Values of the [IMG align=ABSMIDDLE alt=$ \mathfrak{sl}_2$]tex_sm_4986_img1[/IMG] weight system on a family of graphs that are not intersection graphs of chord diagrams

2022 ◽  
Vol 213 (2) ◽  
Author(s):  
Polina Aleksandrovna Filippova
Keyword(s):  
Weight System ◽  
Intersection Graphs ◽  
Chord Diagrams

scholarly journals Vassiliev invariants from symmetric spaces

2016 ◽  
Vol 25 (10) ◽  
pp. 1650055 ◽  
Author(s):  
Indranil Biswas ◽  
Niels Leth Gammelgaard
Keyword(s):  
Symmetric Space ◽  
Lie Algebra ◽  
Curvature Tensor ◽  
Symmetric Spaces ◽  
Weight System ◽  
Riemannian Symmetric Space ◽  
Riemann Curvature ◽  
Riemann Curvature Tensor ◽  
Vassiliev Invariants ◽  
Chord Diagrams

We construct a natural framed weight system on chord diagrams from the curvature tensor of any pseudo-Riemannian symmetric space. These weight systems are of Lie algebra type and realized by the action of the holonomy Lie algebra on a tangent space. Among the Lie algebra weight systems, they are exactly characterized by having the symmetries of the Riemann curvature tensor.

scholarly journals A parity map of framed chord diagrams

2015 ◽  
Vol 24 (13) ◽  
pp. 1541006 ◽  
Author(s):  
Denis Petrovich Ilyutko ◽  
Vassily Olegovich Manturov
Keyword(s):  
Hopf Algebra ◽  
Weight System ◽  
Algebra Structure ◽  
Connected Sum ◽  
Directed Line ◽  
Chord Diagrams ◽  
Hopf Algebra Structure ◽  
Natural Way

We consider framed chord diagrams, i.e. chord diagrams with chords of two types. It is well known that chord diagrams modulo [Formula: see text]T-relations admit a Hopf algebra structure, where the multiplication is given by any connected sum with respect to the orientation. But in the case of framed chord diagrams a natural way to define a multiplication is not known yet. In this paper, we first define a new module [Formula: see text] which is generated by chord diagrams on two circles and factored by [Formula: see text]T-relations. Then we construct a “parity” map from the module of framed chord diagrams into [Formula: see text] and a weight system on [Formula: see text]. Using the map and weight system we show that a connected sum for framed chord diagrams is not a well-defined operation. In the end of the paper we touch linear diagrams, the circle replaced by a directed line.

scholarly journals INTERSECTION GRAPHS FOR STRING LINKS

2006 ◽  
Vol 15 (01) ◽  
pp. 53-72 ◽  
Author(s):  
BLAKE MELLOR
Keyword(s):  
Finite Type ◽  
Adjacency Matrix ◽  
Intersection Graphs ◽  
Knot Invariants ◽  
Chord Diagrams ◽  
String Links ◽  
Homfly Polynomials

We extend the notion of intersection graphs for chord diagrams in the theory of finite type knot invariants to chord diagrams for string links. We use our definition to develop weight systems for string links via the adjacency matrix of the intersection graphs, and show that these weight systems are related to the weight systems induced by the Conway and Homfly polynomials.

Integrating a weight system of order n to an invariant of (n−1)-singular knots

1996 ◽  
Vol 05 (01) ◽  
pp. 23-35 ◽  
Author(s):  
MICHEL DOMERGUE ◽  
PAUL DONATO
Keyword(s):  
Weight System ◽  
Combinatorial Formula ◽  
Chord Diagrams ◽  
Reidemeister Moves ◽  
Singular Knots

Starting from a Weight-System denoted by P and defined on the n-Chord-Diagrams with values in an arbitrary Q–module, we give an explicit combinatorial formula for an invariant of (n–1)-singular knots which has P as its derivative. The formula is defined for regular knot projections. Its invariance under singular Reidemeister moves is then proved.

scholarly journals Turán-type results for intersection graphs of boxes

2021 ◽  
pp. 1-6
Author(s):  
István Tomon ◽  
Dmitriy Zakharov
Keyword(s):  
Short Note ◽  
Intersection Graph ◽  
Intersection Graphs ◽  
Poset Dimension ◽  
Axis Parallel ◽  
Turán Theorem ◽  
Separation Dimension

Abstract In this short note, we prove the following analog of the Kővári–Sós–Turán theorem for intersection graphs of boxes. If G is the intersection graph of n axis-parallel boxes in $${{\mathbb{R}}^d}$$ such that G contains no copy of K t,t , then G has at most ctn( log n)2d+3 edges, where c = c(d)>0 only depends on d. Our proof is based on exploring connections between boxicity, separation dimension and poset dimension. Using this approach, we also show that a construction of Basit, Chernikov, Starchenko, Tao and Tran of K2,2-free incidence graphs of points and rectangles in the plane can be used to disprove a conjecture of Alon, Basavaraju, Chandran, Mathew and Rajendraprasad. We show that there exist graphs of separation dimension 4 having superlinear number of edges.

Lie algebras graded by the weight system (Θ ,sl)

2021 ◽  
Vol 581 ◽  
pp. 1-44
Author(s):  
Alexander Baranov ◽  
Hogir M. Yaseen
Keyword(s):  
Lie Algebras ◽  
Weight System

scholarly journals Chaos on the hypercube

2020 ◽  
Vol 2020 (11) ◽  
Author(s):  
Yiyang Jia ◽  
Jacobus J. M. Verbaarschot
Keyword(s):  
Magnetic Flux ◽  
Spectral Density ◽  
Hermite Polynomials ◽  
Weighted Sums ◽  
Majorana Fermions ◽  
Chord Diagrams ◽  
Magnetic Inversion ◽  
Constant Magnitude ◽  
Kitaev Model ◽  
Spectral Statistics

Abstract We analyze the spectral properties of a d-dimensional HyperCubic (HC) lattice model originally introduced by Parisi. The U(1) gauge links of this model give rise to a magnetic flux of constant magnitude ϕ but random orientation through the faces of the hypercube. The HC model, which also can be written as a model of 2d interacting Majorana fermions, has a spectral flow that is reminiscent of Maldacena-Qi (MQ) model, and its spectrum at ϕ = 0, actually coincides with the coupling term of the MQ model. As was already shown by Parisi, at leading order in 1/d, the spectral density of this model is given by the density function of the Q-Hermite polynomials, which is also the spectral density of the double-scaled Sachdev-Ye-Kitaev model. Parisi demonstrated this by mapping the moments of the HC model to Q-weighted sums on chord diagrams. We point out that the subleading moments of the HC model can also be mapped to weighted sums on chord diagrams, in a manner that descends from the leading moments. The HC model has a magnetic inversion symmetry that depends on both the magnitude and the orientation of the magnetic flux through the faces of the hypercube. The spectrum for fixed quantum number of this symmetry exhibits a transition from regular spectra at ϕ = 0 to chaotic spectra with spectral statistics given by the Gaussian Unitary Ensembles (GUE) for larger values of ϕ. For small magnetic flux, the ground state is gapped and is close to a Thermofield Double (TFD) state.

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