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Varieties of chord diagrams, braid group cohomology and degeneration of equality conditions
Pacific Journal of Mathematics. 2023. Vol. 326. No. 1. P. 135–160.
The topology of the space of subalgebras of the function space specified by collections of equality conditins f(x)=f(y) is studied and applied to the problems of interpolation theory and knot theory
Kazaryan M., Kodaneva N., Lando S., Journal of Geometry and Physics 2026 Vol. 225 Article 105841
Weight systems associated to the Lie algebras 𝔤𝔩(N) for N = 1,2,... can be unified into auniversal one. The construction is based on an extension of the 𝔤𝔩(N) weight systems to permutations. This universal weight system takes values in the algebra of polynomials C[N;C1,C2,...] in infinitely many variables. We show that under the substitution Cm ...
Added: April 23, 2026
Khoroshkin A., Lyskov D., / Series math "arxiv.org". 2025.
Added: October 14, 2025
Lando S., Yang Z., Annales de l'Institut Henri Poincare (D) Combinatorics, Physics and their Interactions 2025
n a recent paper by M. Kazarian and the second author, a recurrence for the Lie algebras so(N) weight systems has been suggested; the recurrence allows one to construct the universal so weight system. The construction is based on an extension of the so weight systems to permutations. Another recent paper, by M. Kazarian, N. Kodaneva, and the first author, shows that under the ...
Added: May 15, 2025
Zinova P., Kazaryan M., Математический сборник 2023 Т. 214 № 6 С. 87–109
В теории Васильева инварианты узлов конечного порядка описываются в терминах весовых систем – функций на хордовых диаграммах, удовлетворяющих четырехчленным соотношениям. В частности, крашеному многочлену Джонса соответствует весовая система, описываемая в терминах алгебры Ли sl2sl2. Согласно теореме Чмутова–Ландо значение этой весовой системы зависит лишь от графа пересечений хордовой диаграммы, что позволяет говорить о ее значениях на графах пересечений.
В настоящей статье мы ...
Added: February 4, 2025
Kodaneva N., Lando S., Journal of Geometry and Physics 2025 Vol. 210 Article 105421
Weight systems are functions on chord diagrams satisfying so-called Vassiliev’s 4-term relations. They are closely related to finite type knot invariants, see [31 Certain weight systems can be derived from graph invariants, see a recent account in [19]. Another main source of weight systems are Lie algebras, the construction due to D. Bar-Natan [3] and ...
Added: January 23, 2025
Yang Z., Journal of Geometry and Physics 2023 Vol. 187 Article 104808
To a finite type knot invariant, a weight system can be associated, which is a function on chord diagrams satisfying so-called 4-term relations. In the opposite direction, each weight system determines a finite type knot invariant. In particular, a weight system can be associated to any metrized Lie algebra, and any metrized Lie superalgebra. However, computation ...
Added: March 24, 2023
Krasilnikov E., Arnold Mathematical Journal 2021 Vol. 7 No. 4 P. 609–618
Chord diagrams and 4-term relations were introduced by Vassiliev in the late 1980. Various constructions of weight systems are known, and each of such constructions gives rise to a knot invariant. In particular, weight systems may be constructed from Lie algebras as well as from the so-called 4-invariants of graphs. A Chmutov–Lando theorem states that ...
Added: September 29, 2021
Zinova P., Функциональный анализ и его приложения 2020 Т. 54 № 3 С. 73–93
A weight system is a function on chord diagrams that satisfies the so-called four-term
relations. Vassiliev’s theory of finite-order knot invariants describes these invariants in terms of
weight systems. In particular, there is a weight system corresponding to the colored Jones polynomial.
This weight system can be easily defined in terms of the Lie algebra sl2, but this ...
Added: December 10, 2020
Alexander Dunaykin, Vyacheslav Zhukov, Moscow Mathematical Journal 2022 Vol. 22 No. 1 P. 69–81
To a singular knot K with n double points, one can associate a chord
diagram with n chords. A chord diagram can also be understood as a 4regular graph endowed with an oriented Euler circuit. L. Traldi introduced
a polynomial invariant for such graphs, called a transition polynomial.
We specialize this polynomial to a multiplicative weight system, that ...
