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\section*{chord diagram}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{combinatorics}{}\paragraph*{{Combinatorics}}\label{combinatorics}

[[!include combinatorics - contents]]

\hypertarget{knot_theory}{}\paragraph*{{Knot theory}}\label{knot_theory}

[[!include knot theory - contents]]

$\backslash$tableofcontents

\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

A \emph{chord diagram} is a graphical representation of a matching on a (cyclically or linearly) ordered set. Chord diagrams are a basic object of study in [[combinatorics]] with applications in diverse areas, notably in [[knot theory]]. They come in both \emph{rooted} and \emph{unrooted} versions.

\hypertarget{illustration}{}\subsubsection*{{Illustration}}\label{illustration}

A typical graphical representation of a chord diagram is as a [[circle]] with some marked [[points]] and [[lines]] connecting those points, something like so:

 If we label the points of the diagram,

 then the chords can be seen as describing a matching (or equivalently, an [[involution]]) on the set of marked points which pairs (or equivalently, \emph{swaps}) $A$ with $C$, $B$ with $D$, and $E$ with $F$. The marked points are also equipped with a cyclic order coming from the orientation of the circle, and the subtle point is that a chord diagram can have non-trivial [[symmetries]] when considered up to this action of rotation. For that reason, in many situations it is useful to equip the circle with a distinguished [[basepoint]] (distinct from the marked points), the result being called a ``rooted'' diagram. Rooted chord diagrams may be conveniently visualized by ``cutting open'' the circle at the basepoint. For example, here we have rooted the chord diagram above at a basepoint lying on the interval AF, and opened up the circle:

 The marked points of a rooted chord diagram are intrinsically equipped with a [[linear order]] ($A \lt B \lt C \lt D \lt E \lt F$) rather than a cyclic order.

Finally, it is also at times useful to consider chord diagrams with marked points which are not attached to any chord; if we think of the diagram as representing an involution, then this allows for the possibility of representing involutions with [[fixed points]].

\hypertarget{definitions}{}\subsection*{{Definitions}}\label{definitions}

Chord diagrams can be defined both in topological terms, which formalize the graphical intuition, as well as in purely combinatorial terms. We present only the combinatorial definition here, starting from \emph{rooted} chord diagrams and using those to define unrooted chord diagrams, rather than the other way around. (For topological definitions of both unrooted and rooted chord diagrams, see for example Chapter 6 of \href{{#LandoZvonkin}}{Lando and Zvonkin}, or Definition 1.5 of \hyperlink{Bar-Natan1995}{Bar-Natan 1995} for the unrooted case.)

\hypertarget{rooted_chord_diagrams}{}\subsubsection*{{Rooted chord diagrams}}\label{rooted_chord_diagrams}

\begin{udefn}
A \textbf{rooted chord diagram} of order $n$ is a [[surjective]] [[monotone function]]

\begin{displaymath}
D : \underset{n\,\text{times}}{\underbrace{[0,1] + \dots + [0,1]}} \twoheadrightarrow [0,2n-1]
\end{displaymath}
from the $n$-fold [[coproduct]] of the [[walking arrow]] (seen as a 2-element [[poset]]) into the [[linear order]] $[0,2n-1] = \{ 0 \lt \dots \lt 2n-1 \}$. The symmetric group $S_n$ acts freely on the domain $[0,1] + \dots + [0,1]$ of a rooted chord diagram of order $n$, and we consider two rooted chord diagrams to be isomorphic if they only differ by precomposition with such a permutation action.

\end{udefn}
Equivalently, a rooted chord diagram of order $n$ is a way of arranging the elements of $[0,2n-1]$ into a collection of (necessarily mutually disjoint) ordered pairs $(D_{10}, D_{11}), \dots, (D_{n0},D_{n1})$, where $D_{i0} \lt D_{i1}$ for each $1 \le i \le n$.

