Quandles

A quandle, $X$, is a 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, [Br88,FeRou92,Joy82,Mat82]. The following are typical examples of quandles. A group $G$ with conjugation as the quandle operation: $a*b=b a b^{-1}$, denoted by $X=$Conj$(G)$, is a quandle. Any subset of $G$ that is closed under such conjugation is also a quandle. Another example is the set of polynomials $M={\mathbb{Z}_p }[t] / f(t)$ with mod $p$ integers, for a prime $p$, and modulo a polynomial $f(t)$. Then the elements of $M$ are represented by remainders of polynomials mod $p$ when divided by $f$, so that $M$ consists of $p^d$ elements. A quandle structure is defined by $a*b=ta+(1-t)b$, $a,b \in M$, and it is called an Alexander quandle. (In general, any $\Lambda (={\mathbb{Z}}[t, t^{-1}])$-module $M$ is a quandle with $a*b=ta+(1-t)b$, $a,b \in M$, that is called an Alexander quandle.) Classification of Alexander quandles are discussed in [Nel03]. Let $n$ be a positive integer, and for elements $i, j \in \{ 0, 1, \ldots , n-1 \}$, define $i\ast j \equiv 2j-i \pmod{n}$. Then $\ast$ defines a quandle structure called the dihedral quandle, $R_n$. This is an Alexander quandle $R_n=\mathbb{Z}_n[t]/(t+1)$.

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.

If $G$ is a group and $s$ is an automorphism, $a*b=s(ab^{-1})b$ defines a quandle structure on $G$, and if $s(h)=h$ for for all $h$ in a subgroup $H$, then $G/H$ inherits the operation: $Ha * Hb = Hs(ab^{-1})b$. This example is due to Joyce [Joy82], where it is proved that any homogeneous quandle, $X$, is isomorphic to such a quandle: Let $G={\rm Aut}(X)$, let an element $x\in X$ be fixed, let $H= \{ f \in {\rm Aut}(X): xf=x \}$. For any element $z \in H$, $z \in Z(H)$ (centralizer). Then the evaluation map $e(g)= xg$ provides the isomorphism.