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\begin{document}

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\section*{quandle}

\hypertarget{quandles}{}\section*{{Quandles}}\label{quandles}

\noindent\hyperlink{the_idea}{The Idea}\dotfill \pageref*{the_idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{related_entries}{Related entries}\dotfill \pageref*{related_entries} \linebreak
\noindent\hyperlink{links}{Links}\dotfill \pageref*{links} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{the_idea}{}\subsection*{{The Idea}}\label{the_idea}

A quandle is a set equipped with a binary operation satisfying axioms analogous to the three [[Reidemeister moves]] in knot theory. A quandle is a special case of a [[rack]].

While mainly used to obtain invariants of [[knot|knots]], quandles are interesting algebraic structures in their own right. In particular, the definition of a quandle axiomatizes the properties of conjugation in a group. More abstractly, we can say that a quandle is an algebraic structure where every element acts as an automorphism of that structure, fixing that element.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

A \textbf{quandle} is a selfdistributive idempotent right [[quasigroup]]. In more detail, a \textbf{quandle} is a [[rack]] obeying the law

\begin{displaymath}
a \triangleright a = a
\end{displaymath}
or equivalently

\begin{displaymath}
a \triangleleft a = a \, .
\end{displaymath}
In other words, a quandle is a set $Q$ equipped with two binary operations, $\triangleright$ and $\triangleleft$, obeying the laws:

\begin{displaymath}
a \triangleright (b \triangleright c) = (a \triangleright b)\triangleright (a \triangleright  c)
\end{displaymath}
\begin{displaymath}
(c \triangleleft b) \triangleleft a = (c \triangleleft a)\triangleleft (b \triangleleft a)
\end{displaymath}
\begin{displaymath}
(a \triangleright b)\triangleleft a = b
\end{displaymath}
\begin{displaymath}
a \triangleright (b \triangleleft a) = b
\end{displaymath}
\begin{displaymath}
a \triangleright a = a
\end{displaymath}
\begin{displaymath}
a \triangleleft a = a
\end{displaymath}
Given laws 3 and 4, the operation $\triangleright$ determines the operation $\triangleleft$, and vice versa, and then law 1 is equivalent to law 2, while law 5 is equivalent to law 6. So, this definition has a certain redundancy built in. See [[rack]] for more discussion of related points.

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

Every [[group]] gives a quandle where the operations come from conjugation:

\begin{displaymath}
a \triangleright b = a b a^{-1}
\end{displaymath}
\begin{displaymath}
b \triangleleft a = a^{-1} b a
\end{displaymath}
In fact, every equational law satisfied by [[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.

Every tame knot in $\mathbb{R}^3$ 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 [[isomorphism|isomorphic]] fundamental quandles then there is a [[homeomorphism]] of $\mathbb{R}^3$, possibly 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 $Q$. 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 $Q$, subject to certain constraints easily read off from a diagram of the knot. More sophisticated invariants of this sort can be constructed with the help of quandle [[cohomology]].

The Alexander quandles are also important, since they can be used to compute the [[Alexander polynomial]] of a knot. Let $A$ be a module over the ring $\mathbb{Z}[t, t^{-1}]$ of [[Laurent polynomial|Laurent polynomials]] in one variable. Then the \textbf{Alexander quandle} consists of $A$ made into a quandle with the left action given by

\begin{displaymath}
a \triangleright b = t a + (1-t)b
\end{displaymath}
Analogous to how evaluating the Alexander polynomial at $t = -1$ (and then taking absolute value) defines the \emph{determinant} of a knot, similarly, instantiating the Alexander quandle at $t = -1$ gives rise to the \textbf{dihedral quandle}

\begin{displaymath}
a \triangleright b = 2b - a
\end{displaymath}
which, when interpreted as an action on the ring $\mathbb{Z}_n$ of integers modulo $n$, may be used to define the classical notion of [[colorable knot|n-colorability]] of a knot.

[[rack|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.

A quandle $Q$ is said to be \textbf{involutory} if it obeys the law

\begin{displaymath}
a \triangleright (a \triangleright b) = b
\end{displaymath}
or equivalently

\begin{displaymath}
(b \triangleleft a) \triangleleft a = b
\end{displaymath}
Any [[symmetric space]] gives an involutory quandle, where $a \triangleright b$ is the result of `reflecting $b$ through $a$'. In fact this leads to an elegant definition of symmetric spaces. Note that involutory quandles are algebras of a certain [[Lawvere theory]], since [[rack|racks]] are already algebras of a Lawvere theory, and involutory quandles are racks obeying some extra equational laws. We may thus define involutory quandle objects in any [[category]] with finite [[products]], such as the category of [[smooth manifolds]]. Loos has shown that a connected symmetric space is the the same as an involutory quandle object $Q$ in the category of smooth manifolds with the additional properties that:

\begin{itemize}%
\item each point $a$ is an \emph{isolated} fixed point of the operation $a \triangleright -$.


