Weight systems and invariants of graphs and embedded graphs

TL;DR

This paper reviews recent advances in the theory of weight systems, focusing on their constructions from graph invariants and Lie algebra representations, including explicit formulas and recurrence relations for key cases.

Contribution

It introduces new constructions of weight systems from graph invariants and Lie algebras, and discusses extending these concepts to embedded graphs and link invariants.

Findings

Explicit generating functions for (2) weight system values

Recurrence relations for (N) weight system computations

Extension of weight systems to arbitrary embedded graphs

Abstract

We describe recent achievements in the theory of weight systems, which are functions on chord diagrams satisfying so-called $4$-term relations. Our main attention is devoted to constructions of weight systems. The two main sources of these constructions that are discussed in the paper are invariants of intersection graphs of chord diagrams that satisfy $4$-term relations for graphs and metrized Lie algebras. For the simplest nontrivial metrized Lie algebra $\mathfrak {sl}(2)$, we present recent results about the explicit form of generating functions for the values of the corresponding weight system on important families of chord diagrams. We also explain another recent result: construction of recurrence relations for computing the values of the $\mathfrak{gl}(N)$-weight system. These relations are based on M. Kazarian's extension of the $\mathfrak{gl}(N)$-weight system to arbitrary permutations. Certain recent papers suggest an approach to extending weight systems and graph invariants to arbitrary embedded graphs, which is based on the study of the corresponding Hopf algebra structures; we describe this approach. Weight systems defined on arbitrary embedded graphs correspond to finite type invariants of links (multicomponent knots).

Equations

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$$f(C)-f(C_{ab}^{\prime })=f({\tilde{C}}_{ab})-f({\tilde{C}}_{ab}^{\prime })$$

$$f(G)-f(G_{ab}^{\prime })=f({\tilde{G}}_{ab})-f({\tilde{G}}_{ab}^{\prime });$$

$${\chi }_G(c)={\chi }_{G_e^{\prime }}(c)-{\chi }_{G_e^{\prime \prime }}(c)$$

$${\chi }_G(c)-{\chi }_{G_{ab}^{\prime }}(c)=-{\chi }_{G_{ab}^{\prime \prime }}(c).$$

$${\chi }_{{\tilde{G}}_{ab}}(c)-{\chi }_{{\tilde{G}}_{ab}^{\prime }}(c)=-{\chi }_{{\tilde{G}}_{ab}^{\prime \prime }}(c).$$

$$W_G=W_{G_e^{\prime }}+W_{G_e^{\prime \prime }},$$

$$S_G(x_1,x_2,\dots )=\sum _{\begin{array}{c}\beta :V(G)\to X\\ \beta \text{ regular}\end{array}}{x}_{\beta },$$

$$p_k=\sum _{i=1}^{\infty }{x}_i^k$$

$$L_G(x)=L_{G\setminus a}(x)+L_{G^{ab}\setminus b}(x),$$

$$T_C(x)=\sum _{s:V(C)\to \{\varphi ,\chi ,\psi \}}\prod _{v\in V(C)}w(s(v))x^{c(s)-1},$$

$$A({\tilde{G}}_{ab})=I_{12}^{t}A(G)I_{12},$$

$$Q_G(u)=\sum _{U\subset V(G)}\nu (G{|}_U)u^{|V(G)|-|U|},$$

$$\mathcal{G}={\mathcal{G}}_0\oplus {\mathcal{G}}_1\oplus {\mathcal{G}}_2\oplus \dots$$

$$m:G_1\otimes G_2\mapsto G_1\bigsqcup G_2$$

$$m:{\mathcal{G}}_k\otimes {\mathcal{G}}_n\to {\mathcal{G}}_{k+n}$$

$$\mu :G\mapsto \sum _{V(G)=U\bigsqcup W}G{|}_U\otimes G{|}_W,$$

$$\mu :p\mapsto 1\otimes p+p\otimes 1.$$

$$H=H_0\oplus H_1\oplus H_2\oplus \dots ,$$

$$m:H_i\otimes H_j\to H_{i+j},\mu :H_n\to \underset{i+j=n}{⨁}{H}_i\otimes H_j.$$

$$\log \text{leavevmode}\varphi ={\varphi }_0-\frac{1}{2}{\varphi }_0\ast {\varphi }_0+\frac{1}{3}{\varphi }_0\ast {\varphi }_0\ast {\varphi }_0-\dots ,$$

$$\pi (G)=G-1!\sum _{U_1\bigsqcup U_2=V(G)}G{|}_{U_1}G{|}_{U_1}+2!\sum _{U_1\bigsqcup U_2\bigsqcup U_3=V(G)}G{|}_{U_1}G{|}_{U_2}G{|}_{U_3}-\dots ,$$

$$G\mapsto \sum _{E^{\prime }\subset E(G)}(-1{)}^{|E(G)|-|E^{\prime }|}G{|}_{E^{\prime }},$$

$${\chi }_G(x+y)=\sum _{U\bigsqcup W=V(G)}{\chi }_{G{|}_U}(x){\chi }_{G{|}_W}(y),$$

$$\frac{{\partial }^{2}F}{\partial p_2^2}$$

$$\frac{{\partial }^{2}F}{\partial p_2\partial p_3}$$

$${\mathcal{I}}^{\circ }(q_1,q_2,\dots )=\sum _G\frac{I_G(q_1,q_2,\dots )}{|\mathrm{Aut}(G)|},$$

$$i_n=n!\sum _{G\text{ connected}}\frac{[q_n]I_G(q_1,q_2,\dots )}{|\mathrm{Aut}(G)|}$$

$$S(p_1,p_2,\dots )=1+2^0{s}_1(p_1)+2^1{s}_2(p_1,p_2)+\cdots +2^{n(n-1)/2}{s}_n(p_1,p_2,\dots ,p_n)+\dots .$$

$$\mu :C\mapsto \sum _{I\bigsqcup J=V(C)}C{|}_I\otimes C{|}_J.$$

$$\mathcal{C}={\mathcal{C}}_0\oplus {\mathcal{C}}_1\oplus {\mathcal{C}}_2\oplus \dots ,$$

Full text

1. 1 $4$-term relations

2. 1.1 Chord diagrams and $4$-term reltions

3. 1.2 Intersection graphs

4. 1.3 $4$-term relations for graphs

5. 1.3.1 Chromatic polynomial

6. 1.3.2 Weighted chromatic polynomial (Stanley’s symmetrized chromatic polynomial)

7. 1.3.3 Interlace polynomial

8. 1.3.4 Transition polynomial

9. 1.3.5 Skew characteristic polynomial

10. 2 Hopf algebra of graphs

11. 2.1 Multiplication and comultiplication of graphs

12. 2.2 Primitive elements, Milnor–Moore theorem, and Hopf algebra structure

13. 2.3 Graph invariants and the Hopf algebra structure

14. 2.4 Hopf algebra of graphs and integrability

15. 2.4.1 KP integrable hierarchy

16. 2.4.2 Umbral graph invariants

17. 2.4.3 Integrability of umbral invariants

18. 3 Hopf algebra of chord diagrams and weight systems associated to Lie algebras

19. 3.1 Structure of the Hopf algebra of chord diagrams

20. 3.2 Universal construction of a weight system associated to a Lie algebra

21. 3.2.1 Definition of the weight system

22. 3.2.2 Jacobi diagrams and the values of the weight system on primitive elements

23. 3.3 $\mathfrak{sl}(2)$ weight system

24. 3.3.1 Chmutov-Varchenko recursion relation

25. 3.3.2 $\mathfrak{sl}(2)$-weight system for graphs

26. 3.3.3 The values of the $\mathfrak{sl}(2)$ weight system on complete graphs

27. 3.3.4 Algebra of shares

28. 3.3.5 Values of the $\mathfrak{sl}(2)$-weight system on complete bipartite graphs

29. 3.3.6 Values of the $\mathfrak{sl}(2)$-weight system on graphs that are not intersection graphs

30. 3.4 $\mathfrak{gl}(N)$-weight system

31. 3.4.1 $\mathfrak{gl}(N)$-weight system on permutation and recurrence relations

32. 3.4.2 Computing the $\mathfrak{gl}(N)$-weight system by means of the Harish-Chandra isomorphism

33. 3.5 $\mathfrak{sl}(N)$ weight system

34. 3.6 The weight system $\mathfrak{gl}(1|1)$

35. 4 Embedded graphs and delta-matroids

36. 4.1 Embedded graphs and $4$-term relations

37. 4.2 Delta-matroids of graphs and embedded graphs, and $4$-term relations for them

38. 4.3 Hopf algebras of delta-matroids

39. 4.4 Extending graph invariants to embedded graphs and delta-matroids

40. 4.4.1 Interlace polynomial

41. 4.4.2 Stanley’s symmetrized chromatic polynomial

42. 4.4.3 Transition polynomial

43. 4.4.4 Skew characteristic polynomial

44. 4.5 Extending to nonorientable surfaces: framed chord diagrams, framed graphs, and not necessarily even delta-matroids

[TABLE]

The theory of finite type invariants of knots and links constructed mainly in 1990’ies originating in the paper [62] leads to a necessity to study weight systems. A weight system is a function on chord diagrams satisfying so-called Vassiliev’s $4$-term relations. Usually one considers weight systems taking values in a commutative ring. In turn, a chord diagram is a combinatorial object, a graph of special kind. One can think of it as of a $3$-regular graph together with a distinguished Hamiltonian circuit. Two such graphs are considered as being isomorphic if there is an isomorphism taking the Hamiltonian circuit in the first graph to the one in the second graph, with orientation preserved.

A $4$-term relation is constructed from a chord diagram and a pair of its vertices that are neighbors along the Hamiltonian circuit; it is a linear relation on the values of a function on four chord diagrams with one and the same number of vertices. Since the relations are linear, all weight systems can be described as solutions to a system of linear equations. However, both the number of variables (chord diagrams), and the number of equations ($4$-term relations) grows very rapidly as the number of vertices grows (thus, there are more than 600 thousand diagrams with $2\cdot 9=18$ vertices, and the number of relatons exceeds $5\cdot 10^{6}$). In contrast, the rank of the equations matrix, that is, the dimension of the space of chord diagrams modulo $4$-term relations, grows much slower, and for $9$ chords the rank is only $104$. There are no universal tools to compute these dimensions, and only rough upper and lower estimates are known.

Studying spaces of all weight systems must be completed by constructing specific weight systems and families of such weight systems. These constructions are used, in particular, to obtain lower bounds for the dimensions. Two main sources of constructions are invariants of intersection graphs of chord diagrams that satisfy $4$-term relations for graphs [45], and metrized Lie algebras [40, 5]. These two constructions are essentially different. Graph invariants usually can be easily computed, but their distinguishing power is rather poor. In contrast, weight systems associated to Lie algebras are powerful, but no approaches to their efficient computation has been known till recently. In the present paper, we describe both approaches, with a stress on recent results.

The problem of computing values of weight systems associated with metrized Lie algebras on chord diagrams is complicated because computations must be done in a highly noncommutative algebra, which is the universal enveloping algebra of the Lie algebra. For the simplest nontrivial case of the Lie algebra $\mathfrak{sl}(2)$, the computations could be simplified by using the Chmutov–Varchenko recurrence relations [22], but they also often prove to be nonsufficient. Thus, for long we have not known the values of the $\mathfrak{sl}(2)$-weight system of chord diagrams in which any two chords intersect one another (that is, whose intersection graph is a complete graph). The final answer has been given by P. Zakorko [65] by proving S. Lando’s 2015 conjecture about the generating function for these values. Another recent achievement is the construction of recurrence relations for the values of the $\mathfrak{gl}(N)$-weight system [64]. These relations are based on an extension of the $\mathfrak{gl}(N)$-weight system to arbitrary permutations suggested by M. Kazarian (while chord diagrams are interpreted as permutations of special kind).

One of the key problems in studying graph invariants is the one about their extension to embedded graphs. This question is studied in many recent papers. In most of them, the invariant of embedded graphs is defined as the invariant of the underlying abstract graph enriched by information about the embedding. We, however, are interested first of all in extending weight systems, which are defined by finite type knot invariants, to weight systems associated to finite type invariants of links (a link, in contrast to a knot, has several connected components). A chord diagram can be interpreted as an embedded graph with a single vertex on an orientable surface, while links correspond to generalized chord diagrams, which are embedded graphs, whose number of vertices coincides with the number of link components, on orientable surfaces.

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. We do not know such a structure on the space spanned by embedded graphs. It exists, however, on the space of binary delta-matroids [47], which are combinatorial objects that accumulate important information about graphs or chord diagrams, and about embedded graphs. However, extending weight systems associated to metrized Lie algebras to embedded graphs with arbitrarily mny vertices remains an open problem.

The paper has the following structure.

In Sec. 1, we recall main definitions related to chord diagrams, their intersection graphs, $4$-term relations for chord diagrams and graphs. We give several key examples of graph invariants that satisfy $4$-term relations and define thus weight systems.

In Sec. 2, the Hopf algebra structure on the space of graphs is described as well as the behavior of the graph invariants described above with respect to this structure. Here we also mention recent results [21] relating the Hopf algebra structure to an integrability property of graph invariants.

Section 3 is devoted to weight systems associated to Lie algebras. In addition to known definitions and results about the $\mathfrak{sl}(2)$-weight system we describe recent results about its values on chord diagrams whose intersection graphs is complete bipartite, due to P. Zinova and M. Kazarian, as well as Lando’s conjecture (P. Zakorko’s theorem) about its values on chord diagrams with complete intersection graphs. This section also describes the extension of the $\mathfrak{gl}(N)$-weight system to arbitrary permutations and recurrence relations for its computation.

Section 4 is devoted to constructing extensions of weight systems from chord diagrams to arbitrary embedded graphs. Here we introduce delta-matroids and describe the Hopf algebra structure on the space of even binary delta-matroids.

All necessary definitions and classical results about weight systems can be found in [18]. We reproduce only those definitions and statements that are necessary to state recent results.

The authors are partially supported by International Laboratory of Cluster Geometry NRU HSE, RF Government grant, ag. № 075-15-2021-608 dated 08.06.2021.