Equivalently, a rooted chord diagram of order $n$ is a [[shuffle]] of $n$ distinct 2-letter words $\bar x x$, $\bar y y$, $\bar z z$, etc., considered up to [[alpha-conversion]]. Under this view, rooted chord diagrams are also sometimes referred to as \textbf{double-occurrence words}. (For example, the rooted chord diagram in the illustration above can be encoded as the double-occurrence word $\bar x \bar y x y \bar z z$.)

\hypertarget{unrooted_chord_diagrams}{}\subsubsection*{{Unrooted chord diagrams}}\label{unrooted_chord_diagrams}

Intuitively, the [[cyclic group]] $C_{2n}$ acts on rooted chord diagrams

\begin{displaymath}
D : \underset{n\,\text{times}}{\underbrace{[0,1] + \dots + [0,1]}} \twoheadrightarrow [0,2n-1]
\end{displaymath}
by acting on the codomain $[0,2n-1]$ via the map $\tau_n = x \mapsto x-1\,(\mod{2n})$. This is not quite well-typed, however, since the composite function $\tau_n \circ D$ is not quite order-preserving. It is easiest to define the action of $C_{2n}$ in terms of the view of a rooted chord diagram $D$ as a collection of ordered pairs $(D_{10}, D_{11}), \dots, (D_{n0},D_{n1})$. The action of $\tau_n$ is defined on each pair by

\begin{displaymath}
\tau_n(x,y) = \begin{cases}(x-1,y-1) & x\gt 0 \\ (y-1,2n-1) & x = 0 \end{cases}
\end{displaymath}
and this extends to an action on rooted chord diagrams by acting independently on the pairs.

\begin{udefn}
An \textbf{unrooted chord diagram} of order $n$ is a rooted chord diagram of order $n$, considered up to the action of $\tau_n$.

\end{udefn}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{representation_as_involutions}{}\subsubsection*{{Representation as involutions}}\label{representation_as_involutions}

Every rooted chord diagram

\begin{displaymath}
D : \underset{n\,\text{times}}{\underbrace{[0,1] + \dots + [0,1]}} \twoheadrightarrow [0,2n-1]
\end{displaymath}
uniquely determines a fixed point free involution

\begin{displaymath}
i_D : \{0,\dots,2n-1\} \to \{0,\dots,2n-1\}
\end{displaymath}
swapping the elements of each chord,

\begin{displaymath}
i_D(x) = \begin{cases}y & (x,y) \in D \\ y & (y,x) \in D\end{cases}
\end{displaymath}
and conversely, any such involution uniquely determines a rooted chord diagram up to isomorphism. In particular, there are a double-factorial number

\begin{displaymath}
(2n-1)!! = (2n-1)(2n-3)\cdots 1
\end{displaymath}
of distinct isomorphism classes of rooted chord diagrams of order $n$.

\hypertarget{symmetries}{}\subsubsection*{{Symmetries}}\label{symmetries}

Every rooted chord diagram uniquely determines a chord diagram simply by forgetting the basepoint. Conversely, an unrooted chord diagram of order $n$ can be rooted in up to $2n$ different ways (corresponding to the $2n$ intervals between the marked points), but might also have fewer rootings in the case of symmetries.

\hypertarget{gauss_diagrams_of_ordinary_virtual_knots}{}\subsection*{{Gauss diagrams of ordinary (virtual) knots}}\label{gauss_diagrams_of_ordinary_virtual_knots}

Any [[knot diagram]] can be represented faithfully by a certain kind of chord diagram with some extra structure, known as a \emph{Gauss diagram} \hyperlink{PV94}{(Polyak and Viro 1994)}.

Recall that classically, a knot is defined as an embedding of the circle $S^1$ into $\mathbb{R}^3$, which can be represented two-dimensionally by projecting onto the plane $\mathbb{R}^2$ and marking each double point as either an over-crossing or an under-crossing. One obtains a chord diagram from this knot diagram by considering the original immersing circle and connecting the [[preimages]] of each double point by a chord. Then, information about crossings can be represented by orienting the chords from over-crossing to under-crossing, while information about the local [[writhe]] of each crossing (in the case of a [[framed knot]]) may be encoded by assigning each chord an additional positive or negative sign. Finally, fixing a [[base point]] on the original knot gives rise to a Gauss diagram whose underlying chord diagram is naturally rooted.

