quandles
Quandles are algebraic gadgets introduced by David Joyce in [1] which can be used to define invarients of links. In the case of knots these invarients are complete^{} up to equivalence, that is up to mirror images.
Definition 1
A quandle is an algebraic structure^{}, specifically it is a set $Q$ with two binary operations^{} on it, $\mathrm{\u22b2}$ and ${\mathrm{\u22b2}}^{\mathrm{}\mathrm{1}}$ and the following axioms.

1.
$q\u22b2q=q\forall q\in Q$

2.
$({q}_{1}\u22b2{q}_{2}){\u22b2}^{1}{q}_{2}=({q}_{1}{\u22b2}^{1}{q}_{2})\u22b2{q}_{2}\forall {q}_{1},{q}_{2}\in Q$

3.
$({q}_{1}\u22b2{q}_{2})\u22b2{q}_{3}=({q}_{1}\u22b2{q}_{3})\u22b2({q}_{2}\u22b2{q}_{3})\forall {q}_{1},{q}_{2},{q}_{3}\in Q$
It is useful to consider ${q}_{1}\u22b2{q}_{2}$ as ’${q}_{2}$ acting on ${q}_{1}$’.
Examples.

1.
Let $Q$ be some group, and let $n$ be some fixed integer. Then let ${g}_{1}\u22b2{g}_{2}={g}_{2}^{n}{g}_{1}{g}_{2}^{n},{g}_{1}{\u22b2}^{1}{g}_{2}={g}_{2}^{n}{g}_{1}{g}_{2}^{n}$.

2.
Let $Q$ be some group. Then let ${g}_{1}\u22b2{g}_{2}={g}_{1}{\u22b2}^{1}{g}_{2}={g}_{2}{g}_{1}^{1}{g}_{2}$.

3.
Let $Q$ be some module, and $T$ some invertable linear operator on $Q$. Then let ${m}_{1}\u22b2{m}_{2}=T({m}_{1}{m}_{2})+{m}_{2},{m}_{1}{\u22b2}^{1}{m}_{2}={T}^{1}({m}_{1}{m}_{2})+{m}_{2}$
Homomorphisms^{}, isomorphisms^{} etc. are defined in the obvious way. Notice that the third axiom gives us that the operation^{} of a quandle element on the quandle given by ${f}_{q}:{q}^{\prime}\mapsto {q}^{\prime}\u22b2q$ is a homomorphism, and the second axiom ensures that this is an isomorphism.
Definition 2
The subgroup^{} of the automorphism group^{} of a quandle $Q$ generated by the quandle operations is the operator group of $Q$.
References
 1 D.Joyce : A Classifying Invariant^{} Of Knots, The Knot Quandle : J.P.App.Alg 23 (1982) 3765
Title  quandles 

Canonical name  Quandles 
Date of creation  20130322 16:42:37 
Last modified on  20130322 16:42:37 
Owner  StevieHair (1420) 
Last modified by  StevieHair (1420) 
Numerical id  8 
Author  StevieHair (1420) 
Entry type  Definition 
Classification  msc 08A99 