The interlace polynomial of binary delta-matroids and link invariants

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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“…Theorem 2.6 (see [24] and [38]). At any primitive element the skew characteristic polynomial of a graph is a constant.…”

Section: 3mentioning

confidence: 99%

“…(see [49]). Theorem 4.6 (see [38]). The interlace polynomials of even binary delta-matroids satisfy the 4-term relations for these matroids.…”

Section: 3mentioning

confidence: 99%

Weight systems and invariants of graphs and embedded graphs

Kazarian

Lando

2022

Russian Math. Surveys

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“…Theorem 2.6 ([25], [39]). The skew characteristic polynomial of a graph on any primitive element is a constant.…”

Section: Graph Invariants and The Hopf Algebra Structurementioning

confidence: 99%

Weight systems and invariants of graphs and embedded graphs

Kazaryan¹,

Lando²

2023

Preprint

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“…For the skew characteristic polynomial, this is not true, and we use only the knowledge of the value of the extended invariant on primitives. Extensions of graph invariants to delta-matroids in their relationship with knot and link invariants are studied also in [2,11].…”

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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The interlace polynomial of binary delta-matroids and link invariants

Abstract: In this work, we study the interlace polynomial as a generalization of a graph invariant to delta-matroids. We prove that the interlace polynomial satisfies the four-term relation for delta-matroids and determines thus a finite type invariant of links in the 3-sphere.

Cited by 4 publications

References 10 publications

Weight systems and invariants of graphs and embedded graphs

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

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