In mathematics, a chord diagram consists of a cyclic order on a set of objects, together with a one-to-one pairing ( perfect matching) of those objects. Chord diagrams are conventionally visualized by arranging the objects in their order around a

circle A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is cal ...

, and drawing the pairs of the matching as

chords Chord or chords may refer to: Art and music * Chord (music), an aggregate of musical pitches sounded simultaneously ** Guitar chord, a chord played on a guitar, which has a particular tuning * The Chords (British band), 1970s British mod ...

of the circle. The number of different chord diagrams that may be given for a set of

2n

cyclically ordered objects is the

double factorial In mathematics, the double factorial of a number , denoted by , is the product of all the positive integers up to that have the same Parity (mathematics), parity (odd or even) as . That is, n!! = \prod_^ (n-2k) = n (n-2) (n-4) \cdots. Restated ...

(2n-1)!!

. There is a

Catalan number The Catalan numbers are a sequence of natural numbers that occur in various Enumeration, counting problems, often involving recursion, recursively defined objects. They are named after Eugène Charles Catalan, Eugène Catalan, though they were p ...

of chord diagrams on a given ordered set in which no two chords cross each other. The crossing pattern of chords in a chord diagram may be described by a

circle graph In graph theory, a circle graph is the intersection graph of a Chord diagram (mathematics), chord diagram. That is, it is an undirected graph whose vertices can be associated with a finite system of Chord (geometry), chords of a circle such tha ...

, the

intersection graph In graph theory, an intersection graph is a graph that represents the pattern of intersections of a family of sets. Any graph can be represented as an intersection graph, but some important special classes of graphs can be defined by the types o ...

of the chords: it has a vertex for each chord and an edge for each two chords that cross. In

knot theory In topology, knot theory is the study of knot (mathematics), mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are joined so it cannot be und ...

, a chord diagram can be used to describe the sequence of crossings along the planar projection of a knot, with each point at which a crossing occurs paired with the point that crosses it. To fully describe the knot, the diagram should be annotated with an extra bit of information for each pair, indicating which point crosses over and which crosses under at that crossing. With this extra information, the chord diagram of a knot is called a Gauss diagram. In the Gauss diagram of a knot, every chord crosses an even number of other chords, or equivalently each pair in the diagram connects a point in an even position of the cyclic order with a point in an odd position, and sometimes this is used as a defining condition of Gauss diagrams. In

algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...

, chord diagrams can be used to represent the singularities of

algebraic plane curve In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane cu ...

References

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European Journal of Combinatorics The ''European Journal of Combinatorics'' is an international peer-reviewed scientific journal that specializes in combinatorics. The journal primarily publishes papers dealing with mathematical structures within combinatorics and/or establishing ...

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See also

References