nLab
linear chord diagram

Contents

Contents

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

Z iZ j=k ij. \big\langle Z_i Z_j \big\rangle = k_{i j} \,.

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

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

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


knotsbraids
chord diagram,
Jacobi diagram
horizontal chord diagram
1T&4T relation2T&4T relation/
infinitesimal braid relations
weight systemhorizontal weight system
Vassiliev knot invariantVassiliev braid invariant
weight systems are associated graded of Vassiliev invariantshorizontal weight systems are cohomology of loop space of configuration space

References

Last revised on January 10, 2020 at 03:26:23. See the history of this page for a list of all contributions to it.