1 $4$-term relations

In 1990, V. Vassiliev [62] introduced the notion of finite type knot invariant and associated to any knot invariant of order at most $n$ a function on chord diagrams with $n$ chords. He showed that any such function satisfies $4$-term relations. In 1993, M. Kontsevich [40] proved the converse: if a function on chord diagrams with $n$ chords taking values in a field of characteristic zero satisfies the $4$-term relations, then it can be obtained by Vassiliev’s construction from some knot invariant of order at most $n$. (In Vassiliev’s theorem, as well as in Kontsevich’s one, the function is subject to an additional requirement; namely, it must satisfy the so-called one-term relation. This requirement, however, vanishes when knots are replaced by framed knots, it does not affect the theory, and we will not mention it below). Functions on chord diagrams satisfying $4$-term relations are called weight systems.

The universal finite type knot invariant constructed by Kontsevich (Kontsevich’s integral) allows one to reconstruct, in principle, a finite type invariant from the corresponding weight system. Therefore, studying weight systems plays a key role in understanding of the nature of finite type knot invariants. Note, however, that explicit computation of the Kontsevich integral of a knot is a computationally complicated problem that does not have a satisfactory solution up to now.

1.1 Chord diagrams and $4$-term reltions

A chord diagram of order $n$ is an oriented circle together with $2n$ pairwise distinct points on it split into $n$ pairs considered up to orientation preserving diffeomorphisms of the circle. We call the circle carrying the diagram its Wilson loop.

Everywhere in the pictures below we assume that the circle is oriented counterclockwise. The points forming a pair are connected by a line or arc segment.

A $4$-term relation is defined by a chord diagram and a pair of chords having neighboring ends in it. We say that a function $f$ on chord diagrams satisfies Vassiliev’s $4$-term relations if for any chord diagram $C$ and any pair of chords $a,b$ in $C$ having neighboring ends the equation shown in Fig. 1 holds.

The equation in the picture has the following meaning. All the four chord diagrams entering it can have an additional tuple of chords whose ends belong to the dashed arcs, which is one and the same in all the four diagrams. Out of the four ends of the two distinguished chords $a,b$, both end of $b$ as well as one end of $a$ are fixed, while the second end of $a$ takes successively all the four positions close to the ends of $b$. The specific position of the second end of $a$ determines the chord diagram in the relation. Note that it does not matter whether the two chords $a$ and $b$ intersect one another or not: the case of intersecting chord reduces to that of nonintersecting ones by multiplying the relation shown in Fig. 1 by $-1$.

Below, we write down the $4$-term relation in the form

[TABLE]

and say that the chord diagram $C^{\prime}_{ab}$ is obtained from $C$ by Vassiliev’s first move on the pair of chords $a,b$, the chord diagram $\widetilde{C}_{ab}$ is the result of Vassiliev’s second move, and $\widetilde{C}^{\prime}_{ab}$ is the result of the composition of the two moves; note that the first and the second move, when elaborated on one and the same pair of chords, commute with one another, and the order in which they are performed is irrelevant.

The definition of the $4$-term relation uses subtraction, and we are planning to multiply chord diagrams in what follows. That is why we will usually consider weight systems with values in a commutative ring, making the ring explicit in specific examples.

1.2 Intersection graphs

Invariants of intersection graphs of chord diagrams serve as one of the main sources of weight systems,

The intersection graph $g(C)$ of a chord diagram $C$ is the simple graph the set of whose vertices is in one-to-one correspondence with the set of chords of $C$, and two vertices are connected by an edge iff the corresponding chords intersect. (Two chords in a chord diagram intersect one another if their ends follow the circle in alternating order; the intersection point of chords in pictures is not a vertex of the chord diagram.) Fig. 2 shows an example of a $4$-term relation for a chord diagram and its intersection graph. Only the chord diagrams and their intersection graphs are depicted; there is no reference to the value of the function.

Any graph with at most $5$ vertices is the intersection graph of some chord diagram. There are two graphs with $6$ vertices that are not intersection graphs; they are shown in Fig. 3. As $n$ grows, the fraction of intersection graphs among simple graphs with $n$ vertices grows rapidly. A variety of complete sets of obstacles for a graph to be an intersection graph is known.

On the other hand, certain intersection graphs can be realized as intersection graphs of several, not pairwise isomorphic chord diagrams. Consider, for example, the chain on $n$ vertices,

\underbrace{\makebox(5.4,0.0)[]{\begin{picture}(5.0,0.2)(0.0,-0.2)\put(0.0,0.0){\circle*{0.15}} \put(0.0,0.0){\line(1,0){1.0}} \put(1.0,0.0){\circle*{0.15}} \put(1.0,0.0){\line(1,0){1.0}} \put(2.0,0.0){\circle*{0.15}} \put(2.0,0.0){\line(1,0){0.5}} \put(3.08,0.0){\makebox(0.0,0.0)[]{\ldots}} \put(3.5,0.0){\line(1,0){0.5}} \put(4.0,0.0){\circle*{0.15}} \put(4.0,0.0){\line(1,0){1.0}} \put(5.0,0.0){\circle*{0.15}} \end{picture}}}_{\scriptstyle n\ {\rm vertices}}

For $n=5$, there are three distinct chord diagrams with this intersection graph:

1.3 $4$-term relations for graphs

$4$-term relations for graphs were introduced in [45]. We say that a graph invariant $f$ satisfies the $4$-term relation if for any graph $G$ and any pair $a,b$ of its vertices we have

[TABLE]

here the graph $G^{\prime}_{ab}$ is obtained from $G$ by switching adjacency between the vertices $a$ and $b$, the graph $\widetilde{G}_{ab}$ is obtained from $G$ by switching adjacency of $a$ to all the vertices in $G$ adjacent to $b$, and the graph $\widetilde{G}^{\prime}_{ab}$ is the result of composition of these two operations.

Remark**.**

The transformation $G\mapsto G^{\prime}_{ab}$ is symmetric with respect to $a$ and $b$, $G^{\prime}_{ab}=G^{\prime}_{ba}$. In turn, the second Vassiliev move for graphs $G\mapsto\widetilde{G}_{ab}$ is not symmetric, and the graph $\widetilde{G}_{ba}$is not, as a rule, isomorphic to $\widetilde{G}_{ba}$.

Graph invariants satisfying $4$-term relations are called $4$-invariants. Similarly to the case of weight systems, we assume that $4$-invariants take values in a commutative ring.

A direct study of the way the intersection graph of a chord diagram changes under Vassiliev moves proves the following statement.

Theorem 1.1** ([45]).**

If $f$ is a $4$-invarint of graphs, then $f\circ g$ is a weight system on chord diagrams.

An example of a $4$-term relation is given in Fig. 2. Note, however, that even if a graph $G$ is an intersection graph, this does not mean that the same is true for the other three graph in a $4$-term relation.

Therefore, any $4$-invariant determines a weight system, and hence, by Kontsevich’s theorem, a knot invariant. Below, we give several examples of $4$-invariants.

The following question posed by S. Lando remains open for the last two decades:

Problem**.**

Is it true that modulo $4$-term relations for graphs any graph is a linear combination of intersection graphs?

Computer computations (I. A. Dynnikov, private communication) show that this is true for all graphs with at most $8$ vertices.

1.3.1 Chromatic polynomial

One of historically first examples of $4$-invariants of simple graphs is the chromatic polynomials. Let $G$ be a graph, and let $V(G)$ be the set of its vertices. The chromatic polynomial $\chi_{G}(c)$ of $G$ is the number of regular colorings of the set $V(G)$ in $c$ colors, that is, the number of mappings $V(G)\to\{1,2,\dots,c\}$ such that for any two adjacent (connected by an edge) vertices the values of the mapping are different.

It is well known that the chromatic polynomial satisfies the contraction–deletion relation:

[TABLE]

for any graph $G$ and any edge $e$ in it; here the graph $G^{\prime}_{e}$ is obtained from $G$ be deleting the edge $e$, and $G^{\prime\prime}_{e}$ is the result of contracting this edge. As an edge is contracted, it is deleted, its two ends become a single new vertex of the graph, and any multiple edge that can appear after that is replaced by a single edge.

The contraction–deletion relation can be proved easily: it corresponds to splitting of the set of regular colorings of the vertices of the graph $G^{\prime}_{e}$ in $c$ colors into two disjoint subsets, namely, those colorings where the ends of $e$ are colored differently (these are exactly the regular colorings of the vertices of $G$) and those where these ends are colored with the same color (the latter correspond one-to-one to regular colorings of the vertices of $G^{\prime\prime}_{e}$). Since the chromatic polynomial of a discrete (edge-free) graph on $n$ vertices is $c^{n}$, the contraction-deletion relation proves that, for a graph $G$ on $n$ vertices, $\chi_{G}(c)$ indeed is a polynomial of degree $n$ having the leading coefficient $1$.

Theorem 1.2** ([17]).**

Chromatic polynomial is a $4$-invariant,

Indeed, applying the contraction–deletion relation to a graph $G$ and an edge $e$ having the ends $a,b$ in it, we conclude that

[TABLE]

In turn,

[TABLE]

Verifying that the natural identification of the sets of vertices of the two graphs $G^{\prime\prime}_{ab}$ and $\widetilde{G}^{\prime\prime}_{ab}$ establishes their isomorphism completes the proof.

1.3.2 Weighted chromatic polynomial

(Stanley’s symmetrized chromatic polynomial)

Chromatic polynomial is an important graph invariant; however, it is not distinguishingly powerful. For example, the chromatic polynomial of any tree on $n$ vertices is $c(c-1)^{n-1}$, while the number of pairwise nonisomorphic trees on $n$ vertices grows very rapidly as $n$ grows. Weighted chromatic polynomial, more widely known under the name of Stanley’s symmetrized chromatic polynomial, is a much finer graph invariant.

In order to define weighted chromatic polynomial, we will need the notion of weighted graph.

Definition 1.3**.**

A weighted graph is a simple graph $G$ together with a positive integer, a weight, assigned to each of its vertices. The weight of a weighted graph is the sum of the weights of its vertices.

Definition 1.4** ([17]).**

The weighted chromatic polynomial is the weighted graph invariant $G\mapsto W_{G}(q_{1},q_{2},\dots)$ taking values in the ring of polynomials in infinitely many variables $q_{1},q_{2},\dots$ and possessing the following properties:

• •

the weighted chromatic polynomial of the graph on one vertex of weight $n$ is $q_{n}$;

• •

the weighted chromatic polynomial is multiplicative, $W_{G_{1}\sqcup G_{2}}=W_{G_{1}}W_{G_{2}}$ for a disjoint union of arbitrary weighted graphs $G_{1},G_{2}$;

• •

the weighted chromatic polynomial satisfies the weighted contraction–deletion relation

[TABLE]

where $e$ is an edge of the weighted graph $G$, $G^{\prime}_{e}$ denotes the result of deleting an edge $e$ from $G$, and $G^{\prime\prime}_{e}$ is the result of contracting $e$; contraction is defined in the same way as for simple graphs, and the new vertex is assigned the weight equal to the sum of the weights of the two ends of $e$.

Theorem 1.5** ([17]).**

There is a unique weighted graph invariant possessing the above properties. The weighted chromatic polynomial $W_{G}$ of a weighted graph $G$ of weight $n$ is a quasihomogeneous polynomial of the variables $q_{1},q_{2},\dots$ of degree $n$, if we set the weight of variables $q_{k}$ equal to $k$, for $k=1,2,\dots$.

A simple graph can be considered as a weighted graph, with the weight of each of its vertices set to be $1$. Thus, any weighted graphs invariant defines therefore a simple graphs invariant.

Theorem 1.6**.**

The weighted chromatic polynomial is a $4$-invariant.

Indeed, similarly to the proof for chromatic polynomial, we remark that the natural identification of the set of vertices of the graphs $G^{\prime\prime}_{ab}$ and $\widetilde{G}^{\prime\prime}_{ab}$ establishes an isomorphism of these graphs as weighted graphs.

In 1995, R. Stanley introduced the notion of symmetrized chromatic polynomial.

Definition 1.7** ([60]).**

Let $G$ be a simple graph. A coloring of the set of vertices $V(G)$ of $G$ into an infinite set of colors is a mapping $\beta:V(G)\to X=\{x_{1},x_{2},\dots\}$. To each coloring $\beta$, one associates a monomial $x_{\beta}$ of degree $n=|V(G)|$ in the variables $x_{1},x_{2},\dots$, which is equal to the product of the values of $\beta$ on all the vertices of $G$. The symmetrized chromatic polynomial $S_{G}(x_{1},x_{2},\dots)$ of $G$ is the infinite sum

[TABLE]

where the coloring $\beta$ is said to be regular if it takes any two adjacent vertices to different elements of $X$.

By obvious reasons, the symmetrized chromatic polynomial is a symmetric function in the variables $X$. Stanley’s conjecture asserts that the symmetrized chromatic polynomial distinguishes between any two trees. It is confirmed for trees with up to $29$ vertices [33], indicating thus that the symmetrized chromatic polynomial is a much finer graph invariant than the ordinary chromatic polynomial $\chi_{G}$.

Being a symmetric polynomial in the variables $X$, any symmetrized chromatic polynomial can be expressed as a polynomial in any basis in the space of symmetric polynomials. In particular, one can choose the power sums

[TABLE]

for this basis. When expressed in this form, the symmetrized chromatic polynomial $S_{G}$ becomes a finite quasihomogeneous polynomial of degree $n=|V(G)|$ in the variables $p_{1},p_{2},\dots$ if we set the degree of the variable $p_{k}$ equal to $k$, for $k=1,2,\dots$. The following statement establishes a relationship between Stanley’s symemtrized chromatic polynomial and the weighted chromatic polynomial.

Theorem 1.8** ([53]).**

Stanley’s symmetrized chromatic polynomial, when expressed in terms of power sums, under the substitution $p_{k}=(-1)^{n-k}q_{k}$, $k=1,2,\dots$, becomes the weighted chromatic polynomial; here $n=|V(G)|$.

As a corollary, Stanley’s symmetrized chromatic polynomial is a $4$-invariant of graphs.

In turn, the ordinary chromatic polynomial is a specialization of Stanley’s symmetrized one: the latter transforms into the former under the substitution $p_{i}=c$, for $i=1,2,3,\dots$.

1.3.3 Interlace polynomial

Initially, the interlace polynomial has been defined as a function on oriented graphs having two-in and two-out edges at each vertex. The definition appeared first in the paper [4], which is devoted to DNA sequencing. Later, the definition has been extended to arbitrary simple graphs. We start with defining the pivot operation.

Let $G$ be a simple graph. For any pair $a,b$ of adjacent vertices of $G$, all the other its vertices are split into four classes, namely:

the vertices adjacent to $a$, and not to $b$; 2. 2.

the vertices adjacent to $b$, and not to $a$; 3. 3.

the vertices adjacent to both $a$, and $b$; 4. 4.

the vertices adjacent neither to $a$, nor to $b$.

