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Algebra of shares, complete bipartite graphs and P. A. Zinovaa, M. E. Kazarianab a National Research University Higher School of Economics, Moscow, Russia b Center for Advanced Studies, Skolkovo Institute of Science and Technology, Moscow, Russia
Russian version:
Matematicheskii Sbornik, 2023, Volume 214, Number 6, DOI: https://doi.org/10.4213/sm9795 
Bibliographic databases:
    § 1. IntroductionWeight systems, which are functions on the chord diagrams that satisfy the so-called four-term relations, play an important role in the study of knot invariants. Invariants of finite-order knots, introduced by Vassiliev [1] around 1990, are expressed in terms of weight systems. On the other hand, as Kontsevich [2] proved, any weight system over a field of characteristic $0$ corresponds to some invariant of finite order. There are several main sources of weight systems. One of the richest constructions is the weight system associated with a finite-dimensional Lie algebra endowed with a nondegenerate invariant bilinear form. This construction was proposed by Bar-Natan [3] and Kontsevich [2]. The values of such a weight system lie in the centre of the universal enveloping algebra of the Lie algebra. The simplest nontrivial case of such a weight system is the weight system corresponding to the Lie algebra $\mathfrak{sl}_2$. The knot invariant to which it corresponds is the coloured Jones polynomial. The value of this weight system at a chord diagram of order $n$ is a monic polynomial of degree $n$ in the Casimir element $c$ of $\mathfrak{sl}_2$. For this weight system we have the Chmutov-Varchenko recurrence relations [4], but even in this simplest case the calculation of the value of this weight system at a particular chord diagram turns out to be a difficult problem, in which there were no significant advances for a long time; such advances were only achieved recently. According to the Chmutov-Lando theorem [5], the value of the $\mathfrak{sl}_2$ weight system at a chord diagram depends only on its intersection graph, so we can speak about the values of the $\mathfrak{sl}_2$ weight system at intersection graphs. A recent result due to Zakorko [6] proves an explicit formula for the values of this weight system at complete graphs, proposed by Lando as a conjecture. The purpose of this paper is to give explicit formulae for the values of the $\mathfrak{sl}_2$ weight system on another large class of graphs, namely, complete bipartite graphs. A chord diagram is an oriented circle with a finite set of chords on it, considered up to orientation-preserving diffeomorphisms of the circle. A share in a chord diagram is a pair of disjoint arcs of the circle such that if one endpoint of a chord lies on one of these arcs, then this is also true for the other endpoint. All possible shares span an (infinite-dimensional) vector space, and we define the $\mathfrak{sl}_2$ weight system on it. The values of this weight system lie in the commutative subalgebra of the tensor square $U(\mathfrak{sl}_2) \otimes U(\mathfrak{sl}_2)$ of the universal enveloping algebra of the Lie algebra $\mathfrak{sl}_2$. This subalgebra is generated by three elements denoted by $c_1$, $c_2$ and $\xi$. For this weight system analogues of the four-term relations and the Chmutov-Varchenko six-term recurrence relations are valid. We consider the quotient space of the vector space of shares modulo these relations. This quotient space is also endowed with the structure of an associative algebra if we define the multiplication of shares as their concatenation. Each element of this space is expressed as a linear combination of shares of simpler form; the value of the weight system at it has the form $(c_1-1)^{k_1}(c_2-1)^{k_2}c_1^{n_1}c_2^{n_2}\xi^N$, $k_1, k_2, n_1, n_2, N \in \mathbb N \cup \{ 0 \}$. It follows from the existence of such an expression that the $\mathfrak{sl}_2$ weight system gives a homomorphism from the algebra of shares into the algebra of polynomials in $\xi$ whose coefficients are polynomials in $c_1$ and $c_2$. Following Zakorko [6], we introduce operators $S_1$ and $X$ on the algebra of shares and the corresponding (in the sense of the above homomorphism) operators on the algebra of polynomials. Using the Chmutov-Varchenko six-term relations and the four-term relations, we derive recurrence formulae for the action of the operators $X$ and $S_1$. Using these recurrence formulae, we obtain the generating function of the matrix coefficients $s_{i,m}$, $m=0,1,2,\dots$ and $i = 0,1,2,\dots,m$, of the operator $S_1$ in the basis $1,\xi,\xi^2,\dots$ . In addition to this, we compute explicitly the leading matrix coefficients $s_{m,m}$. It turns out that $s_{m,m}=c-m(m+1)/2$. To derive explicit formulae we introduce a specialization, that is, a mapping from the algebra of shares to the algebra of generating functions such that monomials in the algebra of generating functions correspond to complete bipartite graphs. Using this specialisation we obtain a formula expressing the generating function $G_m$ of the values of the $\mathfrak{sl}_2$ weight system at the complete bipartite graphs $K_{0,m}, K_{1,m},\dots$ in terms of the generating functions $G_0, G_1, \dots, G_{m-1,m}$ and the matrix coefficients of the operator $S_1$. For $m=0,1,2,3,4$ the generating functions $G_m$ were previously calculated by the first-named author (see [7] and [8]). The methods for the computation of these functions that we develop in this paper for all values of $m$ are much more universal and can be applied to a wide variety of other situations. The generating functions $G_m$ allow us to compute recursively the value of the $\mathfrak{sl}_2$ weight system at any complete bipartite graph. Furthermore, it follows from our computations that the generating functions $G_m$ are some linear combinations of certain explicit geometric progressions. Hence we deduce that the value of the $\mathfrak{sl}_2$ weight system at the projection of a complete bipartite graph onto the space of primitive elements is a polynomial of degree not greater than the number of vertices in the smallest part of the complete bipartite graph. This proves in the particular case of complete bipartite graphs that in the case of an arbitrary intersection graph this degree does not exceed half the circumference (length of the maximal cycle) of the graph. This was conjectured by Lando. The paper is organised as follows. In § 2 we recall the basic terms related to chord diagrams and weight systems. Section 3 is devoted to a description of the $\mathfrak{sl}_2$ weight system and its properties. In § 4 we introduce the algebra of shares and the $\mathfrak{sl}_2$ weight system on it. In § 5 we derive explicit formulae for the generating functions of the values of the $\mathfrak{sl}_2$ weight system at complete bipartite graphs. § 2. Hopf algebras of graphs and chord diagramsIn this section we recall the Hopf algebra structures on the spaces of graphs and chord diagrams. Then we discuss the space of primitive elements which is related to the conjecture a special case of which we prove. By a coassociative counital coalgebra over a field $\mathbb K$ we mean a vector space $C$ over $\mathbb K$ with maps
$$
\begin{equation*}
\mu \colon C\to C\otimes C \quad\text{and}\quad \varepsilon \colon C\to \mathbb K,
\end{equation*}
\notag
$$
such that
$$
\begin{equation*}
(\mathrm{id}_C\otimes \mu)\circ \mu=(\mu \otimes \mathrm{id}_C)\circ \mu \quad\text{and}\quad (\mathrm{id}_C\otimes \varepsilon)\circ \mu=\mathrm{id}_C=(\varepsilon \otimes \mathrm{id}_C)\circ \mu.
