nLab chord diagrams modulo 4T are Jacobi diagrams modulo STU

Ingredients

Consider

1. the set $\mathcal{D}^c$ of chord diagrams;

2. the set $\mathcal{D}^t$ of Jacobi diagrams;

Write

(1)$$\mathcal{D}^c \overset{i}{\hookrightarrow} \mathcal{D}^t$$

for the canonical injection, regarding chord diagrams as Jacobi diagrams without internal vertices, hence with all vertices on the circle.

For $R \in$ CRing a commutative ring, write

1. $R\langle \mathcal{D}^c \rangle$ for the $R$-linear span of chord diagrams;

2. $R\langle \mathcal{D}^t \rangle$ for the $R$linear span of Jacobi diagrams

and finally

(2)$$\mathcal{A}^c \;\coloneqq\; R\langle \mathcal{D}^c \rangle/4T \,, \phantom{AAAA} \mathcal{A}^t \;\coloneqq\; R\langle \mathcal{D}^t\rangle/STU$$

for the respective quotient spaces by the 4T relations

and by the STU relations

respectively.

graphics from Sati-Schreiber 19c

Statement

The linear extension of the canonical inclusion $\mathbb{D}^c \overset{i}{\hookrightarrow} \mathbb{D}^t$ (1) descends to the quotients (2) and yields a linear isomorphism:

$$\mathcal{A}^c \underoverset{\simeq}{i}{\longrightarrow} \mathcal{A}^t$$

This is due to Bar-Natan 95, Theorem 6. See also Chmutov-Duzhin-Mostovoy 11, 5.3

graphics from Sati-Schreiber 19

The key step of the proof is to observe that the STU-relations imply the 4T-relations as follows:

graphics from Sati-Schreiber 19c

Application

Relation to weight systems

The immediate consequence is that the space of framed weight systems $\mathcal{W}$, which by definition is the dual vector space to the linear span of chord diagrams modulo the 4T-relations, is equiuvalently also the dual vector space to the linear span of chord diagrams modulo the 4T-relations:

$$\begin{aligned} \mathcal{W} & \coloneqq (\mathcal{A}^c)^\ast \\ & \simeq (\mathcal{A}^t)^\ast \end{aligned}$$

Related theorems

Facts about chord diagrams and their weight systems:

• chord diagrams modulo 4T are Jacobi diagrams modulo STU

• weight systems are the associated graded objects of Vassiliev invariants

• weight systems are cohomology of loop space of configuration space

• weight systems are cohomology of knot graph complex

• stringy weight systems span classical Lie algebra weight systems

• all horizontal weight systems are partitioned Lie algebra weight systems

Related concepts

• Lie algebra weight system

References

The result is due to

• Dror Bar-Natan, Theorem 6 in: On the Vassiliev knot invariants, Topology Volume 34, Issue 2, April 1995, Pages 423-472 (doi:10.1016/0040-9383(95)93237-2, pdf)

Lecture notes

• Dror Bar-Natan, Alexander Stoimenow, The Fundamental Theorem of Vassiliev Invariants (arXiv:q-alg/9702009)

Textbook accounts:

• Sergei Chmutov, Sergei Duzhin, Jacob Mostovoy, Section 5.3 of Introduction to Vassiliev knot invariants, Cambridge University Press, 2012 (arxiv:1103.5628, doi:10.1017/CBO9781139107846)