The pivot $G^{ab}$ of $G$ along the edge $ab$ is the graph obtained by switching the adjacency between any two vertices in the first three classes iff they belong to different classes.

The interlace polynomial $L_{G}(x)$ is defined by the following recurrence relations:

if $G$ does not have edges, then $L_{G}(x)=x^{n}$, where $n$ is the number of vertices in $G$; 2. 2.

for any edge $ab$ in $G$, we have

[TABLE]

where $G\setminus a$ denotes the graph obtained from $G$ by deleting the vertex $a$ and all the edges incident to it.

In [4], it is proved that the interlace polynomial is well defined: the result of its calculation is independent of the order in which the pivots are applied. The definition immediately implies that $L_{G}(x)$ indeed is a polynomial in $x$ whose degree is $n=|V(G)|$.

Theorem 1.9**.**

The interlace polynomial satisfies $4$-term relations.

This theorem is proved in [52] and, in a different way, in [39].

1.3.4 Transition polynomial

Transition polynomial for $4$-regular graphs endowed with an oriented Eulerian circuit was introduced in [35]. By contracting each chord of a chord diagram $C$ we make the latter into a $4$-regular graph in which the Wilson loop of the chord diagram becomes an oriented Eulerian circuit. The transition polynomial of this graph defines, therefore, a mapping from the set of chord diagrams to a space of polynomials.

Conversely, to a $4$-regular graph with a distinguished oriented Eulerian circuit a chord diagram is associated; the number of chords in it coincides with the number of vertices in the graph. The Eulerian circuit turns into the Wilson loop of the chord diagram, and the vertices of the graph become the chords.

Here we define the transition polynomial for a chord diagram. In order to define it, we will need the notion of transition. Let $C$ be a chord diagram. Each chord in $C$ can be replaced by a ribbon in one of the following three ways; we encode these ways by the Greek letters $\chi$, $\phi$, or $\psi$:

• •

$\phi$ if this an ordinary ribbon;

• •

$\chi$ if this is a half-twisted ribbon;

• •

$\psi$ if this is no ribbon at all, that is, if we simply erase the chord.

A choice of a transition for each ribbon, i.e., a mapping $V(C)\to\{\phi,\chi,\psi\}$ taking the set of chords $V(C)$ to the set of transition types, is called a state of the chord diagram $D$.

Choose a weight function $w:\{\phi,\chi,\psi\}\to K$ that associates to each of the three Greek letters an element of a commutative ring $K$.

The weighted transition polynomial of a chord diagram $C$ is the polynomial

[TABLE]

where the summation is carried over all states of $C$, and $c(s)$ denotes the number of connected components of the boundary of the surface obtained from $C$ by attaching a disk to the Wilson loop with the chords replaced by the corresponding ribbons.

Theorem 1.10** ([26]).**

By choosing $w(\chi)=t$, $w(\phi)=-t$, $w(\psi)=s$, for the weight function, we make the transition polynomial into a weight system taking values in the ring of polynomials $\mathbb{C}[t,s,x]$.

1.3.5 Skew characteristic polynomial

Let $G$ be a simple graph with $n=|V(G)|$ vertices. Number the vertices by the numbers from $1$ to $n$ in an arbitrary way and associate to the chosen numbering the adjacency matrix of $G$. The adjacency matrix $A(G)$ is an $n\times n$-matrix over the two-element field $\mathbb{F}_{2}$, containing $1$ in the cell $(i,j)$ if the vertices numbered $i$ and $j$ are connected by an edge, and containing [math] otherwise. In particular, the matrix $A(G)$ is symmetric and has zeroes on the diagonal. The isomorphism class of $G$ is reconstructed uniquely from its adjacency matrix. Various characteristics of the adjacency matrix, for example, its characteristic polynomial, play a key role in the study of graphs.

Definition 1.11**.**

A graph $G$ is nondegenerate (respectively, degenerate) if its adjacency matrix $A(G)$ is nondegenerate (respectively, degenerate) over $\mathbb{F}_{2}$. The nondegeneracy of a graph is the graph invariant taking values in the field $\mathbb{C}$ and equal to $1$ for a nondegenerate graph and to [math] for a degenerate one.

The adjacency matrix of any graph with an odd number of veertices is degenerate since it is antisymmetric over the field $\mathbb{F}_{2}$, whence the nondegeneracy of any graph with an odd number of vertices is [math]. A graph with an even number of vertices can be either nondegenerate or degenerate. We denote the nondegeneracy of a graph $G$ by $\nu(G)$.

Lemma 1.12**.**

Nondegeneracy is invariant with respect to the second Vassiliev move.

Indeed, let $G$ be graph, and let $a,b$ be its vertices. Pick a numbering of the vertices of $G$ such that $a$ is assigned number $1$ and $b$ is assigned number $2$. Then the adjacency matrix $A(\widetilde{G}_{ab})$ of $\widetilde{G}_{ab}$ is

[TABLE]

where $I_{12}$ is the square $|V(G)|\times|V(G)|$-matrix over $\mathbb{F}_{2}$, which coincides with the identity matrix everywhere with the exception of the cell $(1,2)$ whose value equals $1$, and $I_{12}^{t}$ is its transpose matrix. The assertion of the lemma follows from the fact that both matrices $I_{12}$ and $I_{12}^{t}$ are nondegenerate.

Definition 1.13** ([25]).**

The skew characteristic polynomial of a graph $G$ is the polynomial

[TABLE]

where the summation is carried over all subsets $U$ of the set $V(G)$ of vertices of the graph, and $G|_{U}$ denotes the subgraph of $G$ induced by a subset $U$ of its vertices.

The skew characteristic polynomial of a graph with an even number of vertices is an even function of its argument, and if the number of vertices is odd, then the skew characteristic polynomial is odd as well.

Lemma 1.12 implies

Proposition 1.14**.**

The skew characteristic polynomial is a $4$-invariant of graphs.

Let us explain why the skew characteristic polynomial carries this name. Let $C$ be a chord diagram, and let $g(C)$ be its intersection graph. Pick the following orientation of the graph. Choosing a point $\alpha$ on the Wilson loop that is not an end of any chord, cut the circle at this point and develop it into a horizontal line whose orientation inherits that of the circle. Under this transformation, the chords become half circles (arcs), and two vertices in $g(C)$ are connected by an edge iff the corresponding arcs intersect one another. Orient each edge in $g(C)$ from the vertex corresponding to the arc whose left end precedes the left end of the arc corresponding to the second vertex of the edge. If we number the vertices of the graph in the order of the left ends of the corresponding arcs, then each edge is oriented from the vertex with a smaller number to that with a greater one. The orientation of the intersection graph obtained in this way depends on the cut point $\alpha$ of the circle; denote the corresponding directed graph by $\vec{g}_{\alpha}(C)$.

The intersection matrix of a directed graph with $n$ numbered vertices is the integer-valued $n\times n$-matrix whose $(i,j)$-entry is [math] if the vertices $i$ and $j$ are not connected by an edge, it is $1$ if the arrow goes from $i$ to $j$, and it is $-1$ if there is an arrow going from $j$ to $i$. The adjacency matrix of a directed graph is skew symmetric.

Proposition 1.15** ([25]).**

The characteristic polynomial of the adjacency matrix of the directed graph $\vec{g}_{\alpha}(C)$ is independent of the cut point $\alpha$.

Theorem 1.16** ([25]).**

The characteristic polynomial of the adjacency matrix of the directed intersection graph of a chord diagram $C$ coincides with the skew characteristic polynomial $Q_{g(C)}$ of its intersection graph.

Therefore, the skew characteristic polynomial of an intersection graph coincides with the characteristic polynomial of the oriented graph obtained by a certain orientation of its edges, which belongs to a class of admissible orientations. This fact justifies the choice of the name for the invariant. For a graph that is not an intersection graph, there could be no such orientation.

2 Hopf algebra of graphs

Many natural graph invariants are multiplicative. The value of such an invariant of a graph $G_{1}\sqcup G_{2}$, which is a disjoint union of two graphs, $G_{1}$ and $G_{2}$, is the product of its values on $G_{1}$ and $G_{2}$. In particular, all the invariants defined in Sec. 1.3 are multiplicative. Multiplicativity means that in order to calculate the value of the invariant on a graph it suffices to know its values on the connected components of the graph.

In this section we show that the behaviour of the invariants with respect to comultiplication of graphs is of not less importance. Multiplication and comultiplication of graphs together form a Hopf algebra structure on the vector space spanned by simple graphs. This Hopf algebra is a polynomial algebra in its primitive elements. The primitive elements are linear combinations of graphs, and for many graph invariants their values on primitive elements are much simpler than their values of the graphs themselves. Thus, the value of a chromatic polynomial on any primitive element is a linear polynomial, while the chromatic polynomial of a simple graph on $n$ vertices is a polynomial of degree $n$. Following [21], we call polynomial graph invariants whose values on primitive elements are linear polynomials umbral invariants. Integrability properties of umbral invariants are discussed in Sec. 2.4.

2.1 Multiplication and comultiplication of graphs

Denote by $\mathcal{G}_{n}$ the vector space over $\mathbb{C}$ freely spanned by all graphs on $n$ vertices, $n=0,1,2,\dots$, and let

[TABLE]

be the direct sum of these vector spaces. Note that the vector space $\mathcal{G}_{0}$ is one-dimensional; it is spanned by the empty graph. Introduce a commutative multiplication $m:\mathcal{G}\otimes\mathcal{G}\to\mathcal{G}$ on $\mathcal{G}$ by its values on the generators

[TABLE]

and extending its to linear combinations of graphs by linearity. Obviously, this multiplication is commutative and respects the grading:

[TABLE]

for all $k$ and $n$. The empty graph $e\in\mathcal{G}_{0}$ is the unity of this multiplication.

The action of the comultiplication $\mu:\mathcal{G}\to\mathcal{G}\otimes\mathcal{G}$ on a graph $G$ has the form 111Comultiplication is often denoted by $\Delta$; in the present paper, however, symbol $\Delta$ is heavily used in the different context of delta-matroids.

[TABLE]

where summation in the right-hand side is carried over all ordered partitions of the set of vertices $V(G)$ of $G$ into two disjoint subsets, and $G|_{U}$ denotes the subgraph of $G$ induced by a subset $U$ of its vertices. Comultiplication is extended to linear combinations of graphs by linearity. With respect to this comultiplication, the vector space $\mathcal{G}$ is a coalgebra. The counit with respect to $\mu$ is the linear mapping$\epsilon:\mathbb{C}\to\mathcal{G}_{0}$ taking $1$ to $e$.

2.2 Primitive elements, Milnor–Moore theorem,

and Hopf algebra structure

An element $p$ of a coalgebra with comultiplication $\mu$ is primitive if

[TABLE]

In the coalgebra $\mathcal{G}$, the graph $K_{1}\in\mathcal{G}_{1}$ consisting of a single vertex is primitive. No other graph is primitive, however certain linear combinations of graphs in which more than one graph participate are primitive. Thus, it is easy to check that the difference $K_{2}-K_{1}^{2}$ is primitive. Here and below we denote by $K_{n}$ the complete graph on $n$ vertices.

Any polynomial algebra $\mathbb{C}[y_{1},y_{2},\dots]$ in either finitely or infinitely many variables is endowed with a natural coalgebra structure if we declare the variables $y_{i}$ primitive, $\mu(y_{1})=1\otimes y_{i}+y_{i}\otimes 1$. Comultiplication is extended to monomials and their linear combinations (polynomials) as a ring homomorphism, i.e., $\mu(\prod_{k=1}^{n}y_{i_{k}})=\prod_{k=1}^{n}(1\otimes y_{i_{k}}+y_{i_{k}}\otimes 1)$.

Definition 2.1**.**

A tuple $(H,m,\mu,e,\epsilon,S)$, where

• •

$H$ is a vector space;

• •

$m:H\otimes H\to H$ is a multiplication with a unit $e:\mathbb{C}\to H$;

• •

$\mu:H\to H\otimes H$ is a comultiplication with a counit $\epsilon:H\to\mathbb{C}$;

• •

$S:H\to H$ is a linear mapping;

is called a Hopf algebra if

$m$ is a coalebra homomorphism; 2. 2.

$\mu$ is an algebra homomorphism; 3. 3.

$S$ satisfies the relation $m\circ(S\otimes{\rm Id})\circ\mu=\eta\circ\epsilon=m\circ({\rm Id}\otimes S)\circ\mu$.

The mapping $S$ in a Hopf algebra is called an antipode.

The algebra of polynomials $\mathbb{C}[y_{1},y_{2},\dots]$ becomes a Hopf algebra if we define the antipode $S$ as the mapping $S:y_{i}\mapsto-y_{i}$ for all $i=1,2,\dots$, and extend it to whole space $H$ as an algebra homomorphism.

A Hopf algebra $H$ can be graded. In this case, it is represented as a direct sum of finite dimensional subspaces

[TABLE]

and both multiplication and comultiplication must respect the grading, i.e.,

[TABLE]

A graded Hopf algebra is said to be connected if the vector space $H_{0}$ is one-dimensional.

If a grading (weight) $w(y_{i})\in\mathbb{N}$ of each variable $y_{i}$ in a polynomial Hopf algebra is given such that for each $n=1,2,3,\dots$ there are finitely many variables of weight at most $n$, then the polynomial Hopf algebra $\mathbb{C}[y_{1},y_{2},\dots]$ becomes graded. The subspace of grading $n$ in it is spanned by the monomials of weight $n$, where the weight of a monomial is the sum of the weights of the variables entering it.

The following Milnor–Moore theorem describes the structure of graded commutative cocommutative Hopf algebras.

Theorem 2.2** ([48]).**

Any connected graded commutative cocommutative Hopf algebra is a polynomial Hopf algebra in its primitive elements.

This theorem means that if we choose a basis in each subspace of primitive elements $P(H_{n})\subset H_{n}\subset H$ of grading $n$, then $H$ is a graded polynomial Hopf algebra in the elements of these basis if we set the weight of each element equal to its grading.

Remark**.**

The Milnor–Moore theorem remains true if one assumes that the Hopf algebra is cococmmutative only. However, we are going to use it only in the commutative case, where the statement above is sufficient.

A primitive element is naturally associated to any polynomial in a Hopf algebra of polynomials, namely, its linear part. This mapping from the vector space of polynomials to the vector space of their linear parts is a projection, i.e. its square coincides with itself. Milnor–Moore theorem 2.2 implies that the projection to the subspace of primitive elements is naturally defined in any connected graded commutative cocommutative Hopf algebra $H$. We denote this projection by $\pi$, $\pi:H\to P(H)$.

