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Context

Knot theory

knot theory

knot, link

Examples/classes:

Types

knot invariants

link invariants

Related concepts:

category: knot theory

Contents

Idea

In knot theory a framed weight system is an assignment of numbers to chord diagrams that is invariant under the 4T-relation.

If the assignment is in addition invariant under the 1T-relation, then it is called an unframed weight system, or just weight system, for short.

Similarly, in braid group-theory, a horizontal weight system is an assignment of numbers to horizontal chord diagrams that is invariant under the horizontal 2T relation and horizontal 4T relation.

Definition

Round weight systems

Definition

(linear span of chord diagrams modulo 4T)

Let kk be a field (or just a commutative ring). Write

  1. 𝒟 c\mathcal{D}^c for the set of chord diagrams,

  2. k⟨𝒟 c⟩k\langle \mathcal{D}^c\rangle for its kk-linear span,

  3. 𝒜 c≔k⟨𝒟 c⟩/4T\mathcal{A}^c \;\coloneqq\; k\langle \mathcal{D}^c\rangle/4T for the quotient vector space by the 4T-relations.

Definition

(framed weight system)

A kk-valued framed weight system is a linear function on the linear span of chord diagrams modulo 4T-relations (Def. )

w:𝒜 c⟶k, w \;\colon\; \mathcal{A}^c \longrightarrow k \,,

hence the kk-vector space of framed weight systems is the dual vector space

𝒲≔(𝒜 c) * \mathcal{W} \;\coloneqq\; (\mathcal{A}^c)^\ast

(Bar-Natan 95, Def. 1.6, see Chmutov-Duzhin-Mostovoy 11, Def. 4.1.1, Jackson-Moffat 19, Section 11.7)

Remark

Since chord diagrams modulo 4T are Jacobi diagrams modulo STU, framed weight systems equivalently form the dual vector space

(1)𝒲framedweightsystems ≔(𝒜 cchorddiagramsmodulo4T) * ≃(𝒜 tJacobidiagramsmoduloSTU) * \begin{aligned} \overset{ \mathclap{ \color{blue} { {framed} \atop {{weight}\atop{systems}} } } }{ \mathcal{W} } & \;\coloneqq\; \big( \overset { \color{blue} { { {chord} \atop {diagrams} } \atop { {modulo}\,{4T} } } } { \mathcal{A}^c } \big)^\ast \\ & \;\simeq\; \big( \underset{ \color{blue} { { {Jacobi} \atop {diagrams} } \atop { {modulo}\,{STU} } } }{ \mathcal{A}^t } \big)^\ast \end{aligned}

of the quotient vector space 𝒜 t≔k⟨𝒟 t⟩/STU\mathcal{A}^t \coloneqq k\langle \mathcal{D}^t \rangle/STU of the linear span of Jacobi diagrams by the STU-relation.


Horizontal weight systems

Definition

(linear span of horizontal chord diagrams modulo 2T/4T)

Let kk be a field (or just a commutative ring). Write

  1. 𝒟 pb\mathcal{D}^{pb} for the set of horizontal chord diagrams,

  2. k⟨𝒟 pb⟩k\langle \mathcal{D}^{pb}\rangle for its kk-linear span,

  3. 𝒜 pb≔k⟨𝒟 c⟩/(2T,4T)\mathcal{A}^{pb} \;\coloneqq\; k\langle \mathcal{D}^c\rangle/(2T,4T) for the quotient vector space by the 2T relations and horizontal 4T-relations.

(The superscript “ pb{}^{pb}”in Def. is for pure braids, alluding to the fact that horizontal weight systems are associated graded of Vassiliev braid invaraints.)