\end{equation*}
\notag
$$
A bialgebra over $\mathbb K$ is a vector space $B$ over $\mathbb K$ which is both an associative algebra with multiplication $m$ and unit $\eta$ and a coassociative coalgebra with comultiplication $\mu$ and counit $\varepsilon$ for which the following relations hold:
$$
\begin{equation*}
\begin{gathered} \, \mu \circ m=(m \otimes m) \circ (\mathrm{id}\otimes \tau \otimes \mathrm{id}) \circ (\mu \otimes \mu), \\ \varepsilon \otimes \varepsilon=\varepsilon \circ m, \qquad \eta \otimes \eta=\mu \circ \eta, \qquad \mathrm{id}=\varepsilon \circ \eta. \end{gathered}
\end{equation*}
\notag
$$
Here we denote by $\tau \colon B \otimes B \to B \otimes B$ the map switching the factors. A Hopf algebra over a field $\mathbb K$ is a unital, counital, associative and coassociative bialgebra $H$ with antipode, that is, a $\mathbb K$-linear mapping $S \colon H \to H$ such that (in the above notation) the following relation holds:
$$
\begin{equation*}
m \circ (S \otimes \mathrm{id}) \circ \mu=\eta \circ \varepsilon=m \circ (\mathrm{id} \otimes S) \circ \mu.
\end{equation*}
\notag
$$
In the context of this paper the antipode plays no role, so in fact we deal with bialgebras. Nevertheless, following tradition we use the term ‘Hopf algebra’. However, we assume throughout what follows that the ground field $\mathbb{K}$ has characteristic zero. Moreover, all Hopf algebras we use are commutative and cocommutative. In this case the antipode can be chosen naturally, so the use of the term ‘Hopf algebra’ is justified. 2.1. The Hopf algebra of graphsThe product of two graphs $G_1$ and $G_2$ is their disjoint union: $G_1 G_2 := G_1 \sqcup G_2$. This multiplication extends to the space of graphs by linearity. It is consistent with the grading and defines the structure of a graded algebra on the space of graphs. Denote the vertex set of a graph $G$ by $V(G)$. The action of comultiplication $\mu$ on $G$ is defined by
$$
\begin{equation*}
\mu(G):=\sum_{U\subset V(G)} G|_{U}\otimes G|_{V(G)\setminus U}.
\end{equation*}
\notag
$$
Here $G|_U$ denotes the subgraph of $G$ induced by the subset $U$ of its vertex set. Here and below the term ‘subgraph’ means the subgraph induced by a given subset of vertices. Like multiplication, comultiplication also extends to linear combinations of graphs by linearity and is consistent with the grading, that is, we have introduced the structure of a graded coalgebra on the space of graphs. Moreover, the following assertion holds. Claim 1. Multiplication and comultiplication introduced above, together with the naturally defined unit, counit and antipode, define the structure of a graded commutative and cocommutative Hopf algebra. This Hopf algebra structure on the space of graphs was introduced in [9]. Denote the Hopf algebra of graphs by $\mathscr G$, and let $\mathscr G_n$ denote its homogeneous vector subspace spanned by the graphs with $n$ vertices, $n=0,1,2,\dots$, so that
$$
\begin{equation*}
\mathscr G=\mathscr G_0\oplus \mathscr G_1 \oplus \mathscr G_2 \oplus \dotsb.
\end{equation*}
\notag
$$
A four-term element of the space of graphs is a linear combination of the form where $A$ and $B$ are some two vertices of the graph, $G'_{AB}$ is a graph $G$ in which the incidence of the vertices $A$ and $B$ is reversed; $\widetilde G_{AB}$ is the graph $G$ in which, for each vertex connected with $B$, its incidence to the vertex $A$ is reversed. Note that all graphs involved in a four-term element have the same number of vertices.Denote by $\mathscr F_n$ the quotient space of the vector space $\mathscr G_n$ by the vector subspace spanned by the four-term elements with $n$-vertex graphs. The space
$$
\begin{equation*}
\mathscr F=\mathscr F_0\oplus \mathscr F_1 \oplus \mathscr F_2 \oplus \dotsb
\end{equation*}
\notag
$$
is endowed with the structure of a graded Hopf algebra induced from $\mathscr G$; see [10]. 2.2. The Hopf algebra of chord diagramsDefinition 1. A chord diagram of order $n$ is an oriented circle with $2n$ pairwise distinct points combined into $n$ disjoint pairs, which is considered up to orientation-preserving diffeomorphisms of the circle. For a clearer presentation, in figures we join the points forming a pair by a segment (of a straight line or a curve), which we call a chord. A chord has no common points with the circle apart from the endpoints. The vector space formed by the linear combinations of chord diagrams is graded. Every component of the grading is a vector space spanned by the chord diagrams of the same order. By a four-term element of the space of chord diagrams we mean a linear combination of diagrams shown in Figure 1. All four diagrams in this combination contain the same family of chords distinct from the two shown, and the endpoints of chords in this family can only lie on the parts of the circle indicated by dashed lines. Sections where there are no endpoints of other chords are marked by thick lines. If there are two endpoints of chords in such a section, then we call such endpoints neighbouring. We keep using this notation in what follows. By equating to zero four-term elements of the spaces of graphs and chord diagrams we obtain four-term relations. Definition 2. An arc diagram of order $n$ is an oriented straight line with $2n$ pairwise distinct points on it combined into $n$ pairs, considered up to orientation-preserving diffeomorphisms of the line. Each of these $n$ pairs of points is represented by an arc in the upper half-plane that connects the points. If we choose a point on a chord diagram that differs from the endpoints of the chords and ‘cut’ the chord diagram at this point, then we obtain a representation of the chord diagram in the form of an arc diagram (see Figure 2, for example). A chord diagram of order $n$ can have up to $2n$ different representations in the form of an arc diagram. By contrast, an arc diagram determines uniquely the corresponding chord diagram. The product of chord diagrams $C_1$ and $C_2$ is a chord diagram corresponding to the arc diagram obtained as the concatenation of two arbitrary arc representations of the diagrams $C_1$ and $C_2$ (see Figure 3). The product of chord diagrams is well defined (that is, the result does not depend on the choice of the cut point) modulo the four-term relations. Denote by $V(C)$ the set of chord of a diagram $C$. Comultiplication $\mu$ of chord diagrams is defined by
$$
\begin{equation*}
\mu(C):=\sum_{U\subset V(C)} C|_{U}\otimes C|_{V(C)\setminus U}.
