nLab linear chord diagram

Idea

A linear chord diagram or arc diagram or rooted chord diagram is a trivalent finite undirected graph with an embedded oriented line and all vertices on that line.

Equivalemtly this is just an n-tuple equipped with a partition into pairs.

The following shows a generic example of a linear chord diagram:

The graphics on the right shows all linear chord diagrams with precisely four vertices.

Closing up the line of a linear chord diagram to a circle and remembering the ordering of vertices only op to cyclic permutation, it becomes a round chord diagram, usually just called a chord diagram. Conversely, a linear chord diagram is equivalently a round chord diagram with one of its vertices singled out.

graphics from Sati-Schreiber 19c

Applications

Wick’s theorem

The combinatorics of contractions in Wick's theorem is governed by linear chord diagrams:

Let $\{Z_i\}$ be a set of quantum fields/random variables which are free fields/multivariate normally distributed with

$$\big\langle Z_i Z_j \big\rangle = k_{i j} \,.$$

Then Wick's theorem says that the expectation value of the product of $n \in \mathbb{N}$ of these fields/random variables is the sum over linear chord diagrams with $n$ vertices of the product over the edges $e_i \to e_j$ of the given chord diagram of the factors $k_{i j}$.

For example, for $n = 4$, Wick's theorem says this:

Related concepts

chord diagrams	weight systems
linear chord diagrams, round chord diagrams Jacobi diagrams, Sullivan chord diagrams	Lie algebra weight systems, stringy weight system, Rozansky-Witten weight systems

knots	braids
chord diagram, Jacobi diagram	horizontal chord diagram
1T&4T relation	2T&4T relation/ infinitesimal braid relations
weight system	horizontal weight system
Vassiliev knot invariant	Vassiliev braid invariant
weight systems are associated graded of Vassiliev invariants	horizontal weight systems are cohomology of loop space of configuration space

References

• Everett Sullivan, Linear chord diagrams with long chords (arXiv:1611.02771)