### ?

## Two constructions of weight system for invariants of knots in non-trivial 3-manifolds

Topology and its Applications. 2008. Vol. 150. No. 16. P. 1757–1765.

A new family of weight systems of finite type knot invariants of any positive degree in orientable 3-manifolds with non-trivial first homology group is constructed. The principal part of the Casson invariant of knots in such manifolds is split into the sum of infinitely many independent weight systems. Examples of knots separated by corresponding invariants and not separated by any other known finite type invariants are presented.

Kulakova E., Lando S., Mukhutdinova T. et al., / Cornell University. Series math "arxiv.org". 2013.

We introduce a new series~$R_k$, $k=2,3,4,\dots$, of integer valued weight systems. The value of the weight system~$R_k$ on a chord diagram is a signed number of cycles of even length~$2k$ in the intersection graph of the diagram. We show that this value depends on the intersection graph only. We check that for small orders of ...

Added: November 24, 2013

Kulakova E., Lando S., Mukhutdinova T. et al., / Cornell University. Series math "arxiv.org". 2013. No. 1307.4933.

We introduce a new series Rk, k=2,3,4,…, of integer valued weight systems. The value of the weight system Rk on a chord diagram is a signed number of cycles of even length 2k in the intersection graph of the diagram. We show that this value depends on the intersection graph only. We check that for ...

Added: December 18, 2014

Kulakova E., Lando S., Mukhutdinova T. et al., European Journal of Combinatorics 2014 Vol. 41 P. 266–277

We introduce a new series Rk, k = 2, 3, 4, ..., of integer valued weight systems. The value of the weight system Rk on a chord diagram is a signed number of cycles of even length 2k in the intersection graph of the diagram.Weshow that this value depends on the intersection graph only. We ...

Added: October 26, 2014

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

V.A. Vassiliev, S.A. Grishanov, Topology and its Applications 2009 No. 156 P. 2307–2316

We construct combinatorial formulas of Fiedler type (i.e. composed of oriented Gauss diagrams arranged by homotopy classes of loops in the base manifold, see [T. Fiedler, Gauss Diagram Invariants for Knots and Links, Math. Appl., vol. 552, Kluwer Academic Publishers, 2001; M. Polyak, O. Viro, Gauss diagram formulas for Vassiliev invariants, Int. Math. Res. Not. ...

Added: January 20, 2010

Smirnov E., Kleptsyn V., Journal of Knot Theory and Its Ramifications 2016 Vol. 26 P. 1642006

To each ribbon graph we assign a so-called L-space, which is a Lagrangian subspace in an even-dimensional vector space with the standard symplectic form. This invariant generalizes the notion of the intersection matrix of a chord diagram. Moreover, the actions of Morse perestroikas (or taking a partial dual) and Vassiliev moves on ribbon graphs are ...

Added: January 15, 2016

Yang Z., / Cornell University. Series arXiv "math". 2022.

The present paper has been motivated by an aspiration for understanding the weight system corresponding to the Lie algebra glN. The straightforward approach to computing the values of a Lie algebra weight system on a general chord diagram amounts to elaborating calculations in the noncommutative universal enveloping algebra, in spite of the fact that the result ...

Added: October 6, 2022

Yang Z., / Cornell University. Series arXiv "math". 2021.

Each knot invariant can be extended to singular knots according to the skein rule. A Vassiliev invariant of order at most n is defined as a knot invariant that vanishes identically on knots with more than n double points. A chord diagram encodes the order of double points along a singular knot. A Vassiliev invariant of order n gives rise to ...

Added: October 6, 2022

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

Omelchenko A., Grishanov S., Meshkov V., Journal of Knot Theory and Its Ramifications 2007 Vol. 16 No. 6 P. 779–788

A two-variable polynomial invariant of non-oriented doubly periodic structures is proposed. A possible application of this polynomial for the classification of textile structures is suggested. ...

Added: September 11, 2018

Smirnov E., Kleptsyn V., / Cornell University. Series math "arxiv.org". 2015. No. 1401.6160v2.

To each ribbon graph we assign a so-called L-space, which is a Lagrangian subspace in an even-dimensional vector space with the standard symplectic form. This invariant generalizes the notion of the intersection matrix of a chord diagram. Moreover, the actions of Morse perestroikas (or taking a partial dual) and Vassiliev moves on ribbon graphs are ...

Added: January 24, 2014

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

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

Deviatov R., Journal of Knot Theory and Its Ramifications 2009 Vol. 18 No. 9 P. 1193–1203

We construct a series of combinatorial quandle-like knot invariants. We color regions of a knot diagram rather than lines and assign a weight to each coloring. Sets of these weights are the invariants we construct (colorings and weights depend on several parameters).
Using these invariants, we prove that left and right trefoils are not isotopic using ...

Added: June 28, 2012

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

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

Lando S., Zhukov V., Moscow Mathematical Journal 2017 Vol. 17 No. 4 P. 741–755

Vassiliev (nite 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 dene both the rst and the second Vassiliev moves
for binary delta-matroids and introduce a 4-term relation for them in
such a way that ...

Added: December 15, 2017

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

Omelchenko A., Grishanov S., Meshkov V., Soviet Technical Physics Letters (English Translation of Pis'ma v Zhurnal Tekhnicheskoi Fiziki) 2006 Vol. 32 No. 5 P. 445–448

A new isotopic Kauffman-type polynomial invariant of two variables for doubly periodic braided structures is constructed ...

Added: September 11, 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

191574970, Functional Analysis and Its Applications 2006 Vol. 40 No. 2 P. 81–90

It is well known that every module M over the algebra ℒ(X) of operators on a finite-dimensional space X can be represented as the tensor product of X by some vector space E, M ≅ = E ⊗ X. We generalize this assertion to the case of topological modules by proving that if X is a stereotype space with the stereotype approximation property, then for each stereotype module M over the ...

Added: September 23, 2016

Losev A. S., Slizovskiy S., JETP Letters 2010 Vol. 91 P. 620–624

Added: February 27, 2013

Ilyashenko Y., Яковенко С. Ю., . М.: МЦНМО, 2013.

Предлагаемая книга—первый том двухтомной монографии, посвящённой аналитической теории дифференциальных уравнений.
В первой части этого тома излагается формальная и аналитическая теория нормальных форм и теорема о разрешении особенностей для векторных полей на плоскости.
Вторая часть посвящена алгебраически разрешимым локальным задачам теории аналитических дифференциальных уравнений , квадратичным векторным полям и проблеме локальной классификации ростков векторных полей в комплексной области ...

Added: February 5, 2014

Kalyagin V.A., Koldanov A.P., Koldanov P.A. et al., Physica A: Statistical Mechanics and its Applications 2014 Vol. 413 No. 1 P. 59–70

A general approach to measure statistical uncertainty of different filtration techniques for market network analysis is proposed. Two measures of statistical uncertainty are introduced and discussed. One is based on conditional risk for multiple decision statistical procedures and another one is based on average fraction of errors. It is shown that for some important cases ...

Added: July 19, 2014