\end{equation*}
\notag
$$
Here $C|_{U}$ denotes the chord diagram consisting of a subset $U\subset V(C)$ of the chord set of the chord diagram $C$. Multiplication and comultiplication extend to linear combinations of chord diagrams by linearity and are consistent with the grading. Claim 2 (see [2]). The above operations of multiplication and comultiplication define the structure of a Hopf algebra on the quotient space of the space of chord diagrams by the subspace generated by the four-term elements. We denote this Hopf algebra by $\mathscr C$. We call it the Hopf algebra of chord diagrams, meaning that chord diagrams are considered up to the four-term relations. 2.3. The intersection graphThe intersection graph $\gamma(C)$ of a chord diagram $C$ is a graph whose vertices $V(\gamma(C))$ correspond to the chords $V(C)$ of $C$, and two vertices $v_a$ and $v_b$ of which are connected by an edge if and only if the corresponding chords $a$ and $b$ intersect (that is, their endpoints $a_1$, $a_2$ and $b_1$, $b_2$ are positioned on the circle in the following order: $a_1$, $b_1$, $a_2$ and $b_2$). Claim 3 (see [10]). The map assigning to a chord diagram its intersection graph extends to a graded homomorphism of Hopf algebras $\mathscr C \to \mathscr F$. There exist graphs that are not intersection graphs of any chord diagram. Moreover, note that two different chord diagrams can have the same intersection graph. 2.4. Projection onto the space of primitive elementsDefinition 3. An element $p$ of a bialgebra is called primitive if $\mu(p)=1\otimes p + p\otimes 1$. It can easily be seen that the primitive elements form a vector subspace of the bialgebra. Since every homogeneous component of a primitive element is primitive, this vector subspace of a graded bialgebra is also graded. Claim 4 (Milnor-Moore theorem; see [11]). Over a field of characteristic $0$, any connected commutative and co-commutative graded bialgebra is isomorphic to the polynomial bialgebra generated by its primitive elements. A graded bialgebra is connected if its zero homogeneous component is isomorphic to the field itself. The decomposable elements (that is, products of homogeneous elements of lower degrees) span a vector subspace of any homogeneous space of a graded bialgebra. It follows from the Milnor-Moore theorem that every homogeneous subspace decomposes into a direct sum of the subspace spanned by the decomposable elements and the subspace of primitive elements. Therefore, the projection $\pi$ of every homogeneous subspace onto the subspace of primitive elements along the subspace of decomposable elements is well defined. Claim 5 (see [10] and [12]). The projection $\pi(G)$ of an arbitrary graph $G$ onto the subspace of primitive elements along the subspace of decomposable elements of the Hopf algebra $\mathscr G$ has the form where $V_1, V_2, V_3, \dots$ are disjoint nonempty subsets of the vertex set $V(G)$. § 3. $\mathfrak{sl}_2$ weight system on chord diagramsIn this section we recall the necessary information about the weight system related to the Lie algebra $\mathfrak{sl}_2$. In general, we follow [13]. 3.1. The definition of the $\mathfrak{sl}_2$ weight system on chord diagramsLet $A$ be an algebra over a field $\mathbb K$. A linear function $w \colon \mathscr C \to A$ is called a weight system on $\mathscr C$. In other words, a weight system is a function of chord diagrams that equals zero at all four-term elements. A weight system is called multiplicative if its value at the product of any two chord diagrams is equal to the product of its values at the diagrams, that is, if it is an algebra homomorphism. In what follows we consider only the case of $\mathbb{K} = {\mathbb C}$ and $A={\mathbb C}[c]$ (the ring of polynomials in one variable). Let $\mathfrak g$ be a finite-dimensional Lie algebra over $\mathbb C$ endowed with a nondegenerate bilinear invariant form $(\,\cdot\,{,}\,\cdot\,)$. (A form is invariant if $([x ,y], z) = (x ,[y,z])$ for any $x,y,z \in \mathfrak g$.) Let $X = \{x_1, x_2, \dots, x_m\}$ be an orthonormal basis of $\mathfrak g$ with respect to this form. Denote by $U(\mathfrak g)$ the universal enveloping algebra of $\mathfrak g$. Consider the map $w_{\mathfrak g} \colon \mathscr C \to U(\mathfrak g)$ defined as follows. Let $C$ be a chord diagram, $a$ be one of its arc representations, $V(a)$ be the set of all arcs of $a$ and $\nu$ be a map $\nu \colon V(a) \to \{1, 2, \dots, m\}$. We associate with $a$ and $\nu$ an element $w_X(a,\nu) \in U(\mathfrak g)$ by writing $x_{\nu(v)} \in X$ on both ends of each arc $v\in V(a)$ and taking their product from left to right. We denote this product by $w_X(a, \nu)$ and let $w_X(a)$ denote the sum over all possible maps:
$$
\begin{equation}
w_X(A):=\sum_{\nu\colon V(A)\to\{1,\dots,m\}} w_X(A,\nu).
