Jacobi diagram




A closed Jacobi diagram is a connected undirected graph with oriented trivalent vertices and with an embedded oriented circle, regarded modulo cyclic identifications, if any.

Here is a picture of a typical Jacobi diagram:

Here the internal lines need not form a tree; the following is also a Jacobi diagram:

graphics from Sati-Schreiber 19c

If all the vertices sit on the circle, Jacobi diagrams specialize to chord diagrams.

Half the number of vertices of a Jacobi diagram is called its order

order(Γ)#Vertices(Γ)/2 order(\Gamma) \;\coloneqq\; \#Vertices(\Gamma)/2

For example, these are the two closed Jacobi diagrams with 2 vertices (order 1):

and these are are the 10 closed Jacobi diagrams with 4 vertices (order 2):

graphics grabbed from Chmutov-Duzhin-Mostovoy 11

In applications to Vassiliev invariants in knot theory, Jacobi diagrams play the role of connected Feynman diagrams for Chern-Simons theory in the presence of one Wilson line (the circle is then the knot/Wilson line).

Terminology. Jacobi diagrams have originally been called “chinese character diagrams” (Bar-Natan 95, Def. 1.8) or “Feynman diagrams for Chern-Simons theory” (Kricker-Spence-Aitchison 97) or “circle diagrams” (Kneissler 97) or “round diagrams” (Willerton 99). The term “Jacobi diagram” (Chmutov-Duzhin-Mostovoy 11, Chapter 5, Jackson-Moffat 19, Section 13) alludes to the Jacobi identity – here also called the STU relation – via the fact that these diagrams are naturally labelled by a Lie algebra equipped with a Lie algebra representation (which is really just the interaction-aspect of their interpretation as Chern-Simons theory Feynman diagrams…).


Jacobi diagrams


STU-relations and weight systems

For RR \in CRing a commutative ring, let R𝒟 tR\langle \mathcal{D}^t \rangle denote the RR-linear span of the set 𝒟 t\mathcal{D}^t of Jacobi diagrams.

Then one traditionally writes

(1)𝒜 tR𝒟 c/STU \mathcal{A}^t \;\coloneqq\; R\langle \mathcal{D}^c \rangle/STU

for the quotient spaces of the linear span of Jacobi diagrams by the STU relations:


A linear function

w:𝒜 tR, w \;\colon\; \mathcal{A}^t \longrightarrow R \,,

– hence a plain RR-valued function on the set 𝒟 t\mathcal{D}^t of Jacobi diagrams which is invariant under the STU relations

is called a framed weight system. See there for more.

graphics from Sati-Schreiber 19c


Relation to chord diagrams

chord diagrams modulo 4T are Jacobi diagrams modulo STU:

graphics from Sati-Schreiber 19c

Relation to Adams operations

On the vector space 𝒜\mathcal{A} of Jacobi diagrams modulo STU-relations (equivalently chord diagrams modulo 4T relations) there is a system of linear maps (for qq \in \mathbb{Z}, q0q \neq 0)

ψ q:𝒜𝒜 \psi^q \;\colon\; \mathcal{A} \longrightarrow \mathcal{A}

which respect the coalgebra structure and satisfy

ψ q 2ψ 1 q=ψ q 1q 2 \psi^{q_2} \circ \psi^q_1 \;=\; \psi^{q_1 \cdot q_2}

and as such are (dually) analogous to the Adams operations on topological K-theory.

(Bar-Natan 95, Def. 3.11 & Theorem 7)

In fact, when evaluated in Lie algebra weight systems w Nw_{\mathbf{N}} and under the identification (see here) of the representation ring of a compact Lie group GG with the GG-equivariant K-theory of the point, these Adams operations on Jacobi diagrams correspond to the Adams operations on equivariant K-theory:

w N(ψ qD)=w ψ qN(D). w_{\mathbf{N}}(\psi^q D) \;=\; w_{\psi^q \mathbf{N}}(D) \,.

(Bar-Natan 95, Exc. 6.24 )

For more see at Adams operation on Jacobi diagrams.