Definition

(horizontal weight system)

A kk-valued horizontal weight system is a linear function on the linear span of horizontal chord diagrams modulo 2T&4T-relations (Def. )

w:𝒜 pb⟶k, w \;\colon\; \mathcal{A}^{pb} \longrightarrow k \,,

hence the kk-vector space of horizontal weight systems is the dual vector space

𝒲 pb≔(𝒜 pb) * \mathcal{W}_{pb} \;\coloneqq\; (\mathcal{A}^{pb})^\ast

Constructions of weight systems

Lie algebra weight systems

A large class of weight systems arises from reading a (horizontal) chord diagram as a string diagram in the evident way, and then labelling it by the structure morphisms of a Lie algebra object equipped with a Lie algebra representation internal to a suitable tensor category. This does yield weight systems because the required relations translate exactly to the structural equations satisfied by Lie modules (Jacobi identity and Lie action property).

The weight systems arising this way are called Lie algebra weight systems. See there for more.

Examples of weight systems which are not Lie algebra weight systems are rare. Originally it was conjectured that none exist (Bar-Natan 95, Conjecture 1, Bar-Natan & Stoimenow 97, Conjecture 2.4).

Eventually, a (counter-)example of a weight system which at least does not arise from any finite-dimensional super Lie algebra was given in Vogel 11.

Stringy weight systems

stringy weight systems

Rozansky-Witten weight systems

Rozansky-Witten weight systems

Properties

As the associated graded space of Vassiliev invariants

Proposition

(weight systems are associated graded of Vassiliev invariants)

For ground field k=ℝ,ℂk = \mathbb{R}, \mathbb{C} the real numbers or complex numbers, there is for each natural number n∈ℕn \in \mathbb{N} a canonical linear isomorphism

𝒱 n/𝒱 n−1⟶≃AAAA(𝒜 n u) * \mathcal{V}_n/\mathcal{V}_{n-1} \underoverset{\simeq}{\phantom{AAAA}}{\longrightarrow} \big( \mathcal{A}_n^u \big)^\ast

from

  1. the quotient vector space of order-nn Vassiliev invariants of knots by those of order n−1n-1

  2. to the space of unframed weight systems of order nn.

In other words, in characteristic zero, the graded vector space of unframed weight systems is the associated graded vector space of the filtered vector space of Vassiliev invariants.

(Bar-Natan 95, Theorem 1, following Kontsevich 93)

As cohomology of loop spaces of configuration spaces

weight systems are cohomology of loop space of configuration space:

For horizontal chord diagrams

Proposition

(integral horizontal weight systems are integral cohomology of based loop space of ordered configuration space of points in Euclidean space)

For ground ring R=ℤR = \mathbb{Z} the integers, there is, for each natural number nn, a canonical isomorphism of graded abelian groups between

  1. the integral weight systems

    𝒲 pb≔Hom ℤMod(ℤ⟨𝒟 pb⟩/(2T,4T)⏟𝒜 pb,ℤ) \mathcal{W}_{pb} \;\coloneqq\; Hom_{\mathbb{Z} Mod} \big( \underset{ \mathcal{A}^{pb} }{ \underbrace{ \mathbb{Z} \langle \mathcal{D}^{pb} \rangle /(2T,4T) } } , \mathbb{Z} \big)

    on horizontal chord diagrams of nn strands (elements of the set 𝒟 pb\mathcal{D}^{pb})

  2. the integral cohomology of the based loop space of the ordered configuration space of n points in 3d Euclidean space:

Hℤ •(⊔n∈ℕΩConf {1,⋯,n}(ℝ 3))≃(𝒲 pb) •≃Gr •(𝒱 pb). H \mathbb{Z}^\bullet \big( \underset{n \in \mathbb{N}}{\sqcup} \Omega \underset{ {}^{\{1,\cdots,n\}} }{Conf} (\mathbb{R}^3) \big) \;\simeq\; (\mathcal{W}_{pb})^\bullet \;\simeq\; Gr^\bullet( \mathcal{V}_{pb}) \,.

(the second equivalence on the right is the fact that weight systems are associated graded of Vassiliev invariants).

