View source for Racks and quandles

{{Short description|Sets with binary operations analogous to the Reidemeister moves used on knot diagrams}}
{{Algebraic structures}}
In [[mathematics]], '''racks''' and '''quandles''' are sets with [[binary operation]]s satisfying axioms analogous to the [[Reidemeister move]]s used to manipulate [[knot theory|knot]] diagrams.

While mainly used to obtain invariants of knots, they can be viewed as [[abstract algebra|algebraic]] constructions in their own right.  In particular, the definition of a quandle axiomatizes the properties of [[Inner automorphism|conjugation]] in a [[Group (mathematics)|group]].

==History==
In 1942, {{ill|Mituhisa Takasaki|ja|高崎光久}} introduced an algebraic structure which he called a {{transl|ja|kei}} ({{wikt-lang|ja|圭}}),<ref name="takasaki">{{cite journal|first=Mituhisa |last=Takasaki| title=Abstractions of symmetric functions|year=1943|journal=[[Tohoku Mathematical Journal]]|volume=49|pages=143&ndash;207}}</ref><ref name="elhamdadi">{{citation|last=Elhamdadi|first=Mohamed|year=2020|title=A Survey of Racks and Quandles: Some Recent Developments|doi=10.1142/S1005386720000425|arxiv=1910.07400}}</ref> which would later come to be known as an involutive quandle.<ref name="kamada">{{citation|last=Kamada|first=Seiichi|title=Knot invariants derived from quandles and racks|year=2002|volume=4|pages=103-117|doi=10.2140/gtm.2002.4.103|journal=Geometry & Topology Monographs|arxiv=math/0211096}}</ref> His motivation was to find a nonassociative algebraic structure to capture the notion of a [[reflection (mathematics)|reflection]] in the context of [[finite geometry]].<ref name="elhamdadi" /><ref name="kamada" /> The idea was rediscovered and generalized in an unpublished 1959 correspondence between [[John Horton Conway|John Conway]] and [[Gavin Wraith]], who at the time were undergraduate students at the [[University of Cambridge]]. It is here that the modern definitions of quandles and of racks first appear. Wraith had become interested in these structures (which he initially dubbed '''sequentials''') while at school.<ref name="wraith">{{cite web|first=Gavin|last=Wraith|title=A Personal Story about Knots|url=http://www.wra1th.plus.com/gcw/rants/math/Rack.html|url-status=dead|archiveurl=https://web.archive.org/web/20060313115547/http://www.wra1th.plus.com/gcw/rants/math/Rack.html|archivedate=2006-03-13}}</ref>  Conway renamed them '''wracks''', partly as a pun on his colleague's name, and partly because they arise as the remnants (or 'wrack and ruin') of a [[group (mathematics)|group]] when one discards the multiplicative structure and considers only the [[inner automorphism|conjugation]] structure.  The spelling 'rack' has now become prevalent.

These constructs surfaced again in the 1980s:  in a 1982 paper by [[David E. Joyce (mathematician)|David Joyce]]<ref name="joyce">{{cite journal|first=David|last=Joyce|title=''A classifying invariant of knots: the knot quandle''|journal=[[Journal of Pure and Applied Algebra]]|volume=23|year=1982|pages=37&ndash;65|doi=10.1016/0022-4049(82)90077-9|doi-access=free}}</ref> (where the term '''quandle''', an arbitrary nonsense word, was coined),<ref>{{cite web|last1=Baez|first1=John|title=The Origin of the word 'Quandle'|url=https://golem.ph.utexas.edu/category/2015/05/the_origin_of_the_word_quandle.html|website=The n-Category Cafe|accessdate=5 June 2015}}</ref> in a 1982 paper by [[:ru:Сергей Матвеев|Sergei Matveev]] (under the name '''[[distributive property|distributive]] [[groupoids]]''')<ref name="matveev">{{cite journal|first=Sergei|last=Matveev|title=''Distributive groupoids in knot theory''|journal=[[Math. USSR Sbornik]]|volume=47|year=1984|issue=1 |pages=73&ndash;83|doi=10.1070/SM1984v047n01ABEH002630|bibcode=1984SbMat..47...73M }}</ref> and in a 1986 conference paper by [[Egbert Brieskorn]] (where they were called '''[[automorphism|automorphic]] [[set(mathematics)|sets]]''').<ref name="brieskorn">{{cite book|first=Egbert|last=Brieskorn|title=Braids |chapter=Automorphic sets and braids and singularities |series=Contemporary Mathematics |volume=78|year=1988|pages=45&ndash;115|doi=10.1090/conm/078/975077|isbn=9780821850886 }}</ref>  A detailed overview of racks and their applications in knot theory may be found in the paper by [[Colin Rourke]] and [[Roger Fenn]].<ref name="fr">{{cite journal|first2=Roger|last2=Fenn|first1=Colin|last1=Rourke|journal=[[Journal of Knot Theory and Its Ramifications]]|title=''Racks and links in codimension 2''|volume=1|year=1992|pages=343&ndash;406|doi=10.1142/S0218216592000203|issue=4}}</ref>

==Racks==
A '''rack''' may be defined as a set <math>\mathrm{R}</math> with a binary operation <math>\triangleleft</math> such that for every <math>a, b, c \in \mathrm{R}</math> the '''self-distributive law''' holds:
:<math>a \triangleleft(b \triangleleft c) = (a  \triangleleft b) \triangleleft(a  \triangleleft c)</math>

and for every <math>a, b \in \mathrm{R},</math> there exists a unique <math>c \in \mathrm{R}</math> such that 
:<math>a \triangleleft c = b.</math>