Once again, this geometric object can be abstracted into a purely combinatorial one, called the \emph{Gauss code} of the knot (diagram). To construct the Gauss code, begin by assigning arbitrary labels to the crossings, then run the length of the knot (starting at some arbitrary base point) and output the name of each crossing as you visit it, together with a [[bit]] (or two bits in the case of a framed knot) specifying whether you are at an over-crossing or an under-crossing (of positive or negative writhe). The underlying (rooted) chord diagram of the knot is thus encoded as a double-occurrence word.

Polyak and Viro called these ``Gauss diagrams'' after [[Carl Gauss]], who apparently studied the question of which chord diagrams arise from immersions of circles (see the chapter titled ``Gauss is back: curves in the plane'' of Ghys's \emph{\hyperlink{Ghys16}{A singular mathematical promenade}}). This question also played a foundational role in [[virtual knot theory]] --- indeed, according to \hyperlink{Kauffman1999}{Kauffman (1999)}, the ``fundamental combinatorial motivation'' for the definition of virtual knots was the idea that it should be possible to interpret an \emph{arbitrary} Gauss diagram as encoding a knot, now no longer seen as an embedding of $S^1$ into $\mathbb{R}^3$, but into a thickened surface of arbitrary genus.

\hypertarget{chord_diagrams_of_singular_knots}{}\subsection*{{Chord diagrams of singular knots}}\label{chord_diagrams_of_singular_knots}

By an analogous mechanism, any [[singular knot]] $K$ with $n$ simple double points (i.e., points where the knot intersects itself transversally) gives rise to a chord diagram $c(K)$ with $n$ chords, by connecting the preimages of these double points. This chord diagram $c(K)$ of course does not faithfully represent the original knot $K$ (e.g., it does not include any information about over-crossings and under-crossings). However, such chord diagrams nonetheless play an important role in the theory of [[Vassiliev invariants]], by the theorem that the value of a Vassiliev invariant $v$ of degree $\le n$ on any singular knot $K$ with $n$ simple double points depends only on the chord diagram $c(K)$.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[involution]]
\item [[Vassiliev invariant]]
\item [[combinatorial map]]

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

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

For chord diagrams and arc diagrams from the perspective of [[combinatorics]], see:

\begin{itemize}%
\item Jacques Touchard. Sur un probl\`e{}me de configurations et sur les fractions continues. Canadian Journal of Mathematics 4 (1952), 2-25. (\href{http://dx.doi.org/10.4153/CJM-1952-001-8}{doi})


\item Philippe Flajolet and Marc Noy. Analytic combinatorics of chord diagrams. INRIA technical report 3914, March 2000. (\href{http://algo.inria.fr/flajolet/Publications/FlNo00.pdf}{pdf})


\item A. Khruzin. Enumeration of chord diagrams. arXiv:math/0008209, August 2000. (\href{http://arxiv.org/abs/math/0008209}{arxiv})



\end{itemize}
For the relationship to Gauss diagrams in classical and [[virtual knot theory]], see:

\begin{itemize}%
\item Michael Polyak and Oleg Viro, Gauss diagram formulas for Vassiliev invariants, \emph{International Mathematics Research Notes} 1994:11. \href{http://www.math.stonybrook.edu/~oleg/math/papers/1994-Polyak-Viro.pdf}{pdf}


\item [[Louis Kauffman]], Virtual Knot Theory, \emph{European Journal of Combinatorics} (1999) 20, 663-691. \href{http://homepages.math.uic.edu/~kauffman/VKT.pdf}{pdf}


\item Oleg Viro, Virtual Links, Orientations of Chord Diagrams and Khovanov Homology, Proceedings of 12th G\"o{}kova Geometry-Topology Conference, pp. 184--209, 2005. \href{http://www.pdmi.ras.ru/~olegviro/ggt05-viro.pdf}{pdf}


\item \href{http://katlas.org/wiki/Gauss_Codes}{Gauss codes} at the Knot Atlas.



