nLab 4T relation

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

$$\big[ t_{i k} + t_{j k} \,,\, t_{i j} \big] \;=\; 0 \,.$$

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

Related concepts

• infinitesimal braid relation

• STU relation

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

knots	braids
chord diagram, Jacobi diagram	horizontal chord diagram
1T&4T relation	2T&4T relation/ infinitesimal braid relations
weight system	horizontal weight system
Vassiliev knot invariant	Vassiliev braid invariant
weight systems are associated graded of Vassiliev invariants	horizontal weight systems are cohomology of loop space of configuration space

References

Original articles

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

• Dror Bar-Natan, Vassiliev and Quantum Invariants of Braids, Geom. Topol. Monogr. 4 (2002) 143-160 (arxiv:q-alg/9607001)

  (for horizontal chord diagrams)

Textbook accounts

• Sergei Chmutov, Sergei Duzhin, Jacob Mostovoy, Section 4 of: Introduction to Vassiliev knot invariants, Cambridge University Press, 2012 (arxiv/1103.5628, doi:10.1017/CBO9781139107846)

• David Jackson, Iain Moffat, Section 11 of: An Introduction to Quantum and Vassiliev Knot Invariants, Springer 2019 (doi:10.1007/978-3-030-05213-3)

Lecture notes:

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

The graphics above are taken from Sati-Schreiber 19.