\end{equation}
\tag{3.1}
$$
For example, the value of the weight system corresponding to a Lie algebra with orthonormal basis $x_1,\dots, x_m$ at the arc diagram shown in Figure 4 equals
$$
\begin{equation*}
\sum_{i_1=1}^m\sum_{i_2=1}^m \sum_{i_3=1}^m \sum_{i_4=1}^m \sum_{i_5=1}^m x_{i_1}x_{i_2}x_{i_3}x_{i_2}x_{i_4}x_{i_1}x_{i_5}x_{i_3}x_{i_4}x_{i_5}.
\end{equation*}
\notag
$$
Claim 6 (see [2]). 1) For any $C \in \mathscr C$ the result of the above operation is defined uniquely and does not depend on the choice of an arc representation of $C$. 2) For any $a$, $w_X(a) \in Z(U(\mathfrak g))$, where $Z(U(\mathfrak g))$ is the centre of the universal enveloping algebra. 3) $w_X(a)$ does not depend on the choice of the orthonormal basis. 4) This map from the chord diagrams to $ Z(U(\mathfrak g))$ satisfies the four-term relations. Therefore, it extends to a homomorphism of commutative algebras. Since we define multiplication of chord diagrams as the concatenation of arc representations of these diagrams, any Lie-algebra weight system is multiplicative. Note that the construction described can easily be modified for an arbitrary, not necessarily orthonormal, basis in $\mathfrak g$: one just needs to put an element $x_i$ at the left-hand end of the arc with index $i$, and an element $x_i^*$ of the dual basis at its right-hand end. We will use it for the Lie algebra $\mathfrak{sl}_2$ in this form. For the Lie algebra $\mathfrak{sl}_2$, which is the simplest noncommutative semisimple Lie algebra, this construction has the following form. The algebra $\mathfrak{sl}_2$ is generated by three elements $x$, $y$ and $z$ for which the following relations hold
$$
\begin{equation*}
[x, y]=z, \qquad [y, z ]=x \quad\text{and}\quad [z, x]=y.
\end{equation*}
\notag
$$
The bilinear form is given by the relations
$$
\begin{equation}
\begin{gathered} \, (x,x)=(y,y)=(z,z)=-1, \\ (x,y)=(y,z)=(z,x)=0. \notag \end{gathered}
\end{equation}
\tag{3.2}
$$
The centre of this universal enveloping algebra $Z(U(\mathfrak{sl}_2))$ is isomorphic to the algebra of polynomials in the Casimir element $c=-x^2 - y^2 - z^2\in U(\mathfrak{sl}_2)$. Hence (3.1) defines a map $w_{\mathfrak{sl}_2} \colon \mathscr C \to \mathbb C[c]$. This map is an algebra homomorphism, and it is called the $\mathfrak{sl}_2$ weight system on $\mathscr C$. The statement below follows immediately from this definition. Corollary 1. The value of the $\mathfrak{sl}_2$ weight system at a chord diagram with only one chord equals $c$. In [5] the following nontrivial statement was proved, which links the $\mathfrak{sl}_2$ weight system with polynomial graph invariants. Note that its analogue for more complicated Lie algebras, for instance, $\mathfrak{sl}_3$, turns out to be wrong. Claim 7. The value of the $\mathfrak{sl}_2$ weight system at a chord diagram depends on its intersection graph only. 3.2. The properties of the $\mathfrak{sl}_2$ weight systemClaim 8 (Chmutov-Varchenko relations; see [4]). Let $D$ be a chord diagram. Assume that its intersection graph is connected, that is, $D$ is not the product of two diagrams of lower order. Then 1) if $D$ contains a leaf, that is, a chord intersecting only one chord, then
$$
\begin{equation}
w_{\mathfrak{sl}_2}(D)=(c-1)w_{\mathfrak{sl}_2}(D'),
\end{equation}
\tag{3.3}
$$
where $D'$ is the chord diagram obtained from $D$ by deleting the leaf; 2) if a chord diagram with connected intersection graph contains no leaf, then it contains a triple of chords looking like the triple in the leftmost chord diagram shown in one of figures (3.4) and (3.5); 3) the following equalities hold for these triples: andThe Chmutov-Varchenko relations allow one to compute recursively the value of the $\mathfrak{sl}_2$ weight system at an arbitrary chord diagram. However, their direct application is extremely cumbersome: a six-term relation replaces a chord diagram with five simpler ones, and it is often difficult to predict the final answer. § 4. The $\mathfrak{sl}_2$ weight system on sharesIn this section we introduce the algebra of shares, describe its structure and define the $\mathfrak{sl}_2$ weight system on the shares. 