\end{itemize}
For the connection to [[Vassiliev invariants]] of [[singular knots]], see:

\begin{itemize}%
\item [[Sergei K. Lando]] and [[Alexander K. Zvonkin]], \emph{Graphs on Surfaces and Their Applications}, Springer, 2004. (Chapter 6.)


\item [[Dror Bar-Natan]], On the Vassiliev knot invariants, \emph{Topology} 34 (1995), 423-472. (\href{http://www.math.toronto.edu/~drorbn/papers/OnVassiliev/}{html})


\item [[Simon Willerton]], Vassiliev invariants and the [[Hopf algebra]] of chord diagrams, Math. Proc. Camb. Phil. Soc. (1996), 119, 55.


\item Greg Muller, \href{https://cornellmath.wordpress.com/2007/12/25/chord-diagrams-and-lie-algebras/}{``Chord Diagrams and Lie Algebras''}, blog post at \emph{The Everything Seminar}, 25 December 2007.



\end{itemize}
The following book contains an extensive discussion of chord diagrams associated to singular points of analytic curves:

\begin{itemize}%
\item \'E{}tienne Ghys, \emph{A Singular Mathematical Promenade}, preliminary version of December 2016. (\href{https://arxiv.org/abs/1612.06373}{arXiv:1612.06373}) (\href{http://perso.ens-lyon.fr/ghys/articles/Promenade.pdf}{author pdf})

\end{itemize}
On [[Vassiliev invariants]] of [[braid group|braids]] via [[chord diagrams]]:

\begin{itemize}%
\item [[Dror Bar-Natan]], \emph{Vassiliev and Quantum Invariants of Braids}, Geom. Topol. Monogr. 4 (2002) 143-160 (\href{https://arxiv.org/abs/q-alg/9607001}{arxiv:q-alg/9607001})

\end{itemize}
\hypertarget{as_observables_on_fuzzy_spheres_and_ddbrane_intersections}{}\subsubsection*{{As observables on fuzzy spheres and D$p$-D$(p+2)$brane intersections}}\label{as_observables_on_fuzzy_spheres_and_ddbrane_intersections}

Relation of [[Dp-D(p+2)-brane bound states]] (\href{Dp-Dp+2-brane+bound+states#ReferencesRelationToMonopoles}{hence} [[Yang-Mills monopoles]]) to [[Vassiliev braid invariants]] via [[chord diagrams]] computing [[radii]] of [[fuzzy spheres]]:

\begin{itemize}%
\item [[Sanyaje Ramgoolam]], [[Bill Spence]], S. Thomas, Section 3.2 of: \emph{Resolving brane collapse with $1/N$ corrections in non-Abelian DBI}, Nucl. Phys. B703 (2004) 236-276 (\href{https://arxiv.org/abs/hep-th/0405256}{arxiv:hep-th/0405256})


\item [[Simon McNamara]], [[Constantinos Papageorgakis]], [[Sanyaje Ramgoolam]], [[Bill Spence]], Appendix A of: \emph{Finite $N$ effects on the collapse of fuzzy spheres}, JHEP 0605:060, 2006 (\href{https://arxiv.org/abs/hep-th/0512145}{arxiv:hep-th/0512145})


\item [[Simon McNamara]], Section 4 of: \emph{Twistor Inspired Methods in Perturbative FieldTheory and Fuzzy Funnels}, 2006 (\href{http://inspirehep.net/record/1351861}{spire:1351861}, \href{https://strings.ph.qmul.ac.uk/sites/default/files/Mcnamaraphd.pdf}{pdf}, [[McNamara06.pdf:file]])


\item [[Constantinos Papageorgakis]], p. 161-162 of: \emph{On matrix D-brane dynamics and fuzzy spheres}, 2006 ([[Papageorgakis06.pdf:file]])



\end{itemize}
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