Each homogeneous subspace $H_{n}\subset H$ can be represented as a direct sum $H_{n}=P(H_{n})\oplus D(H_{n})$ of the subspace of primitive elements $P(H_{n})$ and the kernel $D(H_{n})$ of the projection $\pi$. The kernel $D(H_{n})$ consists of decomposable elements, that is, of polynomials in primitive elements of grading smaller than $n$.

The projection $\pi$ can be given by the following formula. Let $\phi,\psi:H\to H$ be linear mappings. Define the convolution product $\phi*\psi:H\to H$ as the linear mapping $(\phi*\psi)(a)=m((\phi\otimes\psi)(\mu(a)))$ for all $a\in H$. The convolution product in graded Hopf algebras can be used to define other operations on linear mappings, that are represented in terms of power series. Thus, if $\phi:H\to H$ is a grading preserving linear mapping of the Hopf algebra $H$ to itself such that $\phi(1)=1$, then the logarithm of $\phi$ can be defined as

[TABLE]

where $\phi_{0}:H\to H$ is the linear mapping given by $\phi_{0}:1\mapsto 0$, and $\phi_{0}$ coincides with $\phi$ on all homogeneous subspaces $H_{k}$ of positive grading $k>0$.

Theorem 2.3** ([60]).**

The projection $\pi:H\to H$ is the logarithm of the identity mapping, $\pi=\log\leavevmode\nobreak\ {\rm Id}$.

To prove this statement, it sufficies to verify that $\log\leavevmode\nobreak\ {\rm Id}(p)=\phi_{0}(p)=p$ for any primitive element $p$, and $\log\leavevmode\nobreak\ {\rm Id}(p_{1}p_{2}\dots)=0$ for any nonlinear monomial in primitive elements.

In particular, in the Hopf algebra of graphs $\mathcal{G}$ the projection $\pi$ to the subspace of primitive elements whose kernel is the subspace of decomposable elements is given by [44]

[TABLE]

where the summations are carried over all unordered partitions of the set $V(G)$ of the vertices of $G$ into $2,3,4,\dots$ nonempty disjoint subsets. For example, $\pi(K_{3})=K_{3}-3K_{1}K_{2}+2K_{1}^{3}$. It is easy to check that the linear combination of graphs in the right hand side indeed is a primitive element.

The projection $\pi:\mathcal{G}\to\mathcal{G}$ takes disconnected graphs to [math] since they are decomposable elements of the Hopf algebra $\mathcal{G}$. In turn, projections of connected graphs with $n$ vertices form a basis in the space $P(\mathcal{G}_{n})$ of primitive elements in grading $n$.

The identity mapping ${\rm Id}:\mathcal{G}\to\mathcal{G}$ preserves $4$-term relations, whence the same is true for its logarithm $\log\leavevmode\nobreak\ {\rm Id}=\pi$. As a corollary, the projection formula works in the Hopf algebra $\mathcal{F}$ as well.

In the Hopf algebra of chord diagrams $\mathcal{C}$, the projection formula looks similarly, with the set $V(C)$ of chords of the chord diagram $C$ replacing the set $V(G)$ of vertices of $G$.

Remark**.**

There is another way to associate to a graph a primitive elements in the Hopf algebra of graphs, see e.g. [3], which, probably, looks simpler. Namely, let’s take $G$ to the sum

[TABLE]

where summation is carried over all spanning subgraphs of $G$. However, in contrast to the projection $\pi$ this operation is specific to the Hopf algebra of graphs and cannot be extended by linearity to a projection to the subspace of primitives.

2.3 Graph invariants and the Hopf algebra structure

The behaviour of many graph invariants, as well as weight systems, is closely related to the structure of the corresponding Hopf algebra. The chromatic polynomial demonstrates a typical example. Let us extend the chromatic polynomial to a mapping $\chi:\mathcal{G}\to\mathbb{C}[c]$ of the whole space $\mathcal{G}$ to the space of polynomials in one variable by linearity.

Theorem 2.4**.**

The value of a chromatic polynomial on any primitive element of the Hopf algebra $\mathcal{G}$ is a linear polynomial, i.e., a monomial of degree $1$.

In fact, this property is nothing but the well known binomiality property of chromatic polynomial, which asserts that $\chi$ is a Hopf algebra homomorphism:

Theorem 2.5**.**

The chromatic polynomial of a graph $G$ satisfies the relation

[TABLE]

where summation on the right hand side is carried over all ordered partitions of the set $V(G)$ of vertices of $G$ into two disjoint nonempty subsets.

Graded homomorphisms of the Hopf algebra of graphs $\mathcal{G}$ to the Hopf algebra of polynomials $\mathbb{C}[p_{1},p_{2},\dots]$ are studied in more detail below, see Sec. 2.4.

Theorem 2.4 means, in particular, that the value of chromatic polynomial on the projection $\pi(G)$ of an arbitrary graph $G$ to the subspace of primitives is a linear polynomial. Equation (6) and the fact that the free term of the chromatic polynomial of any (nonempty) graph is [math], imply that the polynomial $\chi_{\pi(G)}(c)$ coincides with the linear term of $\chi_{G}(c)$ for any graph $G$.

Certain other graph invariants behave similarly.

Theorem 2.6** ([25], [39]).**

The skew characteristic polynomial of a graph on any primitive element is a constant.

The interlace polynomial of a primitive element of order $n$ is a polynomial of degree at most $\left[\frac{n}{2}\right]$.

2.4 Hopf algebra of graphs and integrability

In this section we describe a relation between graph invariants and the theory of integrable systems of mathematical physics. To a graph invariant, one can associate its averaging, namely, the result of summation of the values of the invariant over all graphs taken with the weight inverse to the order of the automorphism group of graphs. If the invariant takes values in the ring of polynomials in infinitely many variables, then the result is a formal power series in these variables.

For graphs on surfaces (embedded graphs) similar constructions, as is well known, lead to solutions of integrable hierarchies. However, for abstract graphs a result of this type has been proved only recently [21]. It asserts that the averaging of an umbral graph invariant becomes, after an appropriate rescaling of the variables, a solution to the Kadomtsev–Petviashvili hierarchy.

2.4.1 KP integrable hierarchy

The **Kadomtsev–Petviashvili integrable hierarchy **(KP hierarchy) is an infinite system of nonlinear partial differential equations for an unknown function $F(p_{1},p_{2},\dots)$ depending on infinitely many variables. The equations of the hierarchy are indexed by partitions of integers $n$, $n\geq 4$, into two parts none of which equals $1$. The first two equations, which correspond to partitions of $4$ and $5$, respectively, are

[TABLE]

The left hand side of each equation corresponds to a partition into two parts none of which equals $1$, while the terms in the right hand side correspond to partitions of the same number $n$, which include parts equal to $1$. For $n=6$, there are two equations corresponding to the partitions $2+4=6$ and $3+3=6$, and so on. Exponents of solutions to the KP hierarchy are called its $\tau$-functions.

2.4.2 Umbral graph invariants

Definition 2.7**.**

A graph invariant with values in the ring of polynomials $\mathbb{C}[q_{1},q_{2},\dots]$ in infinitely many variables is called umbral if its extension to a mapping $\mathcal{G}\to\mathbb{C}[q_{1},q_{2},\dots]$ by linearity is a graded Hopf algebra homomorphism; here the grading in the ring of polynomials is defined by the weights of the variables $w(q_{i})=i$, for $i=1,2,3\dots$.

The definition immediately implies that a graph invariant is umbral iff its value on any primitive element of order $n$ is $cq_{n}$ for some constant $c$.

Example 2.8**.**

Stanley’s symmetrized chromatic polynomial is an example of umbral invariant. Indeed, by theorem 1.5, the Hopf algebra of weighted graphs modulo contraction–deletion relation is graded isomorphic to the Hopf algebra of polynomials.

Other umbral invariants and their various specializations obtained by assigning concrete values to the variables $q_{i}$ contain many well known graph invariants.

Example 2.9**.**

In [21], Abel polynomials of graphs are introduced. Associate to any spanning forest in a graph $G$ the monomial in the variables $q_{i}$ equal to the product $iq_{i}$ over all the trees in the forest, where $i$ denotes the number of vertices in the tree. Multiplication by $i$ here corresponds to the choice of a root in the tree, that is, such a monomial encodes the number of rooted trees corresponding to the chosen spanning forest. The Abel polynomial $A_{G}(q_{1},q_{2},\dots)$ is equal to the sum of these monomials over all spanning forests in $G$.

If $G$ is a graph with $n$ vertices, then $A_{G}$ is a quasihomogeneous polynomial of degree $n$, the coefficient of $q_{n}$ in it being the complexity of $G$ times $n$. (Recall that the complexity of a graph is the number of spanning trees in it.) After the substitution $q_{i}=x$ for all $i=1,2,\dots,n$, $A_{K_{n}}$ of the complete graph on $n$ vertices, it becomes the conventional Abel polynomial [1] $A_{n}(x)=x(x+n)^{n-1}$.

Theorem 2.10** ([21]).**

The Abel polynomial is an umbral graph invariant, i.e., its extension to the Hopf algebra of graphs $\mathcal{G}$ is a graded Hopf algebra homomorphism.

2.4.3 Integrability of umbral invariants

Theorem 2.11**.**

Let $I$ be an umbral graph invariant taking values in the ring of polynomials in infinitely many variables $q_{1},q_{2},\dots$. Define the generating function

[TABLE]

where summation is carried over all isomorphism classes of graphs, and $|\mathop{\rm Aut}(G)|$ denotes the order of the automorphism group of $G$.

Denote the constants $i_{n}$ as the sum

[TABLE]

of all the coefficients of the monomial $q_{n}$.

Suppose the constant $i_{n}$ is nonzero for all $n=1,2,\dots$. Then after rescaling the variables $\displaystyle q_{n}=\frac{2^{n(n-1)/2}(n-1)!}{i_{n}}\cdot p_{n}$ the generating function $\mathcal{I}^{\circ}$ becomes the following linear combination of the one-part Schur polynomials:**

[TABLE]

We underline that after the rescaling in the theorem each umbral graph invariant $I$ becomes *one and the same * generating function.

Summation over connected graphs in the expression for $i_{n}$ can be replaced by summation over all graphs, since for any disconnected graph $G$ the coefficient of the monomial $q_{n}$ in the polynomial $I_{G}(q_{1},q_{2},\dots)$ is [math].

Since any linear combination of one-part Schur polynomials is a $\tau$-function of the KP hierarchy, we have

Corollary 2.12**.**

After rescaling of the variables given in Theorem 2.11, the generating function $\mathcal{I}$ becomes a $\tau$-function for the KP hierarchy.

3 Hopf algebra of chord diagrams

and weight systems associated to Lie algebras

Both the space of chord diagrams and the space of graphs are equipped with natural Hopf algebra structures. We describe in this section the relationship of this structure with the weight systems constructed from Lie algebras. This is a large and important class of weight systems related to quantum knot invariants. However, finding values of weight systems in this class is computationally a rather complicated problem since it requires to make computations in a noncommutative algebra. We recall known results and describe new ones that help to overcome these difficulties and obtain explicit answers in the case of the Lie algebras $\mathfrak{sl}(2)$ and $\mathfrak{gl}(N)$.

3.1 Structure of the Hopf algebra of chord diagrams

Similarly to graphs, chord diagrams generate a Hopf algebra. An essential distinction of this Hoph algebra from the Hopf algebra of graphs is that it is well defined only in the quotient space of chord diagrams modulo the subspace spanned by the $4$-term relations. Without taking this quotient there is no way to introduce multiplication correctly (while the comultiplication is defined in a natural way).

Define the product $C_{1}C_{2}$ of chord diagrams $C_{1},C_{2}$ as the chord diagram obtained by gluing their Wilson loops cutting them at arbitrary points distinct from the endpoint of chords, preserving the orientations of these loops, see Fig. 4. This product is well defined nodulo the $4$-term relations.

The comultiplication takes any chord diagram $C$ to the sum of tensor products of chord diagrams obtained by splitting the set of chords in $C$ into two nonintersecting subsets

[TABLE]

These operations turn the vector space

[TABLE]

where $\mathcal{C}_{n}$ is spanned by chord diagrams with $n$ chords, $4$-term relations moduled out, into graded commutative cocommutative connected Hopf algebra [40].

The map taking a chord diagram to its intersection graph is extended by linearity to a homomorphism of the Hopf algebra of chord diagrams to the Hopf algebra of graphs modulo the $4$-term relations.

The Hopf algebra of chord diagrams is naturally isomorphic to the Hopf algebra of Jacobi diagrams. A Jacobi diagram is an embedded $3$-regular graph with distinguished oriented loop called the Wilson loop. The grading of such a diagarm is equal to half the number of vertices (it is easy to see that the number of vertices in a $3$-regular graph is necessarily even). Denote by $\mathcal{J}_{n}$ the space spanned by Jakobi diagrams of grading $n$ factorized by

• •

skewsymmetry relations

• •

STU relations.

The skewsymmetry relation states that the change of orientation at any vertex of the diagram results in the change of the sign of the corresponding element; the STU relation is depicted on Fig. 5. It allows one to represent any Jacobi diagram as a linear combination of chord diagrams. This correspondence establishes an isomorphism between $\mathcal{J}$ and $\mathcal{D}$ [5].

Similarly to the case of chord diagrams, the product of two Jacobi diagrams is defined by gluing their Wilson loops cut at arbitrary points distinct from the vertices of the factors. After removing the Wilson loop, a Jacobi diagram splits into a number of connected components (the connected components of a chord diagram are its chords). The comultiplication takes a Jacobi diagram to the sum of tensor products of its subdiagrams obtained by splitting the set of its connected components into two subsets in all possible ways.

In particular, connected Jacobi diagrams are primitive elements in the Hoph algebra $\mathcal{J}$. These primitive elements span each of the homogeneous subspaces $P(\mathcal{J}_{n})$ of primitive elements (though usually they are subject to certain linear relations). As a corollary, each of the spaces $P(\mathcal{J}_{n})$, as well as the space $P(\mathcal{D}_{n})$ isomorphic to the former, admits a natural filtration

[TABLE]

where the subspace $P^{(k)}(\mathcal{J}_{n})$ is spanned by connected Jacobi diagrams with at most $k$ vertices on the Wilson loop.