4.1. Shares and the generalised four-term relationDefinition 4. A share is a part of a chord diagram consisting of two disjoint arcs of the circle such that if an endpoint of a chord lies on one of these arcs, then its other endpoint also lies on one of these arcs. Figure 5 shows an example of a share in a chord diagram and an example of a pair of arcs that do not form a share. The complement to a share in a chord diagram is also a share. By ‘cutting’ the circle of a chord diagram at an arbitrary pair of points other than the endpoints of chords one obtains a share whose complement is an empty share. Closing a share (that is, adding the missing arcs) turns the share to a chord diagram. An example of such a chord diagram is shown in Figure 5, c. In the literature (see, for example, [14]) a close algebra $\mathcal A(2)$ of chord diagrams on two strands is known. However, multiplication of shares is different from $\mathcal A(2)$: it differs by the choice of orientation on one of the two strands. Below we consider shares up to the four-term relations, which are obtained from the four-term relations for chord diagrams by ‘cutting’ the circle at two points that are not endpoints of chords and do not lie between neighbouring endpoints of chords. The most important for us example of such a relation is shown in Figure 6. Claim 9 (generalised four-term relation). The four-term relation for shares shown in Figure 6 also holds if the ‘fixed’ chord $B$ in it is replaced by an arbitrary set of chords forming a share while the ‘variable’ chord $A$ remains a single chord. Proof. We sum all four-term relations for the chord $A$ and for each chord of the share $B$. The resulting sum on the left-hand side consists of terms that differ only by the position of one endpoint of the chord $A$. It runs over all the neighbouring arcs of the endpoints of all chords in $B$. The terms in which this endpoint lies on an inner arc annihilate. The remaining terms form the required four-term element. The claim is proved. 4.2. The $\mathfrak{sl}_2$ weight system on sharesLet $S$ be a share, $V(S)$ be its chord set and $\nu \colon V(S) \to \{1, 2, 3\}$ be an arrangement of the indices $1$, $2$, $3$ on the chords of the share $S$. Let us assign to the share $S$ and the labelling mapping $\nu$ an element $w(S,\nu)\in U(\mathfrak{sl}_2) \otimes U(\mathfrak{sl}_2)$ as follows: for each arc $v\in V(S)$, on one of its ends we write an element $x_{\nu(v)}$ of the basis $X=\{x_1, x_2, x_3\}$ of $\mathfrak{sl}_2$, and at the other end we write the element $x_{\nu}^*$ of the dual basis. Denote by $w_X(S,\nu)$ the tensor product of the two monomials such that the first is the product of the elements put on the lower arc from left to right, and the second is the product of the elements put on the upper arc from right to left. We denote the sum of such elements over all possible labelling mappings $\nu$ by $w_X(S)$. The proof of the following statement is the same as the proof of a similar statement for chord diagrams. Claim 10. 1. The value of $w_X(S)$ is independent of the choice of a basis. 2. The mapping of shares to $U(\mathfrak{sl}_2) \otimes U(\mathfrak{sl}_2)$ thus obtained satisfies the four-term relations. Denote this mapping by $w_{\mathfrak{sl}_2}$. Its values at the shares are, respectively, $c_1=c\otimes 1$, $c_2=1\otimes c$ and $\xi=-(x\otimes x+y\otimes y+z\otimes z)$ for the basis $x$, $y$ and $z$ in $\mathfrak{sl}_2$ introduced before and $c=-(x^2+y^2+z^2)$.Claim 11. The elements $c_1$, $c_2$ and $\xi$ generate a commutative subalgebra of the algebra $U(\mathfrak{sl}_2) \otimes U(\mathfrak{sl}_2)$. From the definitions of the weight system on the chord diagrams and on shares, it follows that the following statement is true. Theorem 1. The value of the $\mathfrak{sl}_2$ weight system on the chord diagram obtained by closing a share is equal to the result of substituting $c_1=c_2=\xi=c$ into the value of the $\mathfrak{sl}_2$ weight system at this share. 4.3. The ‘normal form’ of a shareThe formal linear combinations of shares form a vector space. Denote by $\Xi^n$ a share formed by $n$ pairwise disjoint chords whose endpoints lie on different arcs; in particular, denote the share of the form Claim 12. 1. Let there be a chord in the share $S$ such that its endpoints lie on one arc and it does not intersect any chord of the share. Denote by $S'$ the result of removing this chord from $S$. Then 2. (Removing a leaf.) Let there be a leaf in a share $S$, that is, a chord intersecting only one chord and such that the its endpoints lie on one arc. Denote by $S'$ the result of removing this chord from $S$. Then
$$
\begin{equation}
w_{\mathfrak{sl}_2}(S) = \begin{cases} (c_1-1)w_{\mathfrak{sl}_2}(S') &\text{if both endpoints of the leaf removed} \\ &\text{lie on the lower arc}, \\ (c_2-1)w_{\mathfrak{sl}_2}(S') &\text{if both endpoints of the leaf removed} \\ &\text{lie on the upper arc}. \end{cases}
\end{equation}
\tag{4.2}
$$
Proof. Consider only the case of the lower arc. Below we denote by $e_1$ and $e_2$ arbitrary elements of the basis $\{x, y, z\}$, and by $A$, $B$, $C$ and $D$ some monomials in $x$, $y$ and $z$ corresponding to the fixed labelling of the share. Recall that, as follows from (3.2), the dual basis is $\{-x, -y, -z\}$. We fix a labelling of all other chords of the share.
