# nLab horizontal chord diagram

Contents

For discussion of standard round chord diagrams see at chord diagram.

### Context

#### Knot theory

knot theory

Examples/classes:

knot invariants

Related concepts:

category: knot theory

# Contents

## Idea

A horizontal chord diagram on $n$ strands is a finite undirected graph that is obtained from a trivalent graph with $n$ numbered embedded disjoint circles by cutting the circles open (to give the strands), such that the result has all edges not inside the circles (the chords) be vertically ordered (i.e. along the strands) and going between distinct strands.

Here is an example of a horizontal chord diagram on 5 strands: ## Defintions

### Set of horizontal chord diagrams

For $n \in \mathbb{N}$, write $\mathcal{D}^{pb}$ for the set of horizontal chord diagrams on $n$ strands (the superscript $pb$ is for pure braids).

### Trace to round chord diagrams

Given a horizontal chord diagram on $n$ strands and given any choice of cyclic permutation of $n$ elements, the trace of horizontal to round chord diagrams is the round chord diagram obtained by gluing the ends of the strands according to the cyclic permutation, and retaining the chords in the evident way.

The following shows an example of the trace operation for cyclic permutation of strands one step to the left: The following are the analogous traces of the four types of horizontal chord diagrams appearing in the 4T relation: This defines a function

$tr \;\colon\; \mathcal{C}^{pb} \longrightarrow \mathcal{C}^c$

from the set of horizontal chord diagrams to the set of round chord diagrams.

### Algebra of horizontal chord diagrams

For $n \in \mathbb{N}$ and for $R \in$ CRing a commutative ring, the linear span $R \langle \mathcal{D}^{pb} \rangle$ on the set of horizontal chord diagrams on $n$ strands becomes an graded associative algebra

$\big( R\langle \mathcal{D}^{pb}\rangle, \circ \big)$

under concatenation of strands. For example: ### The 2T- and 4T-relations

On the $R$-module $R\langle \mathcal{D}^{pb}\rangle$ of horizontal chord diagrams consider the following relations:

The 2T relations: and the 4T relations: In terms of the commutator Lie algebra of the above algebra $\big( R\langle \mathcal{D}^{pb}\rangle, \circ \big)$ of horizontal chord diagrams, these are the infinitesimal braid relations.

One writes

(1)$\mathcal{A}^{pb} \;\coloneqq\; R\langle \mathcal{D}^{pb}\rangle/(2T,4T)$

for the quotient algebra of horizontal chord diagrams by these relations. ### Horizontal weight systems

An $R$-linear map from the quotient module (1) tr $R$ is called a weight system on horizontal chord diagrams (of $n$ strands), or maybe a horizontal weight systems.

Hence for $R = k$ a field, the vector space of all horizontal weight systems is the degreewise dual vector space

$\mathcal{W}_{pb} \;\coloneqq\; \big( \mathcal{A}^{pb} \big)^\ast$

Original articles

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