This definition, while terse and commonly used, is suboptimal for certain purposes because it contains an existential quantifier which is not really necessary. To avoid this, we may write the unique <math>c \in \mathrm{R}</math> such that <math>a \triangleleft c = b</math> as <math>b \triangleright a.</math>  We then have
:<math> a \triangleleft c = b \iff c = b \triangleright a, </math>

and thus 
:<math> a \triangleleft(b \triangleright a) = b,</math>

and
:<math>(a \triangleleft b) \triangleright a = b.</math>

Using this idea, a rack may be equivalently defined as a set <math>\mathrm{R}</math> with two binary operations <math>\triangleleft </math> and <math>\triangleright</math> such that for all <math>a, b, c \in \mathrm{R}\text{:}</math>
#<math>a \triangleleft(b \triangleleft c) =  (a \triangleleft b) \triangleleft(a \triangleleft c)</math> (left self-distributive law)
#<math>(c \triangleright b) \triangleright a = (c \triangleright a) \triangleright(b \triangleright a)</math> (right self-distributive law)
#<math>(a \triangleleft b) \triangleright a = b</math>
#<math>a \triangleleft(b \triangleright a) = b</math>

It is convenient to say that the element <math>a \in \mathrm{R}</math> is acting from the left in the expression <math>a \triangleleft b,</math> and acting from the right in the expression <math>b \triangleright a.</math>  The third and fourth rack axioms then say that these left and right actions are inverses of each other.  Using this, we can eliminate either one of these actions from the definition of rack.  If we eliminate the right action and keep the left one, we obtain the terse definition given initially.

Many different conventions are used in the literature on racks and quandles.  For example, many authors prefer to work with just the ''right'' action.  Furthermore, the use of the symbols <math>\triangleleft</math> and <math>\triangleright</math> is by no means universal: many authors use exponential notation
:<math>a \triangleleft b = {}^a b</math>

and
:<math>b \triangleright a = b^a,</math>

while many others write
:<math>b \triangleright a = b \star a. </math>

Yet another equivalent definition of a rack is that it is a set where each element acts on the left and right as [[automorphism]]s of the rack, with the left action being the inverse of the right one.  In this definition, the fact that each element acts as automorphisms encodes the left and right self-distributivity laws, and also these laws:

:<math>\begin{align}
  a \triangleleft(b \triangleright c) &= (a \triangleleft b) \triangleright(a\ \triangleleft c) \\
  (c \triangleleft b) \triangleright a &= (c \triangleright a) \triangleleft(b \triangleright a)
 \end{align}</math>

which are consequences of the definition(s) given earlier.

==Quandles==
A '''quandle''' is defined as an [[idempotent]] rack, <math>\mathrm{Q},</math> such that for all <math>a \in \mathrm{Q}</math>
: <math>a \triangleleft a = a,</math>

or equivalently
: <math>a \triangleright a = a.</math>

==Examples and applications==
Every group gives a quandle where the operations come from conjugation:
: <math>\begin{align}
  a \triangleleft b &= a b a^{-1} \\
   b \triangleright a &= a^{-1} b a \\
                     &= a^{-1} \triangleleft b
\end{align}</math>

In fact, every equational law satisfied by [[inner automorphism|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.

{{anchor|Fundamental quandle}}Every [[tame knot]] in [[Three-dimensional space|three-dimensional]] [[Euclidean space]] 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 three-dimensional Euclidean space, which may be [[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 <math>\mathrm{Q}.</math>  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 <math>\mathrm{Q},</math> subject to certain constraints.  More sophisticated invariants of this sort can be constructed with the help of quandle [[cohomology]].

The {{visible anchor|Alexander quandles}} are also important, since they can be used to compute the [[Alexander polynomial]] of a knot. Let <math>\mathrm{A}</math> be a module over the ring <math>\mathbb{Z}[t, t^{-1}]</math> of [[Laurent polynomial]]s in one variable. Then the '''Alexander quandle''' is <math>\mathrm{A}</math> made into a quandle with the left action given by
:<math>a \triangleleft b = tb + (1 - t)a. </math>

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.

{{anchor|Involutory quandles}}A quandle <math>\mathrm{Q}</math> is said to be '''involutory''' if for all <math>a, b \in \mathrm{Q},</math>
: <math> a \triangleleft(a \triangleleft b) = b </math>

or equivalently,
: <math> (b \triangleright a) \triangleright a = b .</math>

Any [[Riemannian symmetric space|symmetric space]] gives an involutory quandle, where <math>a \triangleleft b</math> is the result of 'reflecting <math>b</math> through <math>a</math>'.

==See also==
*[[Biracks and biquandles]]
*[[Laver table]]

==References==
<references/>

==External links==
* [http://www.ams.org/samplings/feature-column/fc-2016-03  Knot Quandaries Quelled by Quandles] - An undergraduate introduction to quandles and another knot invariants
* [https://arxiv.org/abs/1002.4429 A Survey of Quandle Ideas] by Scott Carter
* [https://arxiv.org/abs/math/0211096 Knot Invariants Derived from Quandles and Racks] by Seiichi Kamada
* ''Shelves, Racks, Spindles and Quandles'', p.&nbsp;56 of [https://arxiv.org/abs/math/0409602v1 Lie 2-Algebras] by [[Alissa Crans]]
* https://ncatlab.org/nlab/show/quandle

[[Category:Knot theory]]
[[Category:Non-associative algebra]]