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On a weight system conjecturally related to sl2
Cornell University
,
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 small orders of the diagrams, the value of the weight system Rk on a diagram of order exactly 2k coincides with the coefficient of ck in the value of the sl2-weight system on the projection of the diagram to primitive elements.
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
Korotyaev Evgeny, Saburova N., Mathematische Annalen 2020 Vol. 337 P. 723-758
We consider a Laplacian on periodic discrete graphs. Its spectrum consists of a finite
number of bands. In a class of periodic 1-forms, i.e., functions defined on edges of
the periodic graph, we introduce a subclass of minimal forms with a minimal number
I of edges in their supports on the period. We obtain a specific decomposition of ...
Added: February 5, 2021
Guterman A., Maksaev A., Acta Scientiarum Mathematicarum 2018 Vol. 84 No. 1-2 P. 19-38
We prove that additive transformations on matrices over the binary Boolean semiring that preserve the scrambling index are automatically bijective. As a consequence we characterize such maps for matrices over an arbitrary antinegative semiring with identity and without zero-divisors. ...
Added: October 30, 2020
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
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
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
Guterman A., Maksaev A., Linear and Multilinear Algebra 2021 Vol. 69 No. 11 P. 2143-2168
The notion of the scrambling index is a fundamental invariant in graph theory and in the theory of non-negative matrices and their applications. Namely, a scrambling index of a primitive directed graph G is the smallest positive integer such that for any pair of vertices u,v of G there exists a vertex w of G ...
Added: October 30, 2020
Vikentyeva O., Morozenko V. V., Plotnikova E. G. et al., Пермь : ИЦ «Титул», 2020
Данный учебник и практикум представляет основные разделы дисциплины «Дискретная математика»: множества, комбинаторика, графы. Учебник и практикум содержит необходимый теоретический материал, излагаемый в доступной форме и иллюстрированный большим количеством примеров, а также разнообразные по содержанию и сложности задания для самостоятельного решения.
Учебник и практикум подготовлен на основе многолетнего опыта работы авторов и апробирован на практических занятиях в ...
Added: November 16, 2020
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
Aleskerov F. T., Khabina E. L., Shvarts D., М. : Физматлит, 2017
В учебном пособии излагаются современные математические подходы к описанию дискретных математических объектов, к построению и изучению прикладных дискретных математических моделей, адекватных реалиям и потребностям социально-экономической и общественно-политической жизни современного общества. ...
Added: October 31, 2019
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
Pham S. K., Antipov D., Sirotkin Alexander et al., Journal of Computational Biology 2013 Vol. 20 No. 4 P. 359-371
One of the key advances in genome assembly that has led to a significant improvement in contig lengths has been improved algorithms for utilization of paired reads (mate-pairs). While in most assemblers, mate-pair information is used in a post-processing step, the recently proposed Paired de Bruijn Graph (PDBG) approach incorporates the mate-pair information directly in ...
Added: March 21, 2014
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
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., 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
Nechaev S. K., Kovaleva V., Maximov Y. et al., Journal of Statistical Mechanics: Theory and Experiment 2017 Vol. 7 No. 7 P. 1-20
In this paper we investigate the eigenvalue statistics of exponentially weighted ensembles of full binary trees and p-branching star graphs. We show that spectral densities of corresponding adjacency matrices demonstrate peculiar ultrametric structure inherent to sparse systems. In particular, the tails of the distribution for binary trees share the 'Lifshitz singularity' emerging in the one-dimensional localization, ...
Added: October 20, 2017
Rybakin A. S., Кулонен Г. А., СПб. : ВКАС им. Буденного, 1999
Added: February 10, 2013
Plotnikova E. G., Loginova V. V., Левко С. В. et al., М. : Юрайт, 2018
Серия «Университеты России» позволит высшим учебным заведениям нашей страны использовать в образовательном процессе издания (в том числе учебники и учебные пособия по различным дисциплинам), подготовленные преподавателями лучших университетов России и впервые опубликованные в издательствах университетов. Все представленные в этой серии работы прошли экспертную оценку учебно-методического отдела издательства и публикуются в оригинальной редакции.
Учебное пособие содержит практические ...
Added: June 28, 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
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
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