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 KnotQuandle : J.P.App.Alg 23 (1982) 37-65

单词	Quandles
释义	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 KnotQuandle : J.P.App.Alg 23 (1982) 37-65
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