A quandle is a set equipped with a binary operation satisfying axioms analogous to the three Reidemeister moves in knot theory. A quandle is a special case of a rack.
While mainly used to obtain invariants of knots, quandles are interesting algebraic structures in their own right. In particular, the definition of a quandle axiomatizes the properties of conjugation in a group. More abstractly, we can say that a quandle is an algebraic structure where every element acts as an automorphism of that structure, fixing that element.
A quandle is a selfdistributive idempotent right quasigroup. In more detail, a quandle is a rack obeying the law
or equivalently
In other words, a quandle is a set $Q$ equipped with two binary operations, $\triangleright$ and $\triangleleft$, obeying the laws:
Given laws 3 and 4, the operation $\triangleright$ determines the operation $\triangleleft$, and vice versa, and then law 1 is equivalent to law 2, while law 5 is equivalent to law 6. So, this definition has a certain redundancy built in. See rack for more discussion of related points.
Every group gives a quandle where the operations come from conjugation:
In fact, every equational law satisfied by conjugation in a group follows from the quandle axioms. So, one can think of a quandle as what is left of a group when we forget multiplication, the identity, and inverses, and only remember the operation of conjugation.
Every tame knot in $\mathbb{R}^3$ has a “fundamental quandle”. To define this, one can note that the fundamental group of the knot complement, or knot group, has a presentation (the Wirtinger presentation?) in which the relations only involve conjugation. So, this presentation can also be used as a presentation of a quandle. The fundamental quandle is a very powerful invariant of knots. In particular, if two knots have isomorphic fundamental quandles then there is a homeomorphism of $\mathbb{R}^3$, possibly orientation reversing, taking one knot to the other.
Less powerful but more easily computable invariants of knots may be obtained by counting the homomorphisms from the knot quandle to a fixed quandle $Q$. Since the Wirtinger presentation has one generator for each strand in a knot diagram, these invariants can be computed by counting ways of labelling each strand by an element of $Q$, subject to certain constraints easily read off from a diagram of the knot. More sophisticated invariants of this sort can be constructed with the help of quandle cohomology.
The Alexander quandles are also important, since they can be used to compute the Alexander polynomial of a knot. Let $A$ be a module over the ring $\mathbb{Z}[t, t^{-1}]$ of Laurent polynomials in one variable. Then the Alexander quandle consists of $A$ made into a quandle with the left action given by
Analogous to how evaluating the Alexander polynomial at $t = -1$ (and then taking absolute value) defines the determinant of a knot, similarly, instantiating the Alexander quandle at $t = -1$ gives rise to the dihedral quandle
which, when interpreted as an action on the ring $\mathbb{Z}_n$ of integers modulo $n$, may be used to define the classical notion of n-colorability of a knot.
Racks are a useful generalization of quandles in topology, since while quandles can represent knots on a round linear object (such as rope or a thread), racks can represent ribbons, which may be twisted as well as knotted.
A quandle $Q$ is said to be involutory if it obeys the law
or equivalently
Any symmetric space gives an involutory quandle, where $a \triangleright b$ is the result of ‘reflecting $b$ through $a$’. In fact this leads to an elegant definition of symmetric spaces. Note that involutory quandles are algebras of a certain Lawvere theory, since racks are already algebras of a Lawvere theory, and involutory quandles are racks obeying some extra equational laws. We may thus define involutory quandle objects in any category with finite products, such as the category of smooth manifolds. Loos has shown that a connected symmetric space is the the same as an involutory quandle object $Q$ in the category of smooth manifolds with the additional properties that:
each point $a$ is an isolated fixed point of the operation $a \triangleright -$.
$Q$ is connected.
This fact is Theorem I.4.3. in:
He attributes this result to:
David Joyce, A classifying invariant of knots; the knot quandle, J. Pure Appl. Alg. 23 (1982), 37-65, doi, MR83m:57007; Simple quandles, J. Algebra 79 (1982), no. 2, 307–318, doi, MR84d:20078
Gavin Wraith, A personal story about knots.
J. Scott Carter, A survey of quandle ideas, arxiv.
Seiichi Kamada, Knot invariants derived from quandles and racks, arxiv:math/0211096.
J. Scott Carter, Masahico Saito, Quandle homology theory and cocycle knot invariants, arxiv.
Alissa Crans, Shelves, racks, spindles and quandles, arxiv, in Lie 2-Algebras.
Michael Eisermann, Quandle coverings and their Galois correspondence, pdf
The last reference makes it clear that quandles are algebras of a Lawvere theory, so that quandles may be defined in any cartesian monoidal category (a category with finite products). It also shows that any Lie algebra gives a quandle in the category of cocommutative coalgebras.