This is stated as Kohno 02, Theorem 4.1

For round chord diagrams

Proposition

(weight systems are inside real cohomology of based loop space of ordered configuration space of points in Euclidean space)

For ground field k=ℝk = \mathbb{R} the real numbers, there is a canonical injection of the real vector space 𝒲\mathcal{W} of framed weight systems (here) into the real cohomology of the based loop spaces of the ordered configuration spaces of points in 3-dimensional Euclidean space:

𝒲↪Hℝ •(⊔n∈ℕΩConf {1,⋯,n}(ℝ 3)) \mathcal{W} \;\overset{\;\;\;\;}{\hookrightarrow}\; H\mathbb{R}^\bullet \Big( \underset{n \in \mathbb{N}}{\sqcup} \Omega \underset{{}^{\{1,\cdots,n\}}}{Conf} \big( \mathbb{R}^3 \big) \Big)

This is stated as Kohno 02, Theorem 4.2


Combining the above two propositions

  1. weight systems are associated graded of Vassiliev invariants,

  2. weight systems are cohomology of loop space of configuration space

we get this situation:

As cohomology of the knot graph complex

cohomology of knot graph complex is weight systems on chord diagrams

All horizontal weight systems are partitioned Lie algebra weight systems

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


knotsbraids
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

References

General

Original articles:

Textbook accounts:

Lie algebra weight systems

Discussion of Lie algebra weight systems

From the construction given in Bar-Natan 95, Section 2.4 the interpretation of Lie algebra weight systems in terms of string diagrams for Lie algebra objects in tensor categories is evident, but standard textbooks in knot theory/combinatorics do not pick this up:

The interpretation of Lie algebra weight systems as string diagram-calculus and generalization to Lie algebra objects (motivated by generalization at least to super Lie algebras) is made more explicit in

and fully explicit in

See also

  • Vladimir Hinich, Arkady Vaintrob, Cyclic operads and algebra of chord diagrams, Sel. math., New ser. (2002) 8: 237 (arXiv:math/0005197)

  • E. Kulakova, S. Lando, T. Mukhutdinova, G. Rybnikov, On a weight system conjecturally related to 𝔰𝔩 2\mathfrak{sl}_2, European Journal of Combinatorics Volume 41, October 2014, Pages 266-277 (arXiv:1307.4933)

  • Alexander Schrijver, On Lie algebra weight systems for 3-graphs (arXiv:1412.6923)

As the associated graded space of Vassiliev invariants

On spaces of weight systems as the associated graded spaces of Vassiliev invariants:

As cohomology of loop spaces of configuration spaces

On weight systems as the real cohomology of based loop spaces of ordered configuration spaces of points:

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 (via Hypothesis H) and further pointers, see:

Review:

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.

from Bar-Natan 1991

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)

following:

  • 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)
from Narovlansky 2019

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)

  • Micha Berkooz, Mikhail Isachenkov, Prithvi Narayan, Vladimir Narovlansky, Quantum groups, non-commutative AdS 2AdS_2, and chords in the double-scaled SYK model [arXiv:2212.13668]

  • Herman Verlinde, Double-scaled SYK, Chords and de Sitter Gravity [arXiv:2402.00635]

  • Micha Berkooz, Nadav Brukner, Yiyang Jia, Ohad Mamroud, A Path Integral for Chord Diagrams and Chaotic-Integrable Transitions in Double Scaled SYK [arXiv:2403.05980]

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

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

from Sati and Schreiber 19c

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):

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 Jahn and Eisert 21

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)

  • Paul-Hermann Balduf, Amelia Cantwell, Kurusch Ebrahimi-Fard, Lukas Nabergall, Nicholas Olson-Harris, Karen Yeats, Tubings, chord diagrams, and Dyson-Schwinger equations [arXiv:2302.02019]

Review in:

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

Other

Appearance of horizontal chord diagrams in discussion of neutrino interactions in supernovae:

  • Duff Neill, Hanqing Liu, Joshua Martin, Alessandro Roggero: Scattering Neutrinos, Spin Models, and Permutations [arXiv:2406.18677]

Last revised on January 26, 2020 at 12:16:29. See the history of this page for a list of all contributions to it.

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