nLab
chord diagrams modulo 4T are Jacobi diagrams modulo STU

Contents

Contents

Ingredients

Consider

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

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

Write

(1)𝒟 ci𝒟 t \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 RR \in CRing a commutative ring, write

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

  2. R𝒟 tR\langle \mathcal{D}^t \rangle for the RRlinear span of Jacobi diagrams

and finally

(2)𝒜 cR𝒟 c/4T,AAAA𝒜 tR𝒟 t/STU \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 𝔻 ci𝔻 t\mathbb{D}^c \overset{i}{\hookrightarrow} \mathbb{D}^t (1) descends to the quotients (2) and yields a linear isomorphism:

𝒜 ci𝒜 t \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 𝒲\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:

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

References

The result is due to

Lecture notes

Textbook accounts:

Last revised on December 28, 2019 at 19:31:34. See the history of this page for a list of all contributions to it.