An extension of Stanley's chromatic symmetric function to binary delta-matroids

Nenasheva, M.; Zhukov, Vyacheslav

doi:10.1016/j.disc.2021.112549

Cited by 4 publications

(7 citation statements)

References 9 publications

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“…In [51], a way to extending umbral graph invariants to umbral invariants of delta-matroids has been suggested. This way is based on the combinatorial Hopf algebra structure introduced in [2].…”

Section: Hopf Algebras Of Delta-matroidsmentioning

confidence: 99%

“…This structure allowed one to define [51] Stanley's symmetrized chromatic polynomial of deltamatroids as the graded homomorphism W : B e → C[x; q 1 , q 2 , . .…”

Section: Hopf Algebras Of Delta-matroidsmentioning

confidence: 99%

“…In recent papers [51,25], an approach to extending weight systems and graph invariants to arbitrary embedded graphs, which is based on the study of the structure of the corresponding Hopf algebras. The space of graphs, as well as the spaces of chord diagrams modulo 4-term relations are endowed with natural connected graded Hopf algebra structure.…”

mentioning

confidence: 99%

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Weight systems and invariants of graphs and embedded graphs

Kazaryan¹,

Lando²

2023

Preprint

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Section: Hopf Algebras Of Delta-matroidsmentioning

confidence: 99%

“…This structure allowed one to define [51] Stanley's symmetrized chromatic polynomial of deltamatroids as the graded homomorphism W : B e → C[x; q 1 , q 2 , . .…”

Section: Hopf Algebras Of Delta-matroidsmentioning

confidence: 99%

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Weight systems and invariants of graphs and embedded graphs

Kazaryan¹,

Lando²

2023

Preprint

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“…Our approach to extending the skew characteristic polynomial to embedded graphs and delta-matroids is similar in nature to that of [10], where an extension of Stanley's symmetrized chromatic polynomial has been constructed. M. Nenasheva and V. Zhukov are basing more specifically on combinatorial Hopf algebra structures, and constructed a character for the extended invariant, which is a graded Hopf algebra homomorphism.…”

Section: -Term Relation For Graphsmentioning

confidence: 99%

Skew characteristic polynomial of graphs and embedded graphs

Dogra,

Lando

2023

Communications in Mathematics

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We introduce a new one-variable polynomial invariant of graphs, which we call the skew characteristic polynomial. For an oriented simple graph, this is just the characteristic polynomial of its anti-symmetric adjacency matrix. For nonoriented simple graphs the definition is different, but for a certain class of graphs (namely, for intersection graphs of chord diagrams), it gives the same answer if we endow such a graph with an orientation induced by the chord diagram. We prove that this invariant satisfies Vassiliev's $4$-term relations and determines therefore a finite type knot invariant. We investigate the behaviour of the polynomial with respect to the Hopf algebra structure on the space of graphs and show that it takes a constant value on any primitive element in this Hopf algebra. We also provide a two-variable extension of the skew characteristic polynomial to embedded graphs and delta-matroids. The $4$-term relations for the extended polynomial prove that it determines a finite type invariant of multicomponent links.

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“…In the recent papers [50] and [24] an approach to extending weight systems and graph invariants to arbitrary embedded graphs was proposed, which is based on the study of the structure of the corresponding Hopf algebras. Hopf algebras arise in a natural way in solutions of combinatorial problems of various nature (for instance, see [13]).…”

Section: Introductionmentioning

confidence: 99%

Weight systems and invariants of graphs and embedded graphs

Kazarian

Lando

2022

Russian Math. Surveys

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The recent progress in the theory of weight systems, which are functions on the chord diagrams satisfying the so-called 4-relations, is described. Most attention is given to methods for constructing concrete weight systems. The two main sources of the constructions discussed are invariants of the intersection graphs of chord diagrams that satisfy the 4-term relations for graphs, and metrized Lie algebras. In the simplest non-trivial case of the metrized Lie algebra $\mathfrak{sl}(2)$ the recent results on the explicit form of the generating functions of the values of a weight system on important series of chord diagrams are presented. The computations are based on the Chmutov-Varchenko recurrence relations. Another recent result presented is the construction of recurrence relations for the values of the $\mathfrak{gl}(N)$-weight system. These relations are based on Kazarian's idea of extending the $\mathfrak{gl}(N)$-weight system to arbitrary permutations. In a number of recent papers an approach to the extension of weight systems and graph invariants to arbitrary embedded graphs was proposed, which is based on an analysis of the structure of the relevant Hopf algebras. The main principles of this approach are described. Weight systems defined on embedded graphs correspond to finite-order invariants of links ('knots' with several components). Bibliography: 65 titles.

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An extension of Stanley's chromatic symmetric function to binary delta-matroids

Cited by 4 publications

References 9 publications

Weight systems and invariants of graphs and embedded graphs

Weight systems and invariants of graphs and embedded graphs

Skew characteristic polynomial of graphs and embedded graphs

Weight systems and invariants of graphs and embedded graphs

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