Examples/classes:
Types
Related concepts:
Consider
the set $\mathcal{D}^c$ of chord diagrams;
the set $\mathcal{D}^t$ of Jacobi diagrams;
Write
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
$R\langle \mathcal{D}^c \rangle$ for the $R$-linear span of chord diagrams;
$R\langle \mathcal{D}^t \rangle$ for the $R$linear span of Jacobi diagrams
and finally
for the respective quotient spaces by the 4T relations
and by the STU relations
respectively.
graphics from Sati-Schreiber 19c
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:
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
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:
Facts about chord diagrams and their weight systems:
The result is due to
Lecture notes
Textbook accounts:
Last revised on April 25, 2021 at 03:17:16. See the history of this page for a list of all contributions to it.