The universal enveloping algebra $U\mathfrak{g}$ of the Lie algebra $\mathfrak{g}$ admits a filtration by vector subspaces

[TABLE]

where $U^{(k)}\mathfrak{g}$ is spanned by monomials of degree at most $k$ in the elements of $\mathfrak{g}$. This filtration, in turn, induces a filtration in the center $ZU\mathfrak{g}$ of the universal enveloping algebra. It was shown in [22] that for the weight system corresponding to a given metrized Lie algebra $\mathfrak{g}$, its value on a primitive element lying in the filtration term $P^{(k)}$ belongs to $ZU^{(k)}$.

A conjecture in [64] states that the projection $\pi(C)$ of a given chord diagram $C$ with $n$ chords to the subspace of primitive elements lies in the subspace $P^{(k+1)}(\mathcal{J}_{n})$, where $k$ is half the circumference of the intersection graph $g(C)$. (The circumference of a simple graph is the length of the longest cycle in it.) The results concerning the values of the weight systems associated to Lie algebras on the projections of chord diagrams to the subspace of primitive elements stated below confirm this conjecture.

3.2 Universal construction of a weight system

associated to a Lie algebra

To each metrized Lie algebra a weight system is associated which takes values in the center of the universal enveloping algebra of this Lie algebra. This center is typically a ring of polynomials in several generators which are the Casimir elements of the Lie algebra. We describe below the construction of this weight system and provide various algorithms for its computation leading to explicit formulas.

3.2.1 Definition of the weight system

The universal construction for a weight system obtained from a metrized Lie algebra looks like follows. Let $\mathfrak{g}$ be a Lie algebra equipped with an ${\rm ad}$-invariant scalar product (non-degenerate symmetric bilinear form) $\langle\cdot,\cdot\rangle$; here ${\rm ad}$-invariance means that the equality $\langle[a,b],c\rangle=\langle a,[b,c]\rangle$ holds for any three elements $a,b,c\in\mathfrak{g}$. Pick an arbitrary basis $e_{1},\dots,e_{d}$ in $\mathfrak{g}$, $d=\dim\mathfrak{g}$, and denote by $e_{1}^{*},\dots,e_{d}^{*}$ the elements of the dual basis, $\langle e_{i},e_{j}^{*}\rangle=\delta_{i,j}$, $i,j=1,\dots,d$.

Now let $C$ be a chord diagram with $n$ chords. Choose an arbitrary point on a circle distinct from the endpoints of the chords and call it the cut point. Then, for each chord $a$, pick an index $i_{a}$ from the range $1,\dots,d$, and assign the basic element $e_{i_{a}}$ to one of the endpoints of the chord, and the element $e_{i_{a}}^{*}$ of the dual basis to the other endpoint of the chord. Multiply the resulting $2n$ elements of the Lie algebra in the order prescribed by the orientation of the Wilson loop, starting from the cut point. The resulting monomial is considered as an element of the universal enveloping algebra $U\mathfrak{g}$ of the given Lie algebra. Sum up the obtained monomials over all the $d^{n}$ possible ways to put indices on the chords; denote the resulting sum by $w_{\mathfrak{g}}(C)\in U\mathfrak{g}$. For example, for the arc diagram

obtained by cutting the corresponding chord diagram, the resulting element of the universal enveloping algebra is equal to

[TABLE]

Theorem 3.1** ([40]).**

The element $w_{\mathfrak{g}}(C)$ is independent of the choice of the basis $e_{1},\dots,e_{d}$ in the Lie algebra. 2. 2.

The element $w_{\mathfrak{g}}(C)$ is independent of the choice of a cut point in the chord diagram. 3. 3.

The element $w_{\mathfrak{g}}(C)$ belongs to the center $ZU\mathfrak{g}$ of $U\mathfrak{g}$. 4. 4.

The invariant $w_{\mathfrak{g}}$ satisfies the $4$-term relations, and thus is a weight system. 5. 5.

The constructed weight system taking values in the commutative ring $ZU\mathfrak{g}$ is multiplicative, i.e. $w_{\mathfrak{g}}(C_{1}C_{2})=w_{\mathfrak{g}}(C_{1})w_{\mathfrak{g}}(C_{2})$ for any two chord diagrams $C_{1}$ and $C_{2}$.

3.2.2 Jacobi diagrams and the values of the weight system on primitive elements

The definition of the weight system associated with a Lie algebra has a convenient reformulation in terms of Jacobi diagrams. For a given metrized Lie algebra $\mathfrak{g}$ consider the linear $3$-form $\omega(a,b,c)=\langle[a,b],c\rangle$ treated as an element of the third tensor power $\mathfrak{g}^{*}\otimes\mathfrak{g}^{*}\otimes\mathfrak{g}^{*}$. Denote by $\omega^{*}\in\mathfrak{g}\otimes\mathfrak{g}\otimes\mathfrak{g}$ the corresponding dual tensor obtained by identifying $\mathfrak{g}$ with the dual space $\mathfrak{g}^{*}$ using the scalar product $\langle\cdot,\cdot\rangle$. Note that the tensor $\omega^{*}$ is invariant with respect to cyclic permutations of the tensor factors and it changes the sign under transpositions of any two factors. Take now a Jacobi diagram. We associate the tensor $\omega^{*}$ with each its internal vertex labelling factors of the tensor product with the edges exiting from this vertex. To each internal edge, i.e. an edge connecting two internal vertices of the diagram, we associate the convolution of tensors corresponding to the ends of the edge by applying the scalar product. The result of such convolution is an element of the tensor power of $\mathfrak{g}$ whose factors are labelled by the vertices lying on the Wilson loop. Finally, we apply to this tensor the homomorphism to the universal enveloping algebra sending the tensor product of some collection of elements in $\mathfrak{g}$ to their product in $U\mathfrak{g}$ in the order they follow on the Wilson loop in the counterclockwise order starting from the cut point. The resulting element of $U\mathfrak{g}$ is nothing but the value of the weight system $w_{\mathfrak{g}}$ on the given Jacobi diagram. This definition is particularly useful for efficient computation of the values of the weight system $w_{\mathfrak{g}}$ on primitive elements represented by connected Jacobi diagrams.

3.3 $\mathfrak{sl}(2)$ weight system

The Lie algebra $\mathfrak{sl}(2)$ is the simplest noncommutative Lie algebra with a nondegenerate ${\rm ad}$-invariant scalar product. In contrast to the case of more complicated Lie algebras, the $\mathfrak{sl}(2)$ weight system admits the Chmutov-Varchenko recurrence relation, which simplifies dramatically its explicit computation. Nevertheless, even with this recursion relation the computation remains quite laborious and leads to explicit answers in a limited number of cases only. In particular, no explicit formula for the values of the $w_{\mathfrak{sl}(2)}$ weight system on chord diagrams all whose chords intersect one another pairwise was known until recently. We describe below the corresponding result of P. Zakorko as well as the result of P. Zinova and M. Kazarian about the values of $w_{\mathfrak{sl}(2)}$ on chord diagrams whose intersection graph is the complete bipartite graph.

3.3.1 Chmutov-Varchenko recursion relation

The weight system $w_{\mathfrak{g}}$ associated with a given Lie algebra $\mathfrak{g}$ takes values in the commutative ring $ZU\mathfrak{g}$. However, the summands of an expression for $w_{\mathfrak{g}}(C)$ lie in the complicated noncommutative algebra $U\mathfrak{g}$. Therefore, a direct application of the definition is not efficient in practice, and one needs to find methods for more efficient computation of such a weight system that would deal with commutative rings (say, rings of polynomials) at each step.

The simplest noncommutative Lie algebra is $\mathfrak{sl}(2)$. An efficient algorithm for computing this weight system was suggested by Chmutov and Varchenko [22]. The center of the universal enveloping algebra $U\mathfrak{sl}(2)$ is the ring of polynomials in one variable, which is represented by the Casimir element. We denote this generator by $c$. Thus, the weight system $w_{\mathfrak{sl}(2)}$ takes values in the algebra of polynomials in $c$. The value $w_{\mathfrak{sl}(2)}(C)$ on a chord diagram $C$ with $n$ chords is a polynomial in $c$ of degree $n$ with the leading coefficient $1$ and zero free term (for $n>0$).

Theorem 3.2**.**

The weight system $w_{\mathfrak{sl}(2)}$ associated to the Lie algebra $\mathfrak{sl}(2)$ satisfies the following relations:

$w_{\mathfrak{sl}(2)}$* is a multiplicative weight system, $w_{\mathfrak{sl}(2)}(C_{1}C_{2})=w_{\mathfrak{sl}(2)}(C_{1})w_{\mathfrak{sl}(2)}(C_{2})$ for arbitrary chord diagrams $C_{1}$ and $C_{2}$. As a corollary, for the empty chord diagram the value of the weight system is equal to $1$;* 2. 2.

if the diagram $C$ is obtained from a diagram $C^{\prime}$ by adding one chord having no intersections with the chords of $C^{\prime}$, then we have

[TABLE] 3. 3.

If the diagram $C$ is obtained from a diagram $C^{\prime}$ by adding one chord intersecting exactly one chord of $C^{\prime}$, then we have

[TABLE] 4. 4.

The $6$-term relations depicted on Fig. 7 hold.

As usual, it is assumed in these relations that besides the shown chords all diagrams participating in the relations contain other chords that are the same for all six diagrams. The Chmutov-Varchenko relations are sufficient to compute the value of the weight system on arbitrary chord diagram. Indeed, if the relations 1–3 are not applicable to a given chord diagram, then this diagram necessarily contains a chord to which relation 4 can be applied. All diagrams obtained by applying the relations are simpler than the original one in the following sense: the two diagrams on the right hand side have smaller number of chords than each of the diagrams on the left hand side; all the diagrams on the left hand side have equal number of chords but the last three of them have smaller number of pairs of intersecting chords than the first one. This implies that the value of the weight system is computed in a unique way by applying these relations repeatedly in finitely many steps.

The Chmutov-Varchenko relations can be also used for an axiomatic definition of the weight system $w_{\mathfrak{sl}(2)}$. With this approach, the equalities of the theorem are taken as the definition of the weight system. Then, the assertion that the function is well defined becomes a nontrivial statement, which claims that the result of computation is independent of the order in which the Chmutov-Varchenko relations are applied. The $4$-term relations for the constructed function is a corollary of Chmutov-Varchenko relations, i.e., it is a weight system indeed.

3.3.2 $\mathfrak{sl}(2)$-weight system for graphs

The following result of Chmutov and Lando is specific for the Lie algebra $\mathfrak{sl}(2)$ and the weight system associated to it. Its analog, say, for the Lie algebra $\mathfrak{sl}_{3}$ does not hold.

Theorem 3.3** ([19]).**

The value of the weight system $w_{\mathfrak{sl}(2)}$ on any chord diagram is uniquely determined by its intersection graph.

This theorem implies that the weight system $w_{\mathfrak{sl}(2)}$ defines a function on those graphs that are intersection graphs of chord diagrams. This function vanishes on those combinations of intersection graphs that are involved into $4$-term relations.

Conjecture 3.4** (Lando).**

The weight system $w_{\mathfrak{sl}(2)}$ admits an extension to a function on graphs satisfying the $4$-term relation for graphs. This extension is unique.

Existence of such an extension would mean that $w_{\mathfrak{sl}(2)}$ vanishes on any linear combination of intersection graphs that are corollaries of the $4$-term relations for graphs (such a linear combination is not necessarily a corollary of the $4$-term relation for chord diagrams). Uniqueness of such an extension is not specific for $w_{\mathfrak{sl}(2)}$. It is related to the question whether the mapping sending a chord diagram to its intersection graph is epimorphic, see Sect. 1.3. Validity of the conjecture is checked using computer for graphs with at most $8$ vertices (E. Krasilnikov, [41]). Note that for the extension obtained by Krasilnikov the values of $w_{\mathfrak{sl}(2)}$ on some graphs with $8$ vertices are not integer (all such graphs are not intersection graphs since for all intersection graphs the Chmutov–Varchenko relations guarantee that the values of the weight system on all intersection graphs are polynomials with integer coefficients).

There are also some other pieces of evidence supporting Conjecture 3.4. For instance, the required extension is known for the top coefficient of the values of $w_{\mathfrak{sl}(2)}$ in the projection of the chord diagram to the subspace of primitive elements. The value $w_{\mathfrak{sl}(2)}(\pi(D))$ of the weight system $w_{\mathfrak{sl}(2)}$ on the projection of a given chord diagram $D$ with $2n$ chords to the subspace of primitive elements is a polynomial of degree at most $n$.

Theorem 3.5** ([42], [7]).**

The coefficient of $c^{n}$ in the value $w_{\mathfrak{sl}(2)}(\pi(D))$ of the weight system $w_{\mathfrak{sl}(2)}$ on the projection of the chord diagram $D$ with $2n$ chords to the subspace of primitive elements coincides with the value $2\log\leavevmode\nobreak\ \nu(g(D))$ of doubled logarithm of the nondegeneracy of the intersection graph of the diagram (the logarithm is understood in the sense of convolution in the Hopf algebra).

Since the nondegeneracy of a graph is its $4$-invariant, we obtain a confirmation of the conjecture.

One of the possible approaches to the proof of Conjecture 3.4 consists in an attempt to construct a $4$-invariant of graphs taking values in the ring of polynomials in one variable whose values on intersection graphs coincide with those for $w_{\mathfrak{sl}(2)}$. In order to construct such an invariant, it is important to have a large range of explicitly computed values of $w_{\mathfrak{sl}(2)}$ on graphs: these values might indicate which characteristics of the graph are reflected in $w_{\mathfrak{sl}(2)}$. In many ways, the results provided below are inspired by this problem.

In some respect, the weight system $w_{\mathfrak{sl}(2)}$ resembles very much the chromatic polynomial of a graph:

• •

it is multiplicative;

• •

its value on a chord diagram is a polynomial in one variable whose degree is equal to the number of vertices in the intersection graph;

• •

the leading coefficient of this polynomial is equal to  $1$;

• •

the signs of the coefficients alternate;

• •

the absolute values of the second coefficient of the polynomial is equal to the number of edges in the graph;

• •

the free term of this polynomial is equal to zero if the diagram has at least one chord;

• •

adding a leaf to a chord diagram leads to the multiplication of the value of $w_{\mathfrak{sl}(2)}$ by $c-1$.

