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 completePlanetmathPlanetmathPlanetmathPlanetmath up to equivalence, that is up to mirror images.

Definition 1

A quandle is an algebraic structurePlanetmathPlanetmath, specifically it is a set Q with two binary operationsMathworldPlanetmath on it, and -1 and the following axioms.

  1. 1.


  2. 2.


  3. 3.


It is useful to consider q1q2 as ’q2 acting on q1’.

  1. 1.

    Let Q be some group, and let n be some fixed integer. Then let g1g2=g2-ng1g2n,g1-1g2=g2ng1g2-n.

  2. 2.

    Let Q be some group. Then let g1g2=g1-1g2=g2g1-1g2.

  3. 3.

    Let Q be some module, and T some invertable linear operator on Q. Then let m1m2=T(m1-m2)+m2,m1-1m2=T-1(m1-m2)+m2

HomomorphismsMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath, isomorphismsMathworldPlanetmathPlanetmath etc. are defined in the obvious way. Notice that the third axiom gives us that the operationMathworldPlanetmath of a quandle element on the quandle given by fq:qqq is a homomorphism, and the second axiom ensures that this is an isomorphism.

Definition 2

The subgroupMathworldPlanetmathPlanetmath of the automorphism groupMathworldPlanetmath of a quandle Q generated by the quandle operations is the operator group of Q.


  • 1 D.Joyce : A Classifying InvariantMathworldPlanetmath Of Knots, The Knot Quandle : J.P.App.Alg 23 (1982) 37-65
Title quandles
Canonical name Quandles
Date of creation 2013-03-22 16:42:37
Last modified on 2013-03-22 16:42:37
Owner StevieHair (1420)
Last modified by StevieHair (1420)
Numerical id 8
Author StevieHair (1420)
Entry type Definition
Classification msc 08A99