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This article is cited in 2 scientific papers (total in 2 papers) Values of the P. E. Zakorko Faculty of Mathematics, National Research University Higher School of Economics, Moscow, Russia
Russian version:
Matematicheskii Sbornik, 2023, Volume 214, Number 7, DOI: https://doi.org/10.4213/sm9873 
Bibliographic databases:
    § 1. IntroductionChord diagrams and the $4$-term relation on them arose in Vassiliev’s studies of knot invariants. Vassiliev invariants of finite order are described using weight systems, which are functions on the chord diagrams which satisfy the $4$-term Vassiliev relation (see [1]). As Kontsevich showed (see [2]), the converse is also true: from any weight system one can recover a knot invariant. Kontsevich and Bar-Nathan (see [2] and [3], respectively) proposed a method for constructing a weight system from an arbitrary Lie algebra endowed with an invariant bilinear form. The simplest weight system obtained using this construction is the $\mathfrak{sl}_2$ weight system corresponding to the Lie algebra $\mathfrak{sl}_2$. It takes values in the space of polynomials in one variable, namely, the Casimir element for $\mathfrak{sl}_2$. From a chord diagram one can construct the intersection graph of its chords. As Chmutov and Lando [4] showed, the value of the $\mathfrak{sl}_2$ weight system depends only on the intersection graph of the chord diagram, and thus we can speak about the values of this weight system at intersection graphs. However, not for every graph there exists a chord diagram with this intersection graph. Lando formulated the question about the existence of an extension of the weight system to the whole space of graphs. Krasilnikov [5] proved the existence and uniqueness of such an extension to the set of graphs with at most eight vertices. One way to construct an extension of an $\mathfrak{sl}_2$ weight system to a graph invariant is to guess this invariant by counting first a sufficiently large number of values on the intersection graphs. Filippova [6] calculated the values of the $\mathfrak{sl}_2$ weight system, first, on a special series of chord diagrams, and then, under the assumption of the existence of an extension of the invariant to the set of all graphs, found values on a series of graphs that are not intersection graphs. However, even calculating the values at intersection graphs is difficult, since the calculations are performed in a noncommutative algebra or (provided that we bear in mind the Chmutov-Varchenko recurrence relations; see [7]) involve an exponentially large number of diagrams in comparison to the number of chords and their intersections. A complete graph is an intersection graph; however, before this paper no result on the value of the $\mathfrak{sl}_2$ weight system at the corresponding chord diagram was proved. Lando expressed a conjecture concerning the form of the generating function of these values. Bigeni [8] managed to prove this conjecture in part, for the generating function of the coefficients at the linear terms of values. In this paper we prove the conjecture in full, using the concept of a share, which is the subset of chords of a diagram with endpoints on two fixed arcs of the circle. Shares play the main role in the proof of the Chmutov-Lando theorem stating that the values of the $\mathfrak{sl}_2$ weight system depend only on the intersection graphs of the chord diagrams (see [4]). We introduce linear operators of chord addition on the space of shares and write out several relations connecting them. Using these relations we create a simple algorithm for calculating the values of the $\mathfrak{sl}_2$ weight system at chord diagrams with complete intersection graphs, which enables us to write out explicitly the generating function of these values, which coincides with the one predicted by Lando. The form of the resulting generating function enables us to find two convenient bases in the quotient space of shares modulo the Chmutov-Varchenko recurrence relations and to derive that this space is isomorphic to the space of polynomials in two variables. AcknowledgmentsThe author expresses her gratitude to G. S. Minaev, P. A. Zinova (Filippova), M. E. Kazaryan and S. K. Lando for discussions of the subject of this paper and for their useful advice. § 2. Chord diagrams and the weight system $w_{\mathfrak{sl}_2}$Definition 1. A chord diagram of order $n$ is an oriented circle considered up to orientation-preserving diffeomorphisms, with $2n$ pairwise distinct