nLab
4T relation

Contents

Contents

Idea

For round chord diagrams

In knot theory by the 4-term relations or 4T-relations, for short, one means the following relations in the linear span of chord diagrams:

(Bar-Natan 95, Def. 1.6)

graphics from Sati-Schreiber 19c

These are the relations respected by weight systems on chord diagrams.

For horizontal chord diagrams

For horizontal chord diagrams the 4T relations is the following:

(Bar-Natan 96, p. 3)

graphics from Sati-Schreiber 19c

When the linear span of horizontal chord diagrams is regarded as an associative algebra under concatenation of strands (here), this relation is the infinitesimal braid relation

[t ik+t jk,t ij]=0. \big[ t_{i k} + t_{j k} \,,\, t_{i j} \big] \;=\; 0 \,.

Properties

Relation between horizontal and round 4T relations

The 4T-relations for round chord diagrams are the image of the 4T relations for horizontal chord diagrams under tracing horizontal to round chord diagrams:

Relation to STU-relations

Under the embedding of the set of round chord diagrams into the set of Jacobi diagrams, the STU-relations imply the 4T relations on round chord diagrams:

Using this, one finds that chord diagrams modulo 4T are Jacobi diagrams modulo STU:

graphics from Sati-Schreiber 19c

chord diagramsweight systems
linear chord diagrams,
round chord diagrams
Jacobi diagrams,
Sullivan chord diagrams
Lie algebra weight systems,
stringy weight system,
Rozansky-Witten weight systems


knotsbraids
chord diagram,
Jacobi diagram
horizontal chord diagram
1T&4T relation2T&4T relation/
infinitesimal braid relations
weight systemhorizontal weight system
Vassiliev knot invariantVassiliev braid invariant
weight systems are associated graded of Vassiliev invariantshorizontal weight systems are cohomology of loop space of configuration space

References

Original articles

Textbook accounts

Lecture notes:

The graphics above are taken from Sati-Schreiber 19.

Last revised on December 20, 2019 at 14:34:05. See the history of this page for a list of all contributions to it.