1. The contribution of all possible labellings of the edge $e_1$ to the value of the weight system $w_{\mathfrak{sl}_2}(S')$ is the sum $-\sum_{e_1 = x, y, z} Ae_1^2B\otimes C$, since a direct calculation of $w_{\mathfrak{sl}_2}$ involves labellings of the following form: Now, $-(Ax^2B\otimes C + Ay^2B\otimes C + Az^2B\otimes C) = -A(x^2+y^2+z^2)B\otimes C$. Since $-(x^2+y^2+z^2)=c$ is a central element of $U(\mathfrak{sl}_2)$, this expression is equal to $cAB\otimes C = (c\otimes1)AB\otimes C$, as required.2. It follows from the above that the contribution of $w_{\mathfrak{sl}_2}(S) - c_1 w_{\mathfrak{sl}_2}(S')$ to the value of $w_{\mathfrak{sl}_2}$ is equal to the contribution of the following linear combination of shares: Recall that $e_1^*=-e_1$ and $e_2^*=-e_2$; so the value of the weight system at this linear combination of shares is
$$
\begin{equation*}
\begin{aligned} \, &\sum_{e_1, e_2 \in \lbrace x, y, z \rbrace} Ae_1 e_2 e_1^* B\otimes C e_2^* D-Ae_1 e_1^* e_2 B\otimes Ce_2^* D \\ &\qquad =\sum_{e_1, e_2 \in \lbrace x, y, z \rbrace} Ae_1 [e_2, e_1^* ] B\otimes Ce_2^* D \\ &\qquad =Ax[y,x] B\otimes CyD+ Ay[x,y] B\otimes CxD+ Ax[z,x] B\otimes Cz D \\ &\qquad\qquad +Az[x,z ] B\otimes CxD+ Ay[z,y] B\otimes Cz D+ Az[y,z] B\otimes CyD \\ &\qquad =Ax(-z) B\otimes CyD+ Ayz B\otimes CxD+ AxyB\otimes Cz D \\ &\qquad\qquad +Az (-y) B\otimes CxD+ Ay(-x) B\otimes Cz D+ Az xB\otimes CyD \\ &\qquad =AyB\otimes CyD+AxB\otimes CxD+Az B\otimes Cz D. \end{aligned}
\end{equation*}
\notag
$$
We have obtained the value of the weight system at a share of the form that is, at the share $-S'$. The claim is proved. Claim 13. ‘Cut’ each circle of the six-term relations for chord diagrams (3.4) and (3.5) at two points such that they are not endpoints of chords and do not lie between neighboring endpoints of chords. Then the six-term relations for shares obtained are true for the weight system. (In a similar way four-term relations for shares — see Figure 6 — were obtained from four-term relation for chord diagrams — see Figure 1.) Denote by $\mathscr S$ the quotient space of shares modulo the four-term and six-term relations and the following relations. 1. Let there be a chord in a share $S$ such that its endpoints lie on the same arc and it does not intersect any chord of the share. Denote by $S'$ the result of removing this chord from $S$. Then
$$
\begin{equation*}
S =\begin{cases} c_1 \cdot S' &\text{if both endpoints of the chord removed} \\ & \text{lie on the lower arc}, \\ c_2 \cdot S' &\text{if both endpoints of the chord removed} \\ &\text{lie on the upper arc}. \end{cases}
\end{equation*}
\notag
$$
2. (Removing a leaf.) Let there be a leaf in a share $S$, that is, a chord intersecting only one chord and such that its endpoints lie on the same arc. Denote by $S'$ the result of removing this chord from $S$. Then
$$
\begin{equation*}
S =\begin{cases} (c_1-1)\cdot S' &\text{if both endpoints of the leaf removed} \\ &\text{lie on the lower arc}, \\ (c_2-1) \cdot S' &\text{if both endpoints of the leaf removed} \\ & \text{lie on the upper arc}. \end{cases}
\end{equation*}
\notag
$$
Claim 14. The shares $\Xi^n$ for $n = 0, 1, 2, \dots$ generate the space $\mathscr S$ as an algebra over the ring of polynomials of the variables $c_1$ and $c_2$. Proof. An arbitrary share is expressed as a polynomial of $\Xi$, $c_1$ and $c_2$ in the space $\mathscr S$ as follows. Choose a chord of a share both of whose endpoints lie on the same arc of the share. We call the number of endpoints of other chords lying between the endpoints of the fixed one the length of the fixed chord. We say that a share $S'$ is simpler than a share $S$ if one of the following conditions is satisfied:
1) there are fewer chords in $S'$ than in $S$; 2) the shares $S$ and $S'$ have the same number of chords, but there are chords with endpoints on one arc in $S$, while there are no such chords in $S'$; 3) the shares $S$ and $S'$ have the same number of chords, and there are chords with endpoints on one arc both in $S$ and in $S'$, but the minimum length of such an arc in $S'$ is less than the one for $S$; 4) both shares have no chords with endpoints on one arc, but there are fewer pairs of intersecting chords in $S'$. We want to prove that every share in $\mathscr S$ can be expressed as a polynomial in $\Xi$ with coefficients depending on $c_1$ and $c_2$. We apply the following simplifying transformations to express a share as a linear combination of simpler shares: 1) if a share has chords of length 0 or 1, then it is simplified using the relations in Claim 12; 2) if a share has chords with endpoints on one arc, and the minimum length of such a chord is greater than 1, then such a share is simplified using a suitable six-term relation; 3) if a share has no chords with endpoints on one arc, but there are pairs of intersecting chords in it, then such a share is simplified using a suitable four-term relation. Simplifying transformations are not applicable if all the chords of the share have endpoints on different arcs and do not intersect pairwise, that is, the share has the form $\Xi^n$ for some $n$. Thus, the procedure described expresses every share in $\mathscr S$ as a linear combination of the shares $\Xi^n$ for various $n$. The claim is proved. Thus, we have constructed a mapping from $\mathscr S$ to the module of polynomials of degree $n$ in $\Xi$ over the ring of polynomials in $c_1$ and $c_2$. We denote this mapping by $f$. 4.4. Chord addition operatorsWe define operators $X$ and $S_1$ in $\mathscr S$ such that they add one chord to an arbitrary share as follows (the grey line denotes the share): andAdding a chord on the left is equivalent to adding a chord on the right, since this is true for any share of the form $\Xi^n$, and any share can be expressed as a linear combination of such shares. We introduce a polynomial operator $\widetilde X$ on the space of polynomials such that the diagram is commutative. Since $X$ maps $\Xi^n$ to $\Xi^{n+1}$, the operator $\widetilde X$ acts in the space of polynomials by multiplication by $\xi$. We introduce an operator $\widetilde S_1$ such that the analogous diagram with $X$ replaced by $S_1$ over the top arrow is commutative.The operators $\widetilde X$ and $\widetilde S$ exist since $f$ is an isomorphism. Indeed, $f$ is a surjective homomorphism with respect to multiplication of shares introduced above; it is injective since $c_1, c_2, \xi \in U(\mathfrak{sl}_2)\otimes U(\mathfrak{sl}_2)$ are linearly independent. Remark 1. Zakorko [6] also proved that such a mapping is an isomorphism, but that proof was different. Proof. The first two equations follow directly from Claim 12. In order to prove the last equation, let us prove the corresponding equation for the operator $S_1$. We apply the operator $S_1X^2$ to a given share and write the six-term relation. We obtain All shares on the right-hand side, except the last one, can be expressed in terms of the actions of the operators $X$ and $S_1$. To express the last share in this way we use a four-term relation (relations (4.7)–(4.10)) twice, getting rid of the excessive intersections: and
By substituting (4.8)–(4.10) into (4.7) and then into (4.6) we obtain a relation in $\mathscr S$, which can be rewritten as
$$
\begin{equation*}
S_1X^2=2XS_1X-X^2S_1-(S_1X-c_1X+X^2)+(c_1X-c_1^2+c_1S_1)-(XS_1- c_2S_1).