points chosen on it, which are grouped into $n$ pairs. For clarity we connect each pair of selected points on the circle by a chord, a line segment or an arc lying inside the circle. From a chord diagram one can construct a simple graph without multiple edges and loops, which is called its intersection graph. Definition 2. The intersection graph of a chord diagram is the graph whose vertices correspond to chords of the diagram and whose edges connect those and only those vertices whose chords intersect. Definition 3. An arc diagram of order $n$ is an oriented straight line, considered up to orientation-preserving diffeomorphisms, with $2n$ pairwise distinct points chosen on it and grouped into $n$ pairs. For clarity, each distinguished pair of points on the line is connected by an arc lying in a fixed half-plane. A chord diagram can be represented as an arc diagram by ‘cutting’ the circle of the diagram at a point different from the $2n$ distinguished points (Figure 1). The formal linear combinations of chord diagrams with coefficients in $\mathbb C$ form the vector space of chord diagrams. Similar spaces are formed by graphs and arc diagrams. Definition 4. By a weight system one means a linear function $w$ on the vector space of chord diagrams that satisfies the 4-term relation In this and the following equalities diagrams can also contain chords with endpoints on dotted arcs of the circle, and the position of these chords is the same for all terms of a single equality. Below, in the expressions for values of the weight system we omit the function itself, identifying the diagram with its value. Let us show how, given an arbitrary Lie algebra $H$ with nondegenerate invariant bilinear form $(\,\cdot\,{,}\,\cdot\,)$, we can construct a weight system, where, under the invariance of the bilinear form, we mean the equality $([a, [b, c]])=([a, b], c)$ for all $a,b,c \in H$. Let us choose an arbitrary basis $x_1, x_2, \dots, x_n$ which is orthonormal with respect to $(\,\cdot\,{,}\,\cdot\,)$. We construct as follows a function $\omega$ on the space of arc diagrams, with values in the universal enveloping algebra of $H$. To every arc of the diagram we assign an integer from $1$ to $n$ (in accordance with the number of elements of the basis); two different arcs can obtain the same number. To every endpoint of an arc we assign a basis element with index equal to the number of this arc. We multiply the elements of the basis in the order of their position on the oriented line of the diagram. Let us add the resulting products for all possible maps from the set of arcs to $\{1, \dots, n\}$. The sum of these products lies in the universal enveloping algebra of $H$. For example, we present the expression for the value of $\omega$ at the arc diagram in Figure 1:
$$
\begin{equation*}
\sum_{i_1=1}^{n} \sum_{i_2=1}^{n} \sum_{i_3=1}^{n} \sum_{i_4=1}^{n} \sum_{i_5=1 }^{n} x_{i_1} x_{i_2} x_{i_1} x_{i_3} x_{i_4} x_{i_2} x_{i_5} x_{i_3} x_{i_4} x_{i_5}.
\end{equation*}
\notag
$$
To extend $\omega$ to a weight system on the space of chord diagrams we must check that the values at different arc representations of the same chord diagram coincide and that the function $\omega$ satisfies the 4-term relation. This is guaranteed by the following theorem. Theorem 1 (see [2]). The function $\omega$ has the following properties: 1) $\omega$ does not depend on the choice of the orthonormal basis $e_1, \dots, e_n$; 2) $\omega$ takes the same values at different representations of the same chord diagram; 3) $\omega$ satisfies the $4$-term relation; 4) the image of $\omega$ lies in the centre of the universal enveloping algebra $H$. The simplest case of the above construction corresponds to the Lie algebra $\mathfrak{sl}_2$. In this Lie algebra one can choose an orthonormal basis $x_1$, $x_2$, $x_3$ which satisfies the relation $[x_i, x_j]=\varepsilon_{ijk} x_k$, where $\varepsilon_{ijk}$ is the Levi-Civita symbol. The element $c :=x_1^2+x_2^2+x_3^2$ generates the centre of the universal enveloping algebra of $\mathfrak{sl}_2$. The constructed weight system is denoted by $w_{\mathfrak{sl}_2}$ and called the $\mathfrak{sl}_2$ weight system. It takes values in the space of polynomials in the variable $c$. However, when calculating the values at particular diagrams, it is more convenient to use the combinatorial relations of the $\mathfrak{sl}_2$ weight system, which we present below. Theorem 2. For the weight system $w_{\mathfrak{sl}_2}$, the following is true. $\bullet$ The initial values are: $\bullet$ (Multiplicativity.) Let the set of chords of the diagram be divided into two complementary subsets such that no two chords from different subsets intersect. Then the value of the weight system $w_{\mathfrak{sl}_2}$ at the whole chord diagram is equal to the product of its values at the two diagrams containing only chords from the first and the second subset, respectively: $\bullet$(Chmutov-Varchenko relations; see [7].) Let $D$ be a chord diagram with connected intersection graph. Then 1) either the diagram contains a leaf, which is a chord intersecting a unique chord, and therefore where $D'$ denotes the diagram of $D$ without the leaf;2) or there is one of the two triples of chords shown on the left-hand sides of the equalities and for which the 6-term relations are satisfied.Using only the properties from Theorem 2 we can calculate the value of the weight system $w_{\mathfrak{sl}_2}$ at any chord diagram. On the simplest series of chord diagrams consisting of $n$ disjoint chords the weight system $w_{\mathfrak{sl}_2}$ takes the values $c^{n}$ by the multiplicativity property. In the general case, calculating the value at a diagram with many chords and many intersections of chords is not easy because the number of diagrams increases rapidly when one uses 4-term and 6-term relations. Theorem 3 (see [4], Theorem 4). The value of the $\mathfrak{sl}_2$ weight system at a chord diagram depends only on its intersection graph. We study the values of the weight system $w_{\mathfrak{sl}_2}$ on a series of chord diagrams with complete intersection graphs. Notation 1. Let $K_n(c)$ denote the value of the weight system $w_{\mathfrak{sl}_2}$ at a chord diagram with complete intersection graph with $n$ vertices (Table 1 shows the polynomials $K_n( c)$ for small $n$). Table 1.Values of the weight system at diagrams with complete intersection graphs
The main result of this paper is the proof of Lando’s conjecture concerning the sequence of polynomials $K_n(c)$. Theorem 4 (Lando’s conjecture, 2016; see [8]). The generating function of the sequence of polynomials $K_n(c)$ is the continued fraction whose coefficients have the form The expression on the right-hand side of (2.4) is called the Jacobi infinite continued fraction. This relation must be understood as the equality of formal power series on the right- and left-hand sides. To calculate the first $n$ coefficients of the power series of an infinite continued fraction, it is sufficient to find the first $n$ coefficients of the power series of its approximation by a finite fraction, which can be obtained by leaving only the first $n$ coefficients at $t$ or $t^2$ on all levels of the fraction. The final continued fraction is a rational function of $t$, and for it the required coefficient of the power series is easy to calculate. It was proved in [8] that the coefficients of the monomial $c$ in the polynomials $K_n(c)$ coincide with the corresponding coefficients in the continued fraction. The absolute values of these coefficients form a sequence of normalized median Genocchi numbers A000366. § 3. Decomposing a bunch into trivial sharesThis part of our work is devoted to the construction of a simple algorithm that calculates the values of the $\mathfrak{sl}_2$ weight system at chord diagrams with complete intersection graphs. The resulting algorithm uses the values of only a small number of diagrams with fewer chords or their intersections. Definition 5 (see [4]). We select two disjoint arcs on the circle of the chord diagram. By a share of the chord diagram we mean a family of its chords such that the endpoints of chords in this family and only they are positioned on two selected arcs of the circle. The chords that do not belong to the share are said to be additional. These chords form the complement share. The 4-term and 6-term relations can be applied to shares, as well as to chord diagrams. The values at the three simplest shares, consisting of one chord, two disjoint chords and two intersecting chords, respectively, such that the endpoints of each chord are placed on different arcs of the share, are related by a 3-term relation. Lemma 1 (see [6], Corollary 3). The 3-term relation holds for values of the weight system $w_{\mathfrak{sl}_2}$. This relation is a simple consequence of the 4-term one: it is sufficient to consider relation (2.1) in each of whose terms the lower dotted arc contains no endpoints of chords. The three-term relationship is the motivating factor for the following definition. Definition 6. A trivial share with $n$ chords