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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)$.

next up previous contents
Next: Colorings of Knot Diagrams Up: Quandles and Their Colorings Previous: Quandles and Their Colorings   Contents
Masahico Saito - Quandle Website 2006-09-19