\end{equation*}
\notag
$$
This is the required equality.
The lemma is proved. Denote by $s_{i,m}$ the matrix coefficients of $\widetilde S_1$ in the monomial basis (the coefficients of $S_1$ in the basis $\mathbf 1, \Xi, \Xi^2, \Xi^3, \dots$ are the same): Theorem 2. The matrix coefficients $s_{i,m}$ of operator $S_1$ are given by a rational generating function: Proof.We rewrite the generating function using relation (4.5):
$$
\begin{equation*}
\begin{aligned} \, &\sum_{m=0}^{\infty} S_1(\xi^m)t^m=S_1(\mathbf 1)+S_1(\xi)t+t^2 \sum_{m=0}^{\infty}S_1(\xi^2\cdot \xi^m)t^m \\ &\qquad=S_1(\mathbf 1)+S_1(\xi)t +t^2\biggl(\sum_{m=0}^{\infty} \bigl((2\xi-1)S_1(\xi\cdot\xi^m)t^m \\ &\qquad\qquad+(c_1+c_2-\xi^2-\xi) S_1(\xi^m)t^m-(\xi-c_1)^2\xi^m t^m \bigr) \biggr). \end{aligned}
\end{equation*}
\notag
$$
Now we rewrite the generating function as follows, by shifting by 1 the summation indices in the first sum on the right-hand side:
$$
\begin{equation*}
\begin{aligned} \, \sum_{m=0}^{\infty} S_1(\xi^m)t^m &=c_1+(c_1-1)\xi t+(2\xi-1)t\biggl (\sum_{m=0}^{\infty} S_1(\xi^m)t^m-c_1 \biggr) \\ &\qquad+t^2(c_1+c_2-\xi^2-\xi)\biggl(\sum_{m=0}^{\infty} S_1(\xi^m)t^m \biggr)-\frac{t^2(\xi-c_1)^2}{1-\xi t}. \end{aligned}
\end{equation*}
\notag
$$
Hence we obtain After algebraic transformations, the expression for the generating function takes the required form. The theorem is proved. Proof. We substitute $\xi t$ for $t$ and $\xi^{-1}$ for $\xi$ into the generating function. Then we obtain Both sides of this equality have a well-defined limit as $\xi\to0$. We substitute in $\xi=0$ and obtain Expanding the right-hand side in a series we obtain the required equation.
§ 5. Generating functions of the values of the $\mathfrak{sl}_2$ weight system at complete bipartite graphsIn this section we derive explicit formulae for generating functions such that their coefficients are values of the $\mathfrak{sl}_2$ weight system at complete bipartite graphs. We use these results to study the properties of the values of the $\mathfrak{sl}_2$ weight system at the projections of complete graphs onto the space of primitive elements. 5.1. Deducing explicit formulae for generating functionsThe join of two simple graphs $G$ and $H$ is the graph obtained by adding to the disjoint union $G\sqcup H$ of these graphs all edges connecting vertices of $G$ with vertices of $H$. For $n=0,1,2,\dots$ we let $(G,n)$ denote the graph equal to the join of the graph $G$ and the discrete graph on $n$ vertices. For instance, if $G$ is also a discrete graph on $m$ vertices, then $(G,n)$ is the complete bipartite graph $K_{m,n}$. Denote by $B_{m,n}$ a chord diagram such that its intersection graph is isomorphic to $K_{m,n}$. Denote by $G_m$ the generating function of the values of the $\mathfrak{sl}_2$ weight system at the chord diagrams $B_{m,n}$, $n = 0,1,2,\dots$:
$$
\begin{equation}
G_m(t):=\sum_{n=0}^{\infty} w_{\mathfrak{sl}_2}(B_{n,m}) t^n.
\end{equation}
\tag{5.1}
$$
Denote by $\mathbb{C}[c][\![t]\!]$ the algebra of formal power series in $t$ such that their coefficients are polynomials in $c$. Consider the linear mapping $\psi \colon \mathscr S \to \mathbb{C}[c][\![t]\!]$ that acts on the generators as follows:
$$
\begin{equation*}
\begin{aligned} \, \psi &\colon \Xi^m \mapsto G_m(t), \\ \psi &\colon c_1 \mapsto c, \\ \psi &\colon c_2 \mapsto c. \end{aligned}
\end{equation*}
\notag
$$
Denote by $\widehat{S}_1$ the operator $\widehat S_1 \colon \mathbb{C}[c][\![t]\!] \to \mathbb{C}[c][\![t]\!]$ such that the diagram is commutative. Then
$$
\begin{equation}
\widehat S_1 G_m(t)=\sum_{i=0}^{\infty} w_{\mathfrak{sl}_2}(B_{n+1,m}) t^n,
\end{equation}
\tag{5.2}
$$
since $S_1$ adds a horizontal chord. The operator $\widehat S_1$ exists because the generating functions $G_m$ are linearly independent for different $m$. In turn, this follows from the fact that the value of the $\mathfrak{sl}_2$ weight system at a chord diagram with $k$ chords is a polynomial of degree $k$. Set $\widehat s_{i,m}:=s_{i,m}\bigr|_{c_1=c_2=c}$. Proof. It follows from (5.2) and the fact that
$$
\begin{equation}
\frac{G_m(t)-c^m}{t}=\sum_{n=0}^{\infty} w_{\mathfrak{sl}_2}(K_{m,n+1})t^n
\end{equation}
\tag{5.5}
$$
that
$$
\begin{equation}
\frac{G_m(t)-c^m}{t}=\sum_{i=0}^m \widehat s_{i,m} G_i, \qquad m=0,1,2,\dots\,.