Added: November 10, 2020
V.A.Vassiliev, European Journal of Combinatorics 2020 Vol. 86 P. 1–7
The rational homology group of the order complex of non-even partitions of a finite set is calculated. A twisted version of Goresky–MacPherson approach to similar homology calculations is proposed. ...
Added: March 21, 2020
Краско Е. С., Лабутин И. Н., Omelchenko A., Записки научных семинаров ПОМИ РАН 2019 Т. 488 С. 119–142
We enumerate labelled and unlabelled Hamiltonian cycles in complete $n$-partite graphs $K_{d,d,\ldots,d}$ having exactly $d$ vertices in each part (in other words, Tur\'an graphs $T(nd, n))$. We obtain recurrence relations that allow us to find the exact values $b_{n}^{(d)}$ of such cycles for arbitrary $n$ and $d$. ...
Added: February 6, 2020
Avdeev R., Journal of Knot Theory and Its Ramifications 2006 Vol. 15 No. 7 P. 853–868
An important problem of knot theory is to find or estimate the extreme coefficients of the Jones–Kauffman polynomial for (virtual) links with a given number of classical crossings. This problem has been studied by Morton and Bae [1] and Manchón [11] for the case of classical links. It turns out that the general case can ...
Added: November 11, 2019
V.A.Vassiliev, Doklady Mathematics (Springer, Germany) 2018 Vol. 98 No. 3 P. 629–633
We calculate homology groups with certain twisted coefficients of configuration spaces of projective spaces. This completes a calculation of rational homology groups of spaces of odd maps of spheres S^m \to S^M, m<M, and of the stable homology of spaces of non-resultant polynomial maps R^{m+1} -> R^{M+1}. Also, we calculate the homology of spaces of Z_r-equivariant maps ...
Added: January 9, 2019
Краско Е. С., Записки научных семинаров ПОМИ РАН 2017 Т. 464 С. 77–87
Maximal chord diagrams up to all isomorphisms are enumerated. The enumerating formula
is based on a bijection between rooted one-vertex one-face maps on locally orientable surfaces
and a certain class of symmetric chord diagrams. This result extends the one of Cori and Marcus
regarding maximal chord diagrams enumerated up to rotations. ...
Added: December 28, 2018
V.A.Vassiliev, Doklady Mathematics 2018 Vol. 98 No. 1 P. 330–333
Stable rational cohomology groups of spaces of non-resultant homogeneous polynomial systems of growing degree in R^n are calculated ...
Added: December 7, 2018
A.V.Omelchenko, Bogdanov A., Meshkov V. et al., Journal of Knot Theory and Its Ramifications 2012 Vol. 21 No. 7 P. 1–17
The paper addresses the enumeration problem for k-tangles. We introduce the notion of a cascade diagram of a k-tangle projection and suggest an effective enumeration algorithm for projections based on the cascade representation. Tangle projections and alternating tangles with up to 12 crossings are tabulated. We also provide pictures of alternating k-tangles with at most ...
Added: August 30, 2018
Omelchenko A., Краско Е. С., Electronic Journal of Combinatorics 2017 Vol. 24 No. 3 P. 1–23
We enumerate chord diagrams without loops and without both loops and parallel chords. For labelled diagrams we obtain generating functions, for unlabelled ones we derive recurrence relations. ...
Added: August 29, 2018
В.А.Васильев, Известия РАН. Серия математическая 2016 Т. 80 № 4 С. 163–184
Rational homology groups of spaces of non-resultant (that is, having only trivial common zeros) systems of homogeneous quadratic polynomial systems in R^3 are calculated ...
Added: March 3, 2018
Vassiliev V., Доклады Российской академии наук. Математика, информатика, процессы управления (ранее - Доклады Академии Наук. Математика) 2017 Т. 477 № 6 С. 637–640
The stabilization of cohomology rings of spaces of non-resultant homogeneous polynomial systems of growing degree in $\R^3$ is studied. The rational stable cohomology rings are explicitly calculated, and the instant of stabilization is estimated ...
Added: December 27, 2017
Lando S., Zhukov V., Moscow Mathematical Journal 2017 Vol. 17 No. 4 P. 741–755
Vassiliev (finite type) invariants of knots can be described in terms of weight systems. These are functions on chord diagrams satisfying so-called 4-term relations. The goal of the present paper is to show that one can define both the first and the second Vassiliev moves for binary delta-matroids and introduce a 4-term relation for them ...
Added: December 11, 2017