Racks and Quandles

A quandle, $X$, is a non-empty set with a binary operation $X \times X \rightarrow X$ denoted by $(a, b) \mapsto a * b$ such that (I) For any $a \in X$, $a* a =a$. (II) For any $a,b \in X$, there is a unique $c \in X$ such that $a= c*b$. (III) For any $a,b,c \in X$, we have $(a*b)*c=(a*c)*(b*c).$ A rack is a set with a binary operation that satisfies (II) and (III). Racks and quandles have been studied in, for example, [Brs88,FR92,Joy82,Mat82]. The following are typical examples of quandles. Any set $X(\neq \emptyset)$ with $a*b=a$ for any $a,b \in X$ is a quandle called a trivial quandle. A group $G$ with $n$-fold conjugation as the quandle operation: $a*b=b^n a b^{-n}$ is a quandle for an integer $n$. Any subset of $G$ that is closed under such conjugation is also a quandle.

Another family is modules over Laurent polynomial ring $\Lambda_p=\mathbb{Z}_p[t,t^{-1}]$, with the operation $a*b=ta + (1-t)b$, called Alexander quandles. Often we use quotient rings $M={\mathbb{Z}_p }[t] / f(t)$, for a prime $p$ and modulo a polynomial $f(t)$ of degree $d$. The elements of these rings $M$ are represented by remainders of polynomials when divided by $f$ mod $p$, so that $M$ consists of $p^d$ elements.

Let $n>2$ be a positive integer, and for elements $i, j \in \mathbb{Z}_n$ (identified with representatives $\{ 0, 1, \ldots , n-1 \}$), define $i\ast j \equiv 2j-i \pmod{n}$. Then $\ast$ defines a quandle structure $R_n$ called the dihedral quandle on $\mathbb{Z}_n$. This is an Alexander quandle $R_n=\mathbb{Z}_n[t]/(t+1)$, and also, is the set of reflections of a regular $n$-gon (elements of the dihedral group $D_{2n}$ represented by reflections) with conjugation as the quandle operation.

A map $f: X \rightarrow Y$ between quandles $X, Y$ is a quandle homomorphism if $f(a,b)=f(a)*f(b)$ for any $a,b \in X$, where we abuse notation and use $*$ for quandle operations on different quandles unless confusion arises. A quandle homomorphism is an isomorphism if it is surjective and injective. If there is an isomorphism between two quandles, they are called isomorphic. An isomorphism between the same quandle $X$ is an automorphism, and they form a group $\rm Aut(X)$, the automorphism group of $X$.

Classification of quandles up to isomorphism are discussed in [Gra02b*] for those with order $p^2$ for prime $p$, and in [Nel03] for Alexander quandles, for example. Another related issue is to realize quandles using familiar structures, and this is discussed in [Joy82] as follows. Consider a group $G$ and a group automorphism $s \in \rm Aut(G)$. The operation $x*y=s(xy^{-1})y$ defines a quandle structure $(G; s)$ on $G$, and if $s(h)=h$ for for all $h$ in a subgroup $H$, then $H\setminus G$ inherits the operation: $Hx * Hy = Hs(xy^{-1})y$, forming another quandle $(H \setminus G ; s)$. In particular, for a fixed $z \in G$, the inner-automorphism $s(x)=z^{-1}xz$ can be used, and if $z \in H$ and $H$ satisfies the above condition, then $(H \setminus G ; s)$ has the quandle operation $Hx * Hy =H xy^{-1} z y$. Then it is proved in [Joy82] that any homogeneous quandle is, in fact, represented this way [Joy82], where a quandle $X$ is homogeneous if for any $x, y \in X$ there is an $f \in \rm Aut(X)$ such that $y=f(x)$.