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Types
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In knot theory, by the STU-relations, one means the following relations in the linear span of Jacobi diagrams:
graphics from Sati-Schreiber 19c
By definition, the STU-relations are the relations respected by weight systems on Jacobi diagrams.
When regarded as relations of string diagrams, the STU relations characterize the Jacobi identity for Lie algebra objects, or more generally the Lie action property of Lie modules:
This is the reason behind the existence of Lie algebra weight systems.
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
That the STU-relations characterize the cochain cohomology of the knot-graph complex in the respective degrees is shown in Koytcheff-Munson-Volic 13, Section 3.4
graphics grabbed from Koytcheff-Munson-Volic 13
Original articles:
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:
In terms of the knot-graph complex:
Last revised on December 20, 2019 at 19:32:56. See the history of this page for a list of all contributions to it.