nLab
horizontal chord diagram

Contents

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


Contents

Idea

A horizontal chord diagram on nn strands is a finite undirected graph that is obtained from a trivalent graph with nn 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 nn \in \mathbb{N}, write 𝒟 pb\mathcal{D}^{pb} for the set of horizontal chord diagrams on nn strands (the superscript pbpb is for pure braids).

Trace to round chord diagrams

Given a horizontal chord diagram on nn strands and given any choice of cyclic permutation of nn 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:𝒞 pb𝒞 c 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 nn \in \mathbb{N} and for RR \in CRing a commutative ring, the linear span R𝒟 pbR \langle \mathcal{D}^{pb} \rangle on the set of horizontal chord diagrams on nn strands becomes an graded associative algebra

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

under concatenation of strands. For example:

The 2T- and 4T-relations

On the RR-module R𝒟 pbR\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 (R𝒟 pb,)\big( R\langle \mathcal{D}^{pb}\rangle, \circ \big) of horizontal chord diagrams, these are the infinitesimal braid relations.

One writes

(1)𝒜 pbR𝒟 pb/(2T,4T) \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 RR-linear map from the quotient module (1) tr RR is called a weight system on horizontal chord diagrams (of nn strands), or maybe a horizontal weight systems.

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

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

(Bar-Natan 96, p. 3)

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

Original articles

Textbook accounts:

Last revised on December 4, 2019 at 23:14:01. See the history of this page for a list of all contributions to it.