Examples/classes:
Types
Related concepts:
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:
graphics from Sati-Schreiber 19c
These are the relations respected by weight systems on chord diagrams.
For horizontal chord diagrams the 4T relations is the following:
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
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:
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 diagrams | weight systems |
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linear chord diagrams, round chord diagrams Jacobi diagrams, Sullivan chord diagrams | Lie algebra weight systems, stringy weight system, Rozansky-Witten weight systems |
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)
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:
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.