# Delta-matroids and Vassiliev invariants

@article{Lando2016DeltamatroidsAV, title={Delta-matroids and Vassiliev invariants}, author={Sergey Lando and V. Zhukov}, journal={arXiv: Combinatorics}, year={2016} }

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. In the study of the sl2 weight system, it was shown that its value on a chord diagram depends on the intersection graph of the diagram rather than on the diagram itself. Moreover, it was shown that the value of this weight system on an intersection graph depends on the cy- cle matroid of the graph rather than on the graph itself… Expand

#### Tables from this paper

#### 9 Citations

Lagrangian Subspaces, Delta-Matroids, and Four-Term Relations

- Mathematics
- 2018

Finite-order invariants (Vassiliev invariants) of knots are expressed in terms of weight systems, that is, functions on chord diagrams (embedded graphs with a single vertex) satisfying the four-term… Expand

Invariants of Framed Graphs and the Kadomtsev—Petviashvili Hierarchy

- Mathematics
- Functional Analysis and Its Applications
- 2019

S. V. Chmutov, M. E. Kazarian, and S. K. Lando have recently introduced a class of graph invariants, which they called shadow invariants (these invariants are graded homomorphisms from the Hopf… Expand

Values of the $$\mathfrak{sl}_2$$ Weight System on Complete Bipartite Graphs

- Mathematics
- 2020

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… Expand

Transition polynomial as a weight system for binary delta-matroids

- Mathematics
- 2019

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 4-regular graph endowed with an oriented Euler circuit. For a… Expand

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

- Computer Science, Mathematics
- Discret. Math.
- 2021

This work considers the binary delta-matroid associated to an embedded graph and defines the extended Stanley's chromatic polynomial as an invariant ofDelta-matroids, which satisfies 4-term relations for simple graphs and determines in this way a knot invariant. Expand

Ribbon graphs and bialgebra of Lagrangian subspaces

- Mathematics
- 2015

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… Expand

The interlace polynomial of binary delta-matroids and link invariants

- Mathematics
- 2020

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… Expand

An action of the Coxeter group $BC_n$ on maps on surfaces, Lagrangian matroids and their representations

- Mathematics
- 2015

For a map $\mathcal M$ cellularly embedded on a connected and closed orientable surface, the bases of its Lagrangian (also known as delta-) matroid $\Delta(\mathcal M)$ correspond to the bases of a… Expand

Values of the $\mathfrak{sl}_2$ weight system on complete bipartite graphs

- Mathematics
- 2021

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… Expand

#### References

SHOWING 1-10 OF 14 REFERENCES

A Polynomial Invariant of Graphs On Orientable Surfaces

- Mathematics
- 2001

Our aim in this paper is to construct a polynomial invariant of cyclic graphs, that is, graphs with cyclic orders at the vertices, or, equivalently, of 2-cell embeddings of graphs into closed… Expand

Handle slides for delta-matroids

- Computer Science, Mathematics
- Eur. J. Comb.
- 2017

The delta-matroid results are compatible with the surface results in the sense that they are their ribbon graphic delta- matroidal analogues. Expand

On a weight system conjecturally related to sl2

- Mathematics, Computer Science
- Eur. J. Comb.
- 2014

It is shown that for small orders of the diagrams, the value of the weight system R"k on a diagram of order exactly 2k coincides with the coefficient of c^k in thevalue of the sl"2-weight system on the projection of the diagram to primitive elements. Expand

J-invariants of plane curves and framed chord diagrams

- Mathematics
- 2006

Arnold defined J-invariants of general plane curves as functions on classes of such curves that jump in a prescribed way when passing through curves with self-tangency. The coalgebra of framed chord… Expand

VASSILIEV KNOT INVARIANTS COMING FROM LIE ALGEBRAS AND 4-INVARIANTS

- Mathematics
- 2001

We study the 4-bialgebra of graphs and the bialgebra of 4-invariants introduced by S. K. Lando. Our main goal is the investigation of the relationship between 4-invariants of graphs and weight… Expand

Ribbon graphs and bialgebra of Lagrangian subspaces

- Mathematics
- 2015

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… Expand

Mutant knots and intersection graphs

- Mathematics
- 2007

We prove that if a finite order knot invariant does not distinguish mutant knots, then the corresponding weight system depends on the intersection graph of a chord diagram rather than on the diagram… Expand

Graphs on Surfaces and Their Applications

- Mathematics
- 2003

0 Introduction: What is This Book About.- 1 Constellations, Coverings, and Maps.- 2 Dessins d'Enfants.- 3 Introduction to the Matrix Integrals Method.- 4 Geometry of Moduli Spaces of Complex Curves.-… Expand

A polynomial of graphs on surfaces

- Mathematics
- 2002

ribbon graphs, i.e., graphs realized as disks (vertices) joined together by strips (edges) glued to their boundaries, corresponding to neighbourhoods of graphs embedded into surfaces. We construct a… Expand

A parity map of framed chord diagrams

- Mathematics
- 2015

We consider framed chord diagrams, i.e. chord diagrams with chords of two types. It is well known that chord diagrams modulo 4T-relations admit Hopf algebra structure, where the multiplication is… Expand