linear chord diagram

**Examples/classes:**

**Related concepts:**

category: knot theory

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

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:

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 |

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

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