\end{equation}
\tag{5.6}
$$
In order to prove (5.4) we rewrite (5.6) as follows:
$$
\begin{equation}
G_m=\frac{1}{1-t\cdot \widehat s_{m,m}}\biggl( c^m+t\sum_{i=0}^{m-1}\widehat s_{i,m}G_i\biggr).
\end{equation}
\tag{5.7}
$$
Substituting in the values from Corollary 2 completes the proof. Therefore, the following theorem is true. Theorem 4. An ordinary generating function $G_m(t)$ of the values of the $\mathfrak{sl}_2$ weight system at the complete bipartite graphs $K_{0,m}, K_{1,m}, K_{2,m},\dots$ is a linear combination of the following form of geometric series: where the $p_{m,k}(c)$ are polynomials Corollary 3. An exponential generating function of the values of the $\mathfrak{sl}_2$ weight system at the complete bipartite graphs $K_{0,m}, K_{1,m}, K_{2,m},\dots$ is a linear combination of the form where the $P_{m,k}(c)$ are polynomials. It follows from the proof of Claim 14 and Theorem 4 that a stronger statement is true. Corollary 4. Let $\Gamma$ be a graph such that for any $n=0,1,2,\dots$ the graph $(\Gamma,n)$ is an intersection graph. Then the ordinary generating function $G_{\Gamma}(t) = \sum_{j=0}^{\infty} w_{\mathfrak{sl}_2}((\Gamma,m)) t^{n}$ is a linear combination of geometric series of the form where the $p_{\Gamma,k}(c)$ are polynomials of degree not higher than $|V(\Gamma)|$. Corollary 5. Let $\Gamma$ be a graph such that for any $n=0,1,2,\dots$ the graph $(\Gamma,n)$ is an intersection graph. Then the exponential generating function $\mathcal G_{\Gamma}(t)=\sum_{j=0}^{\infty} w_{\mathfrak{sl}_2}((\Gamma,m))t^{n}/n!$ is a linear combination of the form where $P_{\Gamma,k}(c)$ are polynomials of degree not higher than $|V(\Gamma)|$. 5.2. The values of the $\mathfrak{sl}_2$ weight system at the projections of the complete bipartite graphs onto the space of primitive elementsIn this section we derive from Corollary 3 an important statement about the values of the $\mathfrak{sl}_2$ weight system at the projections of complete bipartite graphs onto the subspace of primitive elements. Theorem 5. The exponential generating function $\mathcal P_m(t)$ of the values of the $\mathfrak{sl}_2$ weight system at the projections of the complete bipartite graphs $K_{0,m}, K_{1,m}, K_{2,m},\dots$ is a linear combination of the form $\sum_{k=0}^m F_{m,k}(c)\exp(kt)$, where the $F_{m,k}(c)$ are polynomials of degree not higher than $m$. Proof. All linear combinations of graphs of the form $(H,k)$, where $H$ is a subgraph of a fixed graph $G$, form a Hopf subalgebra in the Hopf algebra of graphs. Consider the exponential generating functions
$$
\begin{equation*}
{\mathcal G}_G(t):=t^{|V(G)|}\sum_{n=0}^{\infty} (G,n)\frac{t^n}{n!}
\end{equation*}
\notag
$$
for the graphs of the form $(G,n)$ and
$$
\begin{equation*}
{\mathcal P}_G(t):=t^{|V(G)|}\sum_{n=0}^{\infty} \pi((G,n))\frac{t^n}{n!}
\end{equation*}
\notag
$$
for their projections onto the space of primitive elements. In [8] the first-named author of this paper obtained the following formula, which expresses a generating function of the projections onto the space of primitive elements in terms of the generating functions for such graphs themselves:
In the case of complete bipartite graphs the graph $G=K_{0,m}$ and all of its subgraphs are empty graphs. Since $w_{\mathfrak{sl}_2} (K_1)=c$, we have
$$
\begin{equation*}
w_{\mathfrak{sl}_2}(\mathcal G_{|V_i|}) \exp (-w_{\mathfrak{sl}_2}(K_1) t)=\sum_{k=0}^{|V_i|} P_{|V_i|,k}(c)\exp\biggl(-\frac{k(k+1)}{2}t\biggr).
\end{equation*}
\notag
$$
It follows that the value of the weight system at the projection of the graph $K_{j,m}$ is a polynomial of degree not greater than the sum of the powers of the polynomials $P_{|V_i|,k}$. To prove that this sum does not exceed $m$, it suffices to prove that the degree of $P_{|V_i|,k}$ does not exceed $|V_i|$, that is, for every $m$ the degree of the polynomial $P_{m,k}$ in (5.8) does not exceed $m$. This is true because otherwise the degree of the value of the $\mathfrak{sl}_2$ weight system at a graph with $m+n$ vertices would exceed ${m+n}$. The theorem is proved. In a similar way Corollary 5 yields the following. Corollary 6. Let $\Gamma$ be a graph such that for every $n=0,1,2,\dots$ the graph $(\Gamma,n)$ is an intersection graph. Then the exponential generating function $\mathcal P_{\Gamma}(t)$ of the sequence $w_{\mathfrak{sl}_2(\pi((\Gamma,n)))}$, $n = 0,1,2,\dots$, has the form $\sum_{k=0}^{|V(\Gamma)|} F_{\Gamma,k}(c)\exp(kt)$, where $F_{\Gamma,k}(c)$ is a polynomial of degree not higher than $|V(\Gamma)|$. The circumference (length of the longest cycle) of a complete bipartite graph $K_{n,m}$ is equal to double the size of the smaller of its parts. Further, the circumference of the graph $(\Gamma,n)$ for $n>|V(\Gamma)|$ is not less than $|V(\Gamma)|$. Thus, in the special case of complete bipartite graphs and graphs described in Corollary 6 we have proved the following conjecture. Conjecture 1 (Lando). The value of the $\mathfrak{sl}_2$ weight system at the projection of a chord diagram onto the space of primitive elements is a polynomial of degree not exceeding half the circumference of its intersection graph. 
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\Bibitem{ZinKaz23} |
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