It is worth to know, thereby, whether an analogue of a theorem by J. Huh [34] proved by him for the case of chromatic polynomial holds for the weight system $w_{\mathfrak{sl}(2)}$ as well:

Problem**.**

Is it true that the sequence of absolute values of the coefficients of the weight system $w_{\mathfrak{sl}(2)}$ is logarithmically convex for arbitrary chord diagram? In particular, is it true that this sequence of coefficients is unimodular, that is, it increases first and then decreases?

The proof of Huh relates the chromatic polynomial to the geometry of algebraic manifolds. It would be interesting to find a similar relationship for the $\mathfrak{sl}(2)$ weight system.

3.3.3 The values of the $\mathfrak{sl}(2)$ weight system on complete graphs

The Chmutov-Varchenko relations allow one both to compute the values of the weight system $w_{\mathfrak{sl}(2)}$ on specific chord diagrams and to obtain closed formulas for some infinite series of chord diagrams. In particular, consider the chord diagram $K_{n}$ formed by $n$ pairwise intersecting chords. The intersection graph for this diagram is the complete graph on $n$ vertices. For the complete graphs the computation of one or another graph invariant causes usually no difficulty due to high symmetry of complete graphs. However, for the case of the weight system $w_{\mathfrak{sl}(2)}$ this problem proved to be surprisingly difficult and the answer is rather unexpected.

The following statement was suggested as a conjecture by S. Lando around 2015. It was partially proved (for the linear terms of the polynomials) in [8].

Theorem 3.6** ([65]).**

The generating series for the values of the weight system associated to the Lie algebra $\mathfrak{sl}(2)$ on complete graphs admits the following infinite continuous fraction expansion

[TABLE]

where

[TABLE]

It is useful to compare this decomposition to a continuous fraction with a similar decomposition for the generating function for chromatic polynomials of complete graphs:

[TABLE]

[TABLE]

The proof of this theorem given in [65] contains also the following efficient way for computing the values of the weight system from the theorem, appearing as an intermediate step of computations. Consider the linear operator $T$ acting on the space of polynomials in one variable $x$ whose action is defined as follows:

[TABLE]

and also

[TABLE]

for any polynomial $f$.

Theorem 3.7** ([65]).**

The value of the weight system $\mathfrak{sl}(2)$ on the chord diagram with the complete intersection graph is equal to the value of the polynomial $T^{n}(1)$ at the point $x=c$,

[TABLE]

From the known value of the weight system $w_{\mathfrak{sl}(2)}$ on complete graphs it is easy to find its values on their projections to the subspace of primitive elements. Indeed, polynomials in complete graphs form a Hopf subalgebra in the Hopf algebra of graphs $\mathcal{G}$. The order of the automorphism group of the complete graph $K_{n}$ on $n$ vertices is $n!$, which allows one to represent the exponential generating function for the projections of complete graphs to the subspace of primitives as the logarithm of the exponential generating function for complete graphs:

[TABLE]

Applying $w_{\mathfrak{sl}(2)}$ to both parts of the identity and substituting the values $w_{\mathfrak{sl}_{2}}(K_{n})$ we already know, we obtain the first terms of the expansion:

[TABLE]

3.3.4 Algebra of shares

Definition 3.8**.**

A share with $n$ chords is an ordered pair of oriented intervals in which $2n$ pairwise distinct points are given, split into $n$ disjoint pairs, considered up to orientation preserving diffeomorphisms of each of the two intervals.

For the intervals, we often take two arcs of the Wilson loop. In this case a share is a part of a chord diagram formed by the chords whose both ends belong to a chosen pair of arcs, while none of the other chords has an end on these arcs. If a pair of arcs of a chord diagram forms a share, then its complement also is a share. In this case we call the chord diagram the join of two shares and denote it by $S_{1}\#S_{2}$, see Fig. 8. In a join, the end of the first interval of the share $S_{1}$ is attached to the beginning of the first interval in $S_{2}$, the end of the first interval in $S_{2}$ is attached to the beginning of the second interval in $S_{1}$, the end of the second interval in $S_{1}$ is attached to the beginning of the second interval in $S_{2}$, and, finally, the end of the second interval in $S_{2}$ is attached to the beginning of the first interval in $S_{1}$. Join is a noncommutative operation.

To each share $S$, one can associate a chord diagram by attaching the end of each of the two intervals to the beginning of the other interval, thus uniting the intervals into a Wilson loop. We denote this chord diagram by $\overline{S}$ and call it the closure of $S$. We also call the intersection graph $g(\overline{S})$ the intersection graph of the share $S$ and denote it by $g(S)$.

Two shares can be multiplied by attaching their intervals with coinciding numbers, see Fig. 9. This multiplication is noncommutative.

The space spanned by isomorphism classes of shares can be endowed with relations analogous to the Chmutov–Varchenko relations in Theorem 3.2. Namely, we can assume that all the diagrams taking part in the relation are, in fact, shares. There is, however, some subtlety in defining relations 2 and 3 in the Theorem. Namely, suppose a share $S$ is obtained from a share $S^{\prime}$ by adding a single chord whose both ends belong to the same interval number $i,\leavevmode\nobreak\ i\in\{1,2\}$, which intersects at most one of the other chords. Then the relations for the shares have the form

[TABLE]

in the cases when the given chord intersects none of the other chords, or just one of them, respectively.

Denote by $\mathcal{S}$ the quotient space of the vector space spanned by all the shares modulo the subspace spanned by the $6$-term relations and relations (8). This quotient space is endowed with the algebra structure induced by the share multiplication operation.

Note that since the $6$-term relations are not homogeneous, the algebra $\mathcal{S}$ is filtered rather than graded.

Proposition 3.9**.**

The algebra $\mathcal{S}$ is commutative and isomorphic to the algebra of polynomials in the three generators $c_{1},c_{2},x$ shown below:

1$$2$$c_{1}$$1$$2$$c_{2}$$1$$2$$x

The proposition states that modulo $6$-term relations and relations (8), each share admits a unique representation as a linear combination of the shares $1,x,x^{2},x^{3},\dots$, where the share $x^{n}$ is formed by $n$ parallel chords having ends on the two distinct intervals; the coefficients in these linear combinations are polynomials in $c_{1}$ and $c_{2}$.

One of the reasons for introducing the algebra $\mathcal{S}$ is the following remark.

Proposition 3.10**.**

Suppose two shares (or two linear combinations of shares) $S_{1}$ and $S_{2}$ represent one and the same element in $\mathcal{S}$. Then for an arbitrary share $R$, the values of the $\mathfrak{sl}_{2}$ weight system on the chord diagrams obtained by joining the shares $S_{1}$ and $S_{2}$, respectively, with $R$, coincide,

[TABLE]

One can chose for the additional share $R$, for example, the tuple of $n$ mutually parallel chords having ends on distinct arcs, or the tuple of $n$ pairwise intersecting chords having ends on distinct arcs.

3.3.5 Values of the $\mathfrak{sl}(2)$-weight system on

complete bipartite graphs

Chord diagrams whose intersection graph is complete bipartite form another wide class of chord diagrams, for which the values of $w_{\mathfrak{sl}(2)}$ are given by explicit formulas.

Denote by $K_{m,n}$ the chord diagram formed by $m$ parallel horizontal chords and $n$ parallel vertical chords. The intersection graph of such a chord diagram is the complete bipartite graph having $m$ vertices in one part and $n$ vertices in the other part. Equivalently, $K_{m,n}$ is the join of two shares formed by, respectively, $m$ and $n$ parallel chords, each having ends on distinct arcs. For each $m=0,1,2,3,\dots$, consider the generating function for the values of the weight system $w_{\mathfrak{sl}(2)}$ on the chord diagrams $K_{m,n}$,

[TABLE]

Theorem 3.11**.**

The generating functions $G_{m}$ are subject to the relation

[TABLE]

where the coefficients $s_{i,m}$ are given by the generating power series

[TABLE]

Note that $G_{m}$ enters the right hand side of the equation as well, with coefficient $s_{m,m}=c-\frac{m\,(m-1)}{2}$. Moving this term to the left hand side, we can make the recurrence more explicit:

[TABLE]

By induction, this implies

Corollary 3.12**.**

The generating function $G_{m}$ can be represented as a finite linear combination of geometric porgressions $\frac{1}{1-\left(c-\frac{i(i+1)}{2}\right)\,t}$, $i=0,1,\dots,m$. The coefficients of these geometric progressions are polynomials in $c$.

For small  $m$, the recurrence yields

[TABLE]

The Proposition above admits the following generalization. For an arbitrary share $S$, denote by $(S,n)$ the chord diagram which is the join of $S$ and the share $x^{n}$ consisting of  $n$ parallel chords. Introduce the generating function

[TABLE]

Corollary 3.13**.**

The generating function $f_{S}$ can be represented as a finite linear combination of geometric progressions of the form $\frac{1}{1-\left(c-\frac{i(i+1)}{2}\right)\,t}$, $i=0,1,\dots,m$, where  $m$ is the number of those chords of $S$ whose ends belong to distinct arcs. The coefficients of these progressions are polynomials in $c$.

Indeed, if a share $S$ can be presented modulo $6$-term relations and relations (8) as a linear combination of the basis shares $x^{m}$,

[TABLE]

then the series $f_{S}$ can be expressed as the linear combination of the series

[TABLE]

with the same coefficients, that is,

[TABLE]

and we can apply the Corollary above.

Similarly to the case of complete graphs, polynomials in complete bipartite graphs form a Hopf subalgebra in $\mathcal{G}$. There are formulas for the values of the weight system $w_{\mathfrak{sl}(2)}$ on the projections of complete bipartite graphs to the subspace of primitives, which generalize the formula for the logarithm of the exponential generating function for complete graphs, see details in [32].

3.3.6 Values of the $\mathfrak{sl}(2)$-weight system

on graphs that are not intersection graphs

Not each graph is the intersection graph of some chord diagrams, but using $4$-term relations for graphs one can try to express it in terms of intersection graphs and extend to it the $\mathfrak{sl}(2)$-weight system. The formulas from the previous section allow one to obtain such an extention not only to some specific graph, but also for infinite sequences of graphs. For an arbitrary graph $G$, denote by $(G,n)$ its connected sum with $n$ isolated vertices, i.e., the graph obtained by adding $n$ vertices to $G$ and connecting each of the added vertices to each of the vertices of $G$. Now, take for $G$ the graph C_{5}=\raisebox{-0.86108pt}{\includegraphics[scale={.35}]{pics/C5.jpg}}, the cycle on $5$ vertices. For $n\geq 1$, the graph $(C_{5},n)$ is not an intersection graph.

Proposition 3.14** ([31]).**

Suppose Conjecture 3.4 about extendability of the $\mathfrak{sl}(2)$-weight system to the space of graphs is valid. Then the values of an extended weight system on the graphs $(C_{5},n)$ are given by the generating function

[TABLE]

where $G_{m}=\sum_{n=0}^{\infty}w_{\mathfrak{sl}(2)}(K_{m,n})\,t^{n}$ are the generating functions for the values of the $\mathfrak{sl}(2)$-weight system on complete bipartite graphs.

The proof is achieved by applying to the graphs $(C_{5},n)$ the $4$-term relation shown in Fig. 10,

[TABLE]

Each of the graphs in the right hand side is an intersection graph. Moreover, it is the intersection graph of a chord diagram of the form $(S,n)$, for some share $S$,

[TABLE]

By applying the argument from the previous section, we can express each of the shares obtained in terms of the basic shares $x^{m}$, $m=0,1,\dots,5$, and express in this way the generating function for the graphs $(C_{5},n)$ in terms of the corresponding generating functions for complete bipartite graphs.

3.4 $\mathfrak{gl}(N)$-weight system

Until recently, no recursion similar to the Chmutov–Varchenko one has been known for Lie algebras other than $\mathfrak{sl}(2)$, and essentially the only known way of computing the values of the weight system associated to this or that Lie algebra was the straightforward application of the definition, which is extremely laborious. Here we explain the results of the recent paper [64] devoted to computation of the weight systems associated to the Lie algebras $\mathfrak{gl}(N)$.

For the basis in $\mathfrak{gl}(N)$, one can choose the matrix units $E_{ij}$, which have $1$ on the intersection of the $i$ th row and $j$ th column, all whose other entries are zeroes. The commutator of two matrix units has the form $[E_{ij},E_{kl}]=\delta_{jk}E_{il}-\delta_{il}E_{jk}$. The standard invariant scalar product is given by $\langle x,y\rangle=\mathop{\rm Tr}(xy)$, and the dual basis with respect to this scalar product is $E_{ij}^{*}=E_{ji}$. The center $ZU(\mathfrak{gl}(N))$ is generated freely by the Casimir elements $C_{1},C_{2},\dots,C_{N}$, where $C_{1}=\sum_{i=1}^{N}E_{ii}$, $C_{2}=\sum_{i,j=1}^{N}E_{ij}E_{ji}$, and, more generally,

[TABLE]

Casimir elements with the numbers greater than $N$ are defined similarly. They also belong to the center, but they can be expressed as polynomials in the Casimir elements whose numbers are at most $N$. Thus, the value of the weight system associated to the Lie algebra $\mathfrak{gl}(N)$ is a polynomial in the Casimir elements. For example, it is easy to see that for the chord diagram $K_{1}$, which consists of the single chord, we have $w_{\mathfrak{gl}(N)}(K_{1})=C_{2}$.

Theorem 3.15** ([64]).**

There is a universal weight system, which we denote by $w_{\mathfrak{gl}}$, taking values in the ring of polynomials in infinitely many variables $N,C_{1},C_{2},\dots$, whose value for a given positive integer $N$ coincides with the value of the $\mathfrak{gl}(N)$ weight system.

It is easy to see that for each chord diagram $C$ with $n$ chords the value of the weight system $w_{\mathfrak{gl}(N)}(C)$, for $N\geq n$, is a polynomial in $C_{1},\dots,C_{n}$. The theorem asserts that the coefficients of this polynomial are polynomials in $N$. Moreover, the universal formula for $w_{\mathfrak{gl}(N)}(C)$ we obtain remains valid for $N<n$ as well.

Example 3.16**.**

For the chord diagram $K_{2}$, which is formed by two intersecting chords, we have

[TABLE]

Below, we define the weight system $w_{\mathfrak{gl}}$ for arbitrary permutations and give a recurrence relation for its computation.

