Link Invariants
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A 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 rooted and unrooted versions.
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, 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.
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 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 Lando and Zvonkin, or Definition 1.5 of Bar-Natan 1995 for the unrooted case.)
A rooted chord diagram of order $n$ is a surjective monotone function
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.
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 $x x$, $y y$, $z z$, etc., considered up to alpha-conversion. (Under this view, a rooted chord diagram can also be thought of/referred to as a double-occurrence word.)
Intuitively, the cyclic group $C_{2n}$ acts on rooted chord diagrams
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
and this extends to an action on rooted chord diagrams by acting independently on the pairs.
An unrooted chord diagram of order $n$ is a rooted chord diagram of order $n$, considered up to the action of $\tau_n$.
Every rooted chord diagram
uniquely determines a fixed point free involution
swapping the elements of each chord,
and conversely, any such involution uniquely determines a rooted chord diagram up to isomorphism. In particular, there are
distinct isomorphism classes of rooted chord diagrams of order $n$.
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.
Any knot diagram can be represented faithfully by a certain kind of chord diagram with some extra structure, known as a Gauss diagram (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 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). (In this notation, the involution on the points of the underlying chord diagram is encoded implicitly 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 A singular mathematical promenade). This question also played a foundational role in virtual knot theory — indeed, according to Kauffman (1999), the “fundamental combinatorial motivation” for the definition of virtual knots was the idea that it should be possible to interpret an 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).
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 $c(K)$.
For chord diagrams and arc diagrams from the perspective of combinatorics, see:
Jacques Touchard. Sur un problème de configurations et sur les fractions continues. Canadian Journal of Mathematics 4 (1952), 2-25. (doi)
Philippe Flajolet and Marc Noy. Analytic combinatorics of chord diagrams. INRIA technical report 3914, March 2000. (pdf)
A. Khruzin. Enumeration of chord diagrams. arXiv:math/0008209, August 2000. (arxiv)
For the relationship to Gauss diagrams in classical and virtual knot theory, see:
Michael Polyak and Oleg Viro, Gauss diagram formulas for Vassiliev invariants, International Mathematics Research Notes 1994:11. pdf
Louis Kauffman, Virtual Knot Theory, European Journal of Combinatorics (1999) 20, 663-691. pdf
Oleg Viro, Virtual Links, Orientations of Chord Diagrams and Khovanov Homology, Proceedings of 12th Gökova Geometry-Topology Conference, pp. 184–209, 2005. pdf
Gauss codes at the Knot Atlas.
For the connection to Vassiliev invariants of singular knots, see Chapter 6 of:
Sergei K. Lando and Alexander K. Zvonkin, Graphs on Surfaces and Their Applications, Springer, 2004.
Dror Bar-Natan, On the Vassiliev knot invariants, Topology 34 (1995), 423-472. (html)
Simon Willerton, Vassiliev invariants and the Hopf algebra of chord diagrams, Math. Proc. Camb. Phil. Soc. (1996), 119, 55.
Greg Muller, “Chord Diagrams and Lie Algebras”, blog post at The Everything Seminar, 25 December 2007.
The following book contains an extensive discussion of chord diagrams associated to singular points of analytic curves:
Last revised on October 23, 2017 at 10:57:29. See the history of this page for a list of all contributions to it.