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## Ribbon graphs and bialgebra of Lagrangian subspaces

Journal of Knot Theory and Its Ramifications. 2016. Vol. 26. P. 1642006.

Smirnov E., Kleptsyn V.

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 reinterpreted nicely in the language of L-spaces, becoming changes of bases in this vector space. Finally, we define a bialgebra structure on the span of L-spaces, which is analogous to the 4-bialgebra structure on chord diagrams.

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

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

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

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

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

Dunin-Barkowski P., Popolitov A., Popolitova S., International Journal of Modern Physics A 2022 Vol. 37 No. 36 Article 2250216

We look at how evolution method deforms, when one considers Khovanov polynomials instead of Jones polynomials. We do this for the figure-eight-like knots (also known as 'double braid' knots, see arXiv:1306.3197) -- a two-parametric family of knots which "grows" from the figure-eight knot and contains both two-strand torus knots and twist knots. We prove that ...

Added: March 20, 2023

V.A. Vassiliev, 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 ...

Added: December 23, 2009

Omelchenko A., Гришанов С. А., Мешков В. Р., Письма в Журнал технической физики 2006 Т. 32 № 10 С. 61–67

Построен изотопический полиномиальный инвариант кауфмановского типа от двух переменных для двоякопериодических плетеных структур ...

Added: September 11, 2018

Kochetkov Y., / Cornell Univercity. Series arXiv.org e-print archive "arXiv.math". 2024. No. 2405.10594.

We consider the space P of generic complex 5-degree polynomials. Critical values of such polynomial, i.e. four points
in the complex plane, either are vertices of a convex quadrangle Q, or vertices of a triangle T with one point inside T. The inverse image of Q is a tree-like connected structure of five ovals (a cactus). The inverse image of T is also a ...

Added: May 23, 2024

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

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

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

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

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

Kazaryan M., Zograf P., Letters in Mathematical Physics 2015 Vol. 105 No. 8 P. 1057–1084

We compute the number of coverings of CP1∖{0,1,∞} with a given monodromy type over ∞ and given numbers of preimages of 0 and 1. We show that the generating function for these numbers enjoys several remarkable integrability properties: it obeys the Virasoro constraints, an evolution equation, the KP (Kadomtsev–Petviashvili) hierarchy and satisfies a topological recursion ...

Added: January 19, 2016

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

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

Zograf P., Kazaryan M., St Petersburg Mathematical Journal 2018 Vol. 29 P. 439–445

Generating functions for a fixed genus map and hypermap enumeration become rational after a simple explicit change of variables. Their numerators are polynomials with integral coefficients that obey a differential recursion, and the denominators are products of powers of explicit linear functions. ...

Added: November 5, 2020

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

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

Kotelnikova M. V., Aistov A., Вестник Нижегородского университета им. Н.И. Лобачевского. Серия: Социальные науки 2019 Т. 55 № 3 С. 183–189

The article describes a method that allows to improve the content of disciplines of the mathematical cycle by dividing them into invariant (general) and variable parts. The invariants were identified for such disciplines as «Linear algebra», «Mathematical analysis», «Probability theory and mathematical statistics» delivered to Bachelors program students of economics at several universities. Based on ...

Added: January 28, 2020

Borzykh D., ЛЕНАНД, 2021

Книга представляет собой экспресс-курс по теории вероятностей в контексте начального курса эконометрики. В курсе в максимально доступной форме изложен тот минимум, который необходим для осознанного изучения начального курса эконометрики. Данная книга может не только помочь ликвидировать пробелы в знаниях по теории вероятностей, но и позволить в первом приближении выучить предмет «с нуля». При этом, благодаря доступности изложения и небольшому объему книги, ...

Added: February 20, 2021

В. Л. Попов, Математические заметки 2017 Т. 102 № 1 С. 72–80

Мы доказываем, что аффинно-треугольные подгруппы являются борелевскими подгруппами групп Кремоны. ...

Added: May 3, 2017