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

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



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)

  • A. Kricker, B. Spence and I. Aitchison, Cabling the Vassiliev invariants, J. Knot Theory Ramifications 6 (1997) 327–358

  • Jan Kneissler, The number of primitive Vassiliev invariants up to degree 12 (arXiv:q-alg/9706022)

  • Simon Willerton, The Kontsevich integral and algebraic structures on the space of diagrams in: Knots in Hellas ’98. Series on Knots and Everything, vol. 24, World Scientific, 2000, 530–546 (arXiv:math/9909151)

Lecture notes

Textbook accounts

Chord diagrams and weight systems in Physics

The following is a list of references that involve (weight systems on) chord diagrams/Jacobi diagrams in physics:

  1. In Chern-Simons theory

  2. In Dp-D(p+2) brane intersections

  3. In quantum many body models for for holographic brane/bulk correspondence:

    1. In AdS2/CFT1, JT-gravity/SYK-model

    2. As dimer/bit thread codes for holographic entanglement entropy

For a unifying perspective and further pointers see:

In Chern-Simons theory

Since weight systems are the associated graded of Vassiliev invariants, and since Vassiliev invariants are knot invariants arising as certain correlators/Feynman amplitudes of Chern-Simons theory in the presence of Wilson lines, there is a close relation between weight systems and quantum Chern-Simons theory.

Historically this is the original application of chord diagrams/Jacobi diagrams and their weight systems, see also at graph complex and Kontsevich integral.

Reviewed in:

Applied to Gopakumar-Vafa duality:

  • Dave Auckly, Sergiy Koshkin, Introduction to the Gopakumar-Vafa Large NN Duality, Geom. Topol. Monogr. 8 (2006) 195-456 (arXiv:0701568)

See also

For single trace operators in AdS/CFT duality

Interpretation of Lie algebra weight systems on chord diagrams as certain single trace operators, in particular in application to black hole thermodynamics

In AdS 2/CFT 1AdS_2/CFT_1, JT-gravity/SYK-model

Discussion of (Lie algebra-)weight systems on chord diagrams as SYK model single trace operators:

  • Antonio M. García-García, Yiyang Jia, Jacobus J. M. Verbaarschot, Exact moments of the Sachdev-Ye-Kitaev model up to order 1/N 21/N^2, JHEP 04 (2018) 146 (arXiv:1801.02696)

  • Yiyang Jia, Jacobus J. M. Verbaarschot, Section 4 of: Large NN expansion of the moments and free energy of Sachdev-Ye-Kitaev model, and the enumeration of intersection graphs, JHEP 11 (2018) 031 (arXiv:1806.03271)

  • Micha Berkooz, Prithvi Narayan, Joan Simón, Chord diagrams, exact correlators in spin glasses and black hole bulk reconstruction, JHEP 08 (2018) 192 (arxiv:1806.04380)


  • László Erdős, Dominik Schröder, Phase Transition in the Density of States of Quantum Spin Glasses, D. Math Phys Anal Geom (2014) 17: 9164 (arXiv:1407.1552)

which in turn follows

  • Philippe Flajolet, Marc Noy, Analytic Combinatorics of Chord Diagrams, pages 191–201 in Daniel Krob, Alexander A. Mikhalev,and Alexander V. Mikhalev, (eds.), Formal Power Series and Algebraic Combinatorics, Springer 2000 (doi:10.1007/978-3-662-04166-6_17)

With emphasis on the holographic content:

  • Micha Berkooz, Mikhail Isachenkov, Vladimir Narovlansky, Genis Torrents, Section 5 of: Towards a full solution of the large NN double-scaled SYK model, JHEP 03 (2019) 079 (arxiv:1811.02584)

  • Vladimir Narovlansky, Slide 23 (of 28) of: Towards a Solution of Large NN Double-Scaled SYK, 2019 (pdf)

and specifically in relation, under AdS2/CFT1, to Jackiw-Teitelboim gravity:

In Dpp/D(p+2)(p+2)-brane intersections

Discussion of weight systems on chord diagrams as single trace observables for the non-abelian DBI action on the fuzzy funnel/fuzzy sphere non-commutative geometry of Dp-D(p+2)-brane intersections (hence Yang-Mills monopoles):

graphics from Sati-Schreiber 19c

As codes for holographic entanglement entropy

From Yan 20

Chord diagrams encoding Majorana dimer codes and other quantum error correcting codes via tensor networks exhibiting holographic entanglement entropy:

From JGPE 19

For Dyson-Schwinger equations

Discussion of round chord diagrams organizing Dyson-Schwinger equations:

  • Nicolas Marie, Karen Yeats, A chord diagram expansion coming from some Dyson-Schwinger equations, Communications in Number Theory and Physics, 7(2):251291, 2013 (arXiv:1210.5457)

  • Markus Hihn, Karen Yeats, Generalized chord diagram expansions of Dyson-Schwinger equations, Ann. Inst. Henri Poincar Comb. Phys. Interact. 6 no 4:573-605 (arXiv:1602.02550)

Review in:

  • Ali Assem Mahmoud, Section 3 of: On the Enumerative Structures in Quantum Field Theory (arXiv:2008.11661)

Last revised on January 10, 2020 at 04:15:29. See the history of this page for a list of all contributions to it.