nLab 2T relation

Idea

In knot theory the 2-term relations on horizontal chord diagrams are the following relations in the linear span of the set of horizontal chord diagrams:

For pairwise distinct indices $i,j,k,l$, the $(i,j)$-chord generator commutes with the $(k,l)$-chord generator, for instance:

graphics from Sati-Schreiber 19c

Together with the horizontal 4T relations these are the relations respected by horizontal weight systems.

Related concepts

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

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

• Toshitake Kohno, Loop spaces of configuration spaces and finite type invariants, Geom. Topol. Monogr. 4 (2002) 143-160 (arXiv:math/0211056)