For discussion of standard round chord diagrams see at chord diagram.
Examples/classes:
Related concepts:
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:
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).
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
from the set of horizontal chord diagrams to the set of round 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
under concatenation of strands. For example:
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
for the quotient algebra of horizontal chord diagrams by these relations.
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
Original articles
Dror Bar-Natan, Vassiliev and Quantum Invariants of Braids, Geom. Topol. Monogr. 4 (2002) 143-160 (arxiv:q-alg/9607001)
Toshitake Kohno, Loop spaces of configuration spaces and finite type invariants, Geom. Topol. Monogr. 4 (2002) 143-160 (arXiv:math/0211056)
Adrien Brochier, Cyclotomic associators and finite type invariants for tangles in the solid torus, Algebr. Geom. Topol. 13 (2013) 3365-3409 (arXiv:1209.0417)
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.