3.4.1 $\mathfrak{gl}(N)$-weight system on permutation and recurrence relations

Let $m$ be a positive integer, and let $S_{m}$ be the group of all permutations of the elements $\{1,2,\dots,m\}$; for an arbitrary $\sigma\in S_{m}$, we set

[TABLE]

The element thus constructed belongs, in fact, to the center $ZU(\mathfrak{gl}(N))$. In addition, it is invariant under conjugation by the cyclic permutation,

[TABLE]

For example, the Casimir element $C_{m}$ corresponds to the cyclic permutation $1\mapsto 2\mapsto\dots\mapsto m\mapsto 1$. On the other side, any arc diagram with $n$ arcs can be considered as an involution without fixed points on the set of $m=2n$ elements. The value of $w_{\mathfrak{gl}(N)}$ on a permutation given by this involution coincides with the value of the $\mathfrak{gl}(N)$-weight system on the corresponding chord diagram. For example, for the chord diagram $K_{n}$ we have $w_{\mathfrak{gl}(N)}(K_{n})=w_{\mathfrak{gl}(N)}((1\leavevmode\nobreak\ n{+}1)(2\leavevmode\nobreak\ n{+}2)\dots(n\leavevmode\nobreak\ 2n))$.

Any permutation can be represented by a directed graph. The $m$ vertices of this graph correspond to the permuted elements. They are situated along the oriented circle and are ordered cyclically in the direction of the circle. Directed edges of the graph show the action of the permutation (so that at each vertex there is one incoming and one outgoing edge). The directed graph $G(\sigma)$ consists of these $m$ vertices and $m$ edges, for example,

[TABLE]

(the circle containing the vertices of the graph is shown as the horizontal line).

Theorem 3.17** ([64]).**

The value of the weight system $w_{\mathfrak{gl}(N)}$ on a permutation possesses the following properties:

• •

the weight system $w_{\mathfrak{gl}(N)}$ is multiplicative with respect to the connected sum (concatenation) of permutations. It follows, therefore, that for the empty graph (having zero vertices) the value of $w_{\mathfrak{gl}(N)}$ is $1$.

• •

For the cyclic permutation whose cyclic order is consistent with the permutation the value of $w_{\mathfrak{gl}(N)}$ coincides with the corresponding Casimir element, $w_{\mathfrak{gl}(N)}(1\mapsto 2\mapsto\dots\mapsto m\mapsto 1)=C_{m}$.

• •

For an arbitrary permutation $\sigma\in S_{m}$, and for any two neighboring elements $k$, $k+1$ in the set of vertices $\{1,2,\dots,m\}$, the values of the invariant $w_{\mathfrak{gl}_{N}}$ is subject to the identity

[TABLE]

The graphs on the left hand side of the identity show two neighboring vertices and the edges incident to them. In the graphs on the right hand side, these two vertices are replaced with a single one. All the other vertices and the (semi)edges are the same for all the four graphs participating in the identity.

In the exceptional case $\sigma(k+1)=k$, the identity acquires the form

[TABLE]

Applying the identity in the theorem each graph can be reduced to a monomial in the generators $C_{k}$ (that is, concatenation of independent cycles) modulo graphs with smaller number of vertices. This leads to an inductive computation of the values of the invariant $w_{\mathfrak{gl}(N)}$.

Corollary 3.18**.**

The value of $w_{\mathfrak{gl}(N)}$ on an arbitrary permutation belongs to the center $ZU\mathfrak{gl}(N)$ of the universal enveloping algebra, is a polynomial in $N,C_{1},C_{2},\dots$, and this polynomial is universal (one and the same for all the $\mathfrak{gl}(N)$).

We denote the mapping taking a permutation to this universal polynomial by $w_{\mathfrak{gl}}$. Theorem 3.15 specializes this Corollary to the case when the permutation is an involution without fixed points.

Below, we give a table of values of $w_{\mathfrak{gl}}$ on chord diagrams $K_{n}$, for few first values of $n$.

[TABLE]

Similarly to the case of $\mathfrak{sl}(2)$, the values of the weight system $w_{\mathfrak{gl}(N)}$ on projections of complete graphs to the subspace of primitives can be computed by taking the logarithm of the corresponding exponential generating function.

3.4.2 Computing the $\mathfrak{gl}(N)$-weight system

by means of the Harish-Chandra isomorphism

In this section we describe one more way, suggested in [64], to compute the $\mathfrak{gl}(N)$ weight system. The algebra $U(\mathfrak{gl}(N))$ admits the decomposition into the direct sum

[TABLE]

where $\mathfrak{n}_{-}$, $\mathfrak{h}$ and $\mathfrak{n}_{+}$ are, respectively, the subalgebras of low-triangular, diagonal, and upper-triangular matrices in $\mathfrak{gl}(N)$.

Definition 3.19**.**

The Harish-Chandra projection is the linear projection to the second summand in (9),

[TABLE]

Theorem 3.20** (Harish-Chandra isomorphism, [55]).**

The restriction of the Harish-Chandra projection to the center $ZU(\mathfrak{gl}_{N})$ is an injective homomorphism of commutative algebras, and its isomorphic image consists of symmetric functions in the shifted matrix units $x_{i}=E_{ii}+N-i$, $i=1,\dots,N$.

The isomorphism of the theorem allows one to identify the codomain $\mathbb{C}[C_{1},C_{2},\dots,C_{N}]$ of the weight system $w_{\mathfrak{gl}(N)}$ with the ring of symmetric functions in the generators $x_{1},\dots,x_{N}$. The generators $C_{k}$ under this isomorphism admit the following explicit expression:

[TABLE]

Computation of the values of the $\mathfrak{gl}(N)$ weight system on a given chord diagram, by means of the Harish-Chandra isomorphism consists in applying the projection $\phi$ to each monomial entering the definition of the weight system one by one and obtain its contribution to the final value. The total contribution is obtained by summing up elements of a commutative ring. As a result, one can considerably decrease the required amount of random-access memory. Nevertheless, time resources grows rather rapidly as $N$ grows, since the number of monomials entering the definition, which is $N^{2n}$ for a chord diagram with $n$ chords, also grows. Ramark also that the Harish-Chandra projection can be applied to computing the $w_{\mathfrak{gl}(N)}$ weight system for a specific $N$, and it does not lead to a universal (polynomial) dependence on $N$ for its values.

3.5 $\mathfrak{sl}(N)$ weight system

The Lie algebra $\mathfrak{gl}(N)$ is not simple, it can be represented as the direct sum of the simple Lie algebra $\mathfrak{sl}(N)$ and a one-dimensional Abelian Lie algebra. Therefore, the center of the universal enveloping algebra $\mathfrak{gl}(N)$ is the tensor product of the centers of the universal enveloping algebras of $\mathfrak{sl}(N)$ and $\mathbb{C}$, whence it could be identified with the ring of polynomials in $C_{1}$ with coefficients in $ZU(\mathfrak{sl}(N))$. Therefore, the values of the weight system $w_{\mathfrak{sl}(N)}$ can be obtained from that of $w_{\mathfrak{gl}(N)}$ by setting $C_{1}=0$ and $C_{k}=\tilde{C}_{k}$, $k\geq 2$, where $\tilde{C}_{k}$ is the projection of the corresponding Casimir element in $ZU(\mathfrak{gl}(N))$ to $ZU(\mathfrak{sl}(N))$. The result of this substitution is a polynomial in $\tilde{C}_{2},\tilde{C}_{3},\dots$. More explicitly, set $\tilde{E}_{i,j}=E_{i,j}-\delta_{i,j}N^{-1}C_{1}\in\mathfrak{sl}(N)\subset\mathfrak{gl}(N)$. Then

[TABLE]

where we set $C_{0}=N$ and

[TABLE]

Alternatively, the weight system $w_{\mathfrak{sl}(N)}$ can be computed by extending it to permutations by means of recurrence relations similar to that for the case $\mathfrak{gl}(N)$: just replace there $w_{\mathfrak{gl}(N)}$ with $w_{\mathfrak{sl}(N)}$, $C_{k}$ with $\tilde{C}_{k}$ and set $\tilde{C}_{1}=0$. Similarly to the $w_{\mathfrak{gl}}$ weight system, the result is a universal polynomial in $N,\tilde{C}_{2},\tilde{C}_{3},\dots$, which we denote by $w_{\mathfrak{sl}}$. Since the number of generators is smaller, computation of $w_{\mathfrak{sl}(N)}$ is more efficient, and the answer is more compact. For example, for the chord diagrams $K_{n}$ we obtain

[TABLE]

The values of the weight system $w_{\mathfrak{gl}}$ can be reconstructed from that of $w_{\mathfrak{sl}}$ by using the Hopf algebra structure of the space of chord diagrams. Namely, for any primitive element $P$ of degree greater than $1$ in the algebra of chord diagrams we have $w_{\mathfrak{gl}(N)}(P)=w_{\mathfrak{sl}(N)}(P)\in ZU(\mathfrak{sl}(n))$, i.e., $w_{\mathfrak{gl}(N)}(P)$ can be expressed as a polynomial in $\tilde{C}_{2},\tilde{C}_{3},\dots$. For $P=K_{1}$, we have

[TABLE]

This allows one to reconstruct the values of $w_{\mathfrak{gl}}$ from the known values of $w_{\mathfrak{sl}}$ by presenting a given chord diagram as a polynomial in primitive elements. More explicitly, for a given chord diagram $C$ we have

[TABLE]

In particular, exponential generating functions for the values of $w_{\mathfrak{gl}}$ and $w_{\mathfrak{sl}}$ on the chord diagrams $K_{n}$ are subject to the relation

[TABLE]

3.6 The weight system $\mathfrak{gl}(1|1)$

In addition to metrized Lie algebras, weight systems can be constructed from metrized Lie superalgebras. Such a weight system takes values on the center of the universal enveloping algebra of the corresponding Lie superalgebra. The general construction of such a weight system is very similar to that for Lie algebras, and we are not presenting it here. Similarly to the case of a Lie algebra, computation of its values for a more or less complicated Lie superalgebra is rather laborious.

In [30], this construction is elaborated for the simplest Lie superalgebra $\mathfrak{gl}(1|1)$. The corresponding weight system $w_{\mathfrak{gl}(1|1)}$ takes values in the ring of polynomials in two variables, the subring in the center $ZU\mathfrak{gl}(1|1)$ of the universal enveloping algebra, which is generated by the Casimir elements of grading $1$ and $2$.

In [30], recurrence relations for computing the values of this weight system, which are similar to the Chmutov–Varchenko relations for $w_{\mathfrak{sl}(2)}$ are deduced. In [19], it is proved that the values of this weight system depend on the intersection graph of the chord diagram only. The skew characteristic polynomial of graphs, see Sec. 1.3.5, is nothing but the extension of the weight system $w_{\mathfrak{gl}(1|1)}$ to a $4$-invariant of graphs.

4 Embedded graphs and delta-matroids

A chord diagram can be treated as an embedded graph with a single vertex on an orientable surface. One of the key problems of the theory of weight systems is how one can extend a given weight system to embedded graphs with arbitrarily many vertices. Such generalized weight systems are associated to finite type invariants of links in the same way as ordinary weight systems are associated to finite type invariants of knots.

There are various approaches to extending weight systems and graph invariants to embedded graphs. The most well developed approach consists in interpreting an embedded graph as an ordinary graph to which information about the embedding is added, see, e.g. [28]. Various extensions of the classical Tutte polynomial has been constructed in this way [10, 29].

In this section we describe another approach to constructing extensions, which is under development in the last few years. This approach is based on the analysis of the Hopf algebra structure on the space of delta-matroids. It leads to invariants satisfying $4$-term relations and being generalized weight systems, therefore. Delta-matroids (also known under the name of Lagrangian matroids) where introduced by Bouchét around 1990 [13, 12]. These combinatorial structures isolate equally well essential features of both abstract and embedded graphs. The Hopf algebra structure on various spaces of delta-matroids and $4$-term relations for them were introduced in [47].

4.1 Embedded graphs and $4$-term relations

Embedded graphs, also known under the name of ribbon graphs, are the subject of topological graph theory. An orientable embedded graph can be defined as an abstract graph (having, probably, loops and multiple edges) endowed with a cyclic order of semiedges incident to each vertex. Below, we consider only orientable connected embedded graphs. Such a graph admits a presentation as an orientable surface with boundary formed by disks associated to the verticies and ribbons associated to the edges; each ribbon is attached to one or two disks by its shorter sides, see Fig. 11.

To each chord diagram, one can associate a ribbon graph with a single vertex by making the Wilson loop into the vertex by attaching a disk to it and replacing each chord with a ribbon in such a way that the surface remains orientable.

The $4$-term relations (1) can be interpreted as valid for an arbitrary embedded graph, not necessarily a single-vertex one. In this more general case the ends of the two edges taking part in the relation may be attached to the boundaries of different vertices (two or even three). We call these relations generalized $4$-term relations, and we call the embedded graph invariants satisfying them generalized weight systems. Vassiliev’s first move for embedded graphs exchanges the two neighboring ends of two ribbons, and the second Vassiliev move consists in sliding an end of a ribbon (a handle) along the second ribbon.

Below, we will be interested first of all in the question how to extend a weight system to a generalized weight system. The weight systems we are going to construct will be based on the Hopf algebra structure on the space of delta-matroids.

4.2 Delta-matroids of graphs and embedded graphs,

and $4$-term relations for them

A set system is a pair $(E;S)$ consisting of a finite set $E$ and a subset $S\subset 2^{E}$ of the set of its subsets. A set system $(E;S)$ is said to be proper if the set $S$ is nonempty; below, we consider only proper set systems.

Definition 4.1**.**

A proper set system $(E;S)$ is called a delta-matroid if the following symmetric exchange axiom is valid for it:

for any pair of subsets $X,Y\in S$ and any element $a\in X\Delta Y$ there is an element $b\in Y\Delta X$ such that $(X\Delta\{a,b\})\in S$.

Here $\Delta$ denotes the symmetric set difference operation, $X\Delta Y=(X\setminus Y)\cup(Y\setminus X)$. The element $b$ in the symmetric exchange axiom can either coincide with $a$ or be different from it. The elements of $S$ are called admissible sets for the delta-matroid $(E;S)$.

To each simple graph $G$ and any embedded graph $\Gamma$, one can associate a delta-matroid. For a graph, the construction procceds as follows. Recall that a graph $G$ is said to be nondegenerate if the determinant of its adjacency matrix over the two-element field $\mathbb{F}_{2}$ is $1$. The delta-matroid $\delta(G)=(V(G);S(G))$ of a graph $G$ is the set system consisting of the set $V(G)$ of vertices of $G$ and a set $S(G)\subset 2^{V(G)}$ of its subsets, where a subset $U\subset V(G)$ of the set of vertices of the graph is admissible (i.e., $U\in S(G)$) iff the induced subgraph $G|_{U}$ is nondegenerate.

