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:
weight systems are associated graded of Vassiliev invariants
weight systems are cohomology of loop space of configuration space
stringy weight systems span classical Lie algebra weight systems
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.