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, $\lhd$ and $\lhd^{-1}$ and the following axioms.

1.

$q\lhd q=q\ \forall q\in Q$
2.

$(q_{1}\lhd q_{2})\lhd^{-1}q_{2}=(q_{1}\lhd^{-1}q_{2})\lhd q_{2}\ \forall q_{1}% ,q_{2}\in Q$
3.

$(q_{1}\lhd q_{2})\lhd q_{3}=(q_{1}\lhd q_{3})\lhd(q_{2}\lhd q_{3})\ \forall q_% {1},q_{2},q_{3}\in Q$

It is useful to consider $q_{1}\lhd q_{2}$ as ’ $q_{2}$ acting on $q_{1}$ ’.
Examples.

1.

Let $Q$ be some group, and let $n$ be some fixed integer. Then let $g_{1}\lhd g_{2}=g_{2}^{-n}g_{1}g_{2}^{n},\hskip 8.0ptg_{1}\lhd^{-1}g_{2}=g_{2}% ^{n}g_{1}g_{2}^{-n}$ .
2.

Let $Q$ be some group. Then let $g_{1}\lhd g_{2}=g_{1}\lhd^{-1}g_{2}=g_{2}g_{1}^{-1}g_{2}$ .
3.

Let $Q$ be some module, and $T$ some invertable linear operator on $Q$ . Then let $m_{1}\lhd m_{2}=T(m_{1}-m_{2})+m_{2},\hskip 8.0ptm_{1}\lhd^{-1}m_{2}=T^{-1}(m_% {1}-m_{2})+m_{2}$

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 $f_{q}\colon q^{\prime}\mapsto q^{\prime}\lhd q$ 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) 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