial}_{n}^{\prime}$ is the induced homomorphism, is called thequandle chain complex ofX . Hereafter, we denote all boundary maps by ∂_{n} . 
(3) Let
A be an abelian group. Define the chain and cochain complexes$$\begin{array}{l}{C}_{*}^{W}(X;A)={C}_{*}^{W}(X)\otimes A,\partial =\partial \otimes \text{Id},\\ {C}_{W}^{*}(X;A)=\text{Hom}({C}_{*}^{W}(X),A),\delta =\text{Hom}(\partial ,\text{Id})\end{array}$$ for W=R, D, and Q. The yielded homology groups
$${H}_{n}^{W}(X;A)={H}_{n}({C}_{*}^{W}(X;A))\hspace{0.17em}\text{and\hspace{0.17em}}{H}_{W}^{n}(X;A)={H}^{n}({C}_{W}^{*}(X;A))$$ for W=R, D, and Q are called the
n thrack, degenerate, and quandle homology groups and then thrack, degenerate, and quandle cohomology groups of a rack/quandleX with coefficient groupA .
The free parts of the rack and quandle homology groups of a finite quandle
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
where
if
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

(1)
α _{a} _{,}_{a} (s ,s ) =s for alla ∈X ands ∈S , 
(2)
α _{a} _{,}_{b} (−,t ):S →S is a bijection for eacha ,b ∈X and for eacht ∈S , 
(3)
α _{a} _{*}_{b} _{,}_{c} (α _{a} _{,}_{b} (s ,t ),u ) =α _{a} _{*}_{c} _{,}_{b} _{*}_{c} (α _{a} _{,}_{c} (s ,u ),α _{b} _{,}_{c} (t ,u )) for alla ,b ,c ∈X ands ,t ,u ∈S .
Such a function
We first discuss annihilation of rack and quandle homology groups of quandle extensions using certain dynamic cocycles.
Theorem 2.2
Denote an element (
Using the following chain homotopies
Note that (
Define face maps
i.e.,
Let us first consider the chain homotopy
(1) Assume that
Note that the formula above does not depend on *, in particular
which is the same as
(2) When
On the other hand, if
i.e.,
(3) Suppose that i =
Note that
Therefore, we obtain the following equality:
By (1), (2), and (3),
hence,
We next consider the chain homotopy
(4) If
so this formula does not depend on *, in particular
Moreover, if
which is the same as
(5) Note that
If
Then
On the other hand, if
Thus,
(6) Assume that
Since
By (4), (5), and (6),
thus,
Finally, we obtain the following sequence of chain homotopic chain maps:
Let
Clearly
where
for every
Furthermore, since the rack homology of a quandle splits into the quandle homology and the degenerate homology [9], i.e.,
Using Theorem 2.2, we partially prove Conjecture 1.1.
Corollary 2.3
If
Suppose that
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 “
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
An open problem of whether Inn(
Problem 3.2
Tables
Dihedral quandle of order 6
*  0  2  4  3  5  1 

0  0  4  2  0  4  2 
2  4  2  0  4  2  0 
4  2  0  4  2  0  4 
3  3  1  5  3  1  5 
5  1  5  3  1  5  3 
1  5  3  1  5  3  1 
Homology of dihedral quandles of order 2
1  2  3  

ℤ^{2}  ℤ^{4}  ℤ^{8}  
ℤ^{2}  ℤ^{2}  ℤ^{2}  
0  ℤ^{2}  ℤ^{6}  
ℤ^{2}  ℤ^{4}  
ℤ^{2}  ℤ^{2}  
0  ℤ^{2}  ℤ^{6}  
ℤ^{2}  ℤ^{4}  
ℤ^{2}  ℤ^{2}  
0  ℤ^{2}  ℤ^{6} 
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