\item $Q$ is connected.



\end{itemize}
This fact is Theorem I.4.3. in:

\begin{itemize}%
\item Wolgang Bertram, \emph{The Geometry of Jordan and Lie Structures}, Lecture Notes in Mathematics \textbf{1754}, Springer, Berlin, 2000.

\end{itemize}
He attributes this result to:

\begin{itemize}%
\item Ottmar Loos, \emph{Symmetric Spaces I}, Chapter II, Benjamin, New York, 1969.

\end{itemize}
\hypertarget{related_entries}{}\subsection*{{Related entries}}\label{related_entries}

\begin{itemize}%
\item [[rack]]


\item [[shelf]]



\end{itemize}
\hypertarget{links}{}\subsection*{{Links}}\label{links}

\begin{itemize}%
\item Wikipedia: \href{http://en.wikipedia.org/wiki/Racks_and_quandles}{Racks and quandles}


\item Gavin Wraith, \href{http://www.wra1th.plus.com/gcw/rants/math/Rack.html}{A personal story about knots}.



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

Surveys are

\begin{itemize}%
\item S. Nelson, \emph{What is a Quandle?} , Notices AMS \textbf{63} no.4 (2016) pp.378-380. (\href{http://www.ams.org/publications/journals/notices/201604/rnoti-p378.pdf}{pdf})


\item J. Scott Carter, \emph{A survey of quandle ideas}, (\href{http://arxiv.org/abs/1002.4429}{arxiv}).



\end{itemize}
A monograph is

\begin{itemize}%
\item M. Elhamdadi, S. Nelson, \emph{Quandles: An Introduction to the Algebra of Knots} , AMS Providence 2015. (\href{http://bookstore.ams.org/stml-74}{link})

\end{itemize}
The paper by Crans makes it clear that quandles are algebras of a [[Lawvere theory]], so that quandles may be defined in any [[cartesian monoidal category]] (a category with finite [[products]]). It also shows that any Lie algebra gives a quandle in the category of cocommutative [[coalgebra|coalgebras]].

\begin{itemize}%
\item Alissa Crans, \emph{Shelves, racks, spindles and quandles}, (\href{http://arxiv.org/PS_cache/math/pdf/0409/0409602v1.pdf#page=56}{arXiv}, in \emph{Lie 2-Algebras}).

\end{itemize}
Other research papers:

\begin{itemize}%
\item David Joyce, \emph{A classifying invariant of knots; the knot quandle}, J. Pure Appl. Alg. \textbf{23} (1982), 37-65, , \href{http://www.ams.org/mathscinet-getitem?mr=638121}{MR83m:57007}, (\href{http://www.sciencedirect.com/science/article/pii/0022404982900779/pdf?md5=cc81b2cf5a01afc9277d58d10128878a&pid=1-s2.0-0022404982900779-main.pdf}{pdf}).


\item David Joyce, \emph{Simple quandles}, J. Algebra \textbf{79} (1982), no. 2, 307--318, , \href{http://www.ams.org/mathscinet-getitem?mr=682881}{MR84d:20078}


\item Seiichi Kamada, \emph{Knot invariants derived from quandles and racks}, (\href{http://arxiv.org/abs/math/0211096}{arXiv}).


\item J. Scott Carter, Masahico Saito, \emph{Quandle homology theory and cocycle knot invariants}, (\href{http://arxiv.org/abs/math/0112026}{arXiv}).


\item Michael Eisermann, \emph{Quandle coverings and their Galois correspondence}, (\href{http://www.igt.uni-stuttgart.de/eiserm/publications/qcovering.pdf}{pdf}).


\item Aaron Kaestner, Sam Nelson, Leo Selker, \emph{Parity biquandle invariants of virtual knots}, \href{http://arxiv.org/abs/1507.05583}{arxiv/1507.05583}


\item Dominique Bourn, \emph{Partial Mal'tsevness and category of quandles}, Talk at CT2015, \href{http://ct2015.web.ua.pt/abstracts/bourn_d.pdf}{abstract}, \href{http://ct2015.web.ua.pt/slides/Bourn.pdf}{slides pdf} [[!redirects quandle]] [[!redirects quandles]]



\end{itemize}


\end{document}