In turn, the delta-matroid $\delta(\Gamma)=(E(\Gamma);S(\Gamma))$ of an embedded graph $\Gamma$ is the set system consisting of the set $E(\Gamma)$ of the edges of $\Gamma$ and a set $S(\Gamma)\subset 2^{E(\Gamma)}$ of its subsets, where a subset $U\subset E(\Gamma)$ of the set of edges is admissible (i.e., $U\in S(\Gamma)$) iff the spanning embedded subgraph $\Gamma|_{U}$ has a connected boundary.

If the genus of an embedded graph $\Gamma$ is [math] (that is, this graph is embedded into the sphere), a set of its edges is admissible iff the corresponding spanning subgraph $\Gamma|_{U}$ is a tree. For an arbitrary genus, the admisssible subsets in the delta-matroid $\delta(\Gamma)$ also are called quasitrees.

A. Bouchét proved that the set systems $\delta(G)$ and $\delta(\Gamma)$ indeed are delta-matroids. In addition, if $\Gamma$ is an embedded graph with a single vertex, that is, a chord diagram, the corresponding definitions are compatible:

Theorem 4.2**.**

If $C$ is a chord diagram, that is, an embedded graph with a single vertex, then the delta-matroid $\delta(C)$ is naturally isomorphic to the delta-matroid $\delta(g(C))$ of the intersection graph of $C$.

In other words, the boundary of a chord diagram $C$ is connected iff its intersection graph is nondegenerate. The last assertion has been rediscovered many times, see, e.g., [61].

The delta-matroid of a graph, as well as the delta-matroid of an orientable embedded graph is even. A delta-matroid $(E;S)$ is said to be even if the number of elements in all its admissible sets has the same parity, $|X|\equiv|Y|\leavevmode\nobreak\ {\rm mod}\leavevmode\nobreak\ 2$ for all $X,Y\in S$. Since we are interested first of all in graphs embedded into orientable surfaces, below, we talk only about even delta-matroids, although most of the constructions can be as well extended to delta-matroids that are not even, see Sec. 4.5.

Introduce the twist (or partial dual) of a delta-matroid in the following way. Let $U\subset E$ be a subset of the base set $E$ of a delta-matroid $D=(E;S)$. The twist $D*U$ of a delta-matroid $D$ in the subset $U$ is the set system $D*U=(E;S*U)$, where $S*U=\{X\Delta U|X\in S\}$. For delta-matroids of embedded graphs, the twist operation corresponds to partial duality introduced in [20]. The twist of a delta-matroid in an arbitrary subset of its base set is a delta-matroid [12].

Definition 4.3**.**

The delta-matroid $\delta(G)$ of a graph $G$ is called graphic. The twist of a graphic delta-matroid in a subset of its base set is called a binary delta-matroid.

In particular, the delta-matroid $\delta(\Gamma)$ of any embedded graph $\Gamma$ on an orientable surface is even binary.

For delta-matroids, a $4$-term relation is defined. Its definition requires defining two operations on delta-matroids, namely, the first and the second Vassiliev moves. Each of these operations is specified by a chosen ordered pair of elements of the delta-matroid.

The result $\widetilde{D_{ab}}$ of sliding the handle $a$ along the handle $b$ in a delta-matroid $D=(E;S)$, $a,b\in E$, is defined as $\widetilde{D_{ab}}=(E;\widetilde{S_{ab}})$, where

[TABLE]

The result ${D^{\prime}_{ab}}$ of exchange of the ends of the handles $a$ and $b$ in a delta-matroid $D=(E;S)$, $a,b\in E$, is defined by ${D^{\prime}_{ab}}=(E;{S^{\prime}_{ab}})$, where

[TABLE]

Both the first and the second Vassiliev move take a binary delta-matroid to a delta-matroid of the same type.

A function $f$ on even binary delta-matroids satisfies $4$-term relations if for any such delta-matroid $D=(E;S)$ and any pair $a,b\in E$ of elements of the base set we have

[TABLE]

If a function on even binary delta-matroids satisfies $4$-term relations, then its restriction to delta-matroids of embedded graphs on orientable surfaces satisfies generalized $4$-term relations for them.

4.3 Hopf algebras of delta-matroids

Two set systems $(E_{1};S_{1})$ and $(E_{2};S_{2})$ are said to be isomorphic if there is a one-to-one mapping of the set $E_{1}$ to the set $E_{2}$ that takes the set of subsets $S_{1}$ one-to-one to the set of subsets $S_{2}$.

Denote by $\mathcal{B}^{e}_{n}$ the vector space spanned by isomorphism classes of even binary delta-matroids on $n$-element sets and define

[TABLE]

The vector space $\mathcal{B}^{e}$ can be endowed with a natural structure of graded commutative cocommutative Hopf algebra. Multiplication in this algebra is given by the disjoint union, while comultiplication is given by splitting of the base set of a delta-matroid into two disjoint subsets in all possible ways. A delta-matroid is said to be connected if it cannot be represented as a product of two delta-matroids with smaller base sets.

The quotient of the vector space $\mathcal{B}^{e}$ modulo the subspace of $4$-term relations inherits a Hopf algebra structure. We are not going to use this Hopf algebra, and do not introduce a notation for it.

Remark**.**

In contrast to graphs and delta-matroids, embedded graphs, as far as we know, do not generate a natural Hopf algebra.

4.4 Extending graph invariants to embedded

graphs and delta-matroids

In this section, we explain how to extend to embedded graphs the $4$-invariants of graphs described in Sec. 2. The extension obtained happen to be defined on binary delta-matroids, and their definition uses the Hopf algebra structure on the vector space of binary delta-matroids.

4.4.1 Interlace polynomial

The interlace polynomial of graphs defined above admits a natural extension to binary delta-matroids.

Definition 4.4**.**

The distance from a set system $D=(E;S)$ to a subset $U\subset E$ is the number

[TABLE]

The interlace polynomial $L_{D}(x)$ of a delta-matroid $D=(E;S)$ is the polynomial

[TABLE]

Theorem 4.5**.**

If $D=D(G)$ is the delta-matroid of a graph $G$, then the interlace polynomial $L_{D}$ of $D$ is related to the interlace polynomial of $G$ in the following way:

[TABLE]

Theorem 4.6** ([39]).**

The interlace polynomial of even binary delta-matroids satisfies $4$-term relations for them.

Theorem 4.7**.**

The space of values of the interlace polynomial on the subspace of primitive elements in $\mathcal{B}^{e}_{n}$ has dimension $n$.

This implies that the dimension of the space of primitive elements in $\mathcal{B}^{e}_{n}$ is at least $n$.

4.4.2 Stanley’s symmetrized chromatic polynomial

The notion of umbral invariant of graphs admits a natural extension to invariants of delta-matroids. To define such an invariant, it suffices to specify, for each connected delta-matroid $D=(E;S)$ the coefficient of $q_{n}$, where $n=|E|$, in the value of the umbral invariant of this delta-matroid.

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]. Below, we restrict ourselves with commutative cocommutative Hopf algebras.

A combinatorial Hopf algebra is a pair $(\mathcal{H},\xi)$ consisting of a connected graded Hopf algebra $\mathcal{H}$ and a character (i.e., a multiplicative mapping) $\xi:\mathcal{H}\to\mathbb{C}$.

The polynomial Hopf algebra $\mathbb{C}[q_{1},q_{2},\dots]$ becomes a combinatorial Hopf algebra if we endow it with the canonical character $\zeta$ which takes each variable $q_{k}$ to $1$. This combinatorial Hopf algebra is a terminal object in the category of combinatorial Hopf algebras, that is, to each combinatorial Hopf algebra $(\mathcal{H},\xi)$ a unique umbral invariant (a graded Hopf algebra homomorphism) $\mathcal{H}\to\mathbb{C}[q_{1},q_{2},\dots]$ that takes $\xi$ to $\zeta$ is associated.

It is proved in [2] that Stanley’s symmetrized chromatic polynomial is the graded Hopf algebra homomorphism $\mathcal{G}\to\mathbb{C}[q_{1},q_{2},\dots]$, which corresponds to the character $\xi:\mathcal{G}\to\mathbb{C}$, which takes the graph with a single vertex to $1$, and any connected graph with $n\geq 2$ vertices to [math].

This structure allowed one to define [51] Stanley’s symmetrized chromatic polynomial of delta-matroids as the graded homomorphism $W:\mathcal{B}^{e}\to\mathbb{C}[x;q_{1},q_{2},\dots]$ from the Hopf algebra of even binary delta-matroids to the Hopf algebra of polynomials in the variables $q$ with coefficients in the ring of polynomials in a variable $x$, which is associated to the character taking the delta-matroid $(\{1\};\{\{\emptyset\}\})$ to $1$, the delta-matroid $(\{1\};\{\{1\}\})$ to $x$, and any connected delta-matroid of grading $n\geq 2$ to [math]. These two delta-matroids on $1$-element set span the homogeneous subspace $\mathcal{B}^{e}_{1}$.

Theorem 4.8**.**

Stanley’s symmetrized chromatic polynomial of delta-matroids satisfies $4$-term relations for delta-matroids.

4.4.3 Transition polynomial

Transition polynomial also admits an extension to delta-matroids, which satisfies $4$-term relations for them. In order to define transition polynomial, we will need few further definitions. Let $D=(E;S)$ be a delta-matroid, and let $U\subset E$ be a subset of its base set.

The polynomial

[TABLE]

is called the transition polynomial of the delta-matroid $D=(E;S)$. Here the operation $+$, for an element $u\in E$, is defined by

[TABLE]

The operation $\bar{*}$ is given by

[TABLE]

These two operations are extended to an arbitrary subset $A\subset E$.

This definition is a specialization of the definition of weighted transition polynomial in [14].

Theorem 4.9** ( [26]).**

The transition polynomial of even binary delta-matroids satisfies $4$-term relations for them.

4.4.4 Skew characteristic polynomial

Graph nondegeneracy naturally extends to embedded graphs and delta-matroids. A delta-matroid $(E;S)$ is said to be nondegenerate if $E\in S$, i.e., if the whole set $E$ is admissible. For embedded graphs, this means that the embedded graph is nondegenerate iff its boundary is connected. For chord diagram, this definition is consistent with the one we already know. Nondegeneracy $\nu(D)$ of a delta-matroid is $1$ if $D$ is nondegenerate, and it is [math] otherwise.

We can define the skew characteristic polynomial of a delta-matroid $D=(E;S)$ by

[TABLE]

Theorem 4.10**.**

The skew characteristic polynomial of delta-matroids satisfies $4$-term relations.

Similarly to the case of graphs, this statement follows from the fact that nondegeneracy is preserved under the second Vassiliev move.

4.5 Extending to nonorientable surfaces:

framed chord diagrams, framed graphs,

and not necessarily even delta-matroids

$4$-term relations for chord diagrams, graphs, and embedded graphs have a nonorientable version, which we describe here briefly. It should not be separated from the orientable one: all the constructions, and first of all the Hopf algebra structures, require simultaneous participation of both orientable and nonorientable objects. In each case, orientable objects generate Hopf subalgebras in the Hopf algebras of objects without restrictions to their orientability.

A framed chord diagram is a pair $(C,\varphi)$ consisting of a chord diagram $C$ and a framing $\varphi:V(C)\to\mathbb{F}_{2}$ taking each chord of $C$ to an element of the field $\mathbb{F}_{2}$. An isomorphism of framed chord diagrams is an isomorphism of the underlying chord diagrams taking the framing of the first of them to that of the second one.

A framed graph is a pair $(G,\varphi)$ consisting of a simple graph $G$ and a framing $\varphi:V(G)\to\mathbb{F}_{2}$, which associates to any vertex of $G$ an element of the field $\mathbb{F}_{2}$. An isomorphism of framed graphs is an isomorphism of the underlying graphs taking the framing of the first of them to that of the second one. By numbering the vertices of $G$ with the numbers $1,\dots,n=|V(G)|$, one associates to the framed graph its adjacency matrix $A(G,\varphi)$. Nondiagonal elements of this matrix coincide with those of the adjacency matrix $A(G)$ of the simple graph $G$, and its diagonal elements coincide with the framings of the corresponding vertices. Isomorphism classes of framed graphs coincide with symmetric matrices over the field $\mathbb{F}_{2}$ considered up to simultaneous renumbering of the columns and the rows.

A framed graph is associated to any framed chord diagram, which is its intersection graph. A framed chord diagram can be considered as a ribbon graph with a single vertex: an ordinary ribbon is associated to a chord with framing [math], while a chord with framing $1$ is replaced with a half-twisted ribbon. If there is a chord with framing $1$, then the surface corresponding to the framed chord diagram is nonorientable. A framed graph invariant $f$ is called a $4$-invariant if for any pair of vertices $a,b\in V(G)$ it satisfies the $4$-term relation

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There is a natural one-to-one correspondence between the sets of vertices of any two graphs participating in the relation, therefore, both framings $\varphi$ and $\tilde{\varphi}_{ab}$ have a common domain. Here $\widetilde{\varphi}_{ab}$ denotes the framing of the graph $\widetilde{G}_{ab}$, which coincides with $\varphi$ on all the vertices but $a$, and $\widetilde{\varphi}_{ab}(a)=\varphi(a)+\varphi(b)$. In other words, the adjacency matrix $A(\widetilde{G}_{ab})$ of the framed graph $\widetilde{G}_{ab}$ can be obtained from $A(G)$ by means of the same transformation (5) as in the unframed case.

Nondegeneracy and skew characteristic polynomial for framed graphs are defined in the same way as for ordinary graphs. The nondegenracy of a graph is preserved under the second Vassiliev move. This implies that the skew characteristic polynomial is a $4$-invariant of framed graphs. In contrast to skew characteristic polynomial of simple graphs the skew characteristic polynomial of a framed graph can be neither even, nor odd.

Any $4$-invariant of framed graphs defines a framed weight system, which takes a framed chord diagram to the value of the invariant on its framed intersection graph. In particular, nondegenercy and skew characteristic polynomial both define framed weight systems.

The delta-matroid of a framed graph and the delta-matroid of an arbitary embedded graph, may be nonorientable, are defined in the same way as in the orientable case. If a framed graph has a vertex with framing $1$, then the corresponding delta-matroid will not be even: its admissible sets will contain even as well as odd number of elements. The delta-matroid of an embedded graph on a nonorientable surface also is not even. Isomorphism classes of binary delta-matroids generate a Hopf algebra containing the Hopf subalgebra $\mathcal{B}^{e}$.