is a share consisting of $n$ pairwise disjoint chords such that the endpoints of each of these chords lie on different arcs of the share. A bunch of $n$ chords is a share with $n$ pairwise intersecting chords such that the endpoints of each chord also lie on different arcs. (Examples of chord diagrams containing these types of a share are shown in Figure 2.) Lemma 1 claims that the value of the weight system $w_{\mathfrak{sl}_2}$ at the diagram with a bunch of two chords can be represented as the difference of its values at two diagrams with trivial shares consisting of two chords and one chord, where the complement shares are identical in all three diagrams. This result extends to bunches with an arbitrary number of chords (see Proposition 1). Definition 7. By a decomposition of a share into trivial shares we mean a linear combination of trivial shares with coefficients that are polynomials in $c$ such that the values of the $ {\mathfrak{sl}_2}$ weight system at the share in question and at this linear combination coincide for any coinciding complement shares. A share for which a decomposition into trivial shares exists is said to be decomposable. Proposition 1. A bunch of $n$ chords is decomposable into trivial shares, and there is a decomposition in which the trivial share with the largest number of chords consists of precisely $n$ chords and enters the expansion with coefficient $1$. As we will see below, a decomposition into trivial shares exists for a share of an arbitrary form, and it is even unique. This decomposition can be obtained by applying combinatorial relations to the chords of the share (see Theorem 6). To calculate the value of the weight system $w_{\mathfrak{sl}_2}$ at a chord diagram with complete intersection graph on $n$ vertices it is sufficient to know the concrete form of at least one decomposition of a bunch of $n$ chords. For shorter calculations we introduce some convenient notation. Notation 2. Let $I$ be a share decomposable into trivial shares $I_k$: $I=\sum_{k=0}^{n} p_k(c)I_k$ for some polynomials $p_k(c)$. To the share $I$ with this decomposition we assign the polynomial $I(x, c)= \sum_{k=0}^{n} p_k(c) x^n$, which we call the polynomial of the decomposition of the share into trivial shares. For example, $x^n$ is the polynomial of the decomposition of the trivial share $I_n$ with $n$ chords. We identify the trivial share with its decomposition polynomial: $I_n=x^n$. Notation 3. Consider the shares $I$ and $I'$ which differ by one chord belonging to $I$, disjoint from the other chords of the share, and with endpoints on different distinguished arcs. Then we write $I=xI'$: Remark 1. If $I'$ is decomposable into trivial shares with decomposition polynomial $I'(x,c)$, then $I$ is also decomposable into a linear combination of trivial shares, and $I(x, c) :=x I'( x,c)$ is the decomposition polynomial of $I$. Definition 8. By the closure of a share we mean the chord diagram consisting of this share and an empty complement share. Lemma 2. Let $I$ be a share decomposable into trivial shares which has a decomposition polynomial $I(x, c)$. Then the value of the weight system $w_{\mathfrak{sl}_2}$ at the closure $D$ of $I$ can be calculated by substituting in $x=c$: Proof. For the trivial shares $I_n$ with $n$ chords we have $I_n(x, c)=x^n$, and the value at the diagram with $n$ disjoint chords is $c^n$; for an arbitrary decomposable chord $I$, the statement holds due to the linearity of $w_{\mathfrak{sl}_2}$. Definition 9. The linear operators $\widetilde{T}$ and $\widetilde{S}$ on the space of shares which act on an arbitrary share by adding one chord are defined by The endpoints of the red chords added coincide with the endpoints of the selected arcs. Remark 2. The definition of the operator $\widetilde{S}$ prescribes the addition of a new chord with endpoints on the lower selected arc to a share. If we are only interested in the value of the $ {\mathfrak{sl}_2}$ weight system at the share rather than in the concrete positions of chords in it, then the endpoints of the added chord can be placed on the upper arc: the intersection graph does not change in this case, and thus the value does not change by Theorem 3. In a similar way, one can change the position of the chord added by the operator $\widetilde{T}$ (see (3.2)): Remark 3. The operators $\widetilde{T}$ and $\widetilde{S}$ can be applied to both sides of an equation for the values of the $ {\mathfrak{sl}_2}$ weight system at shares: the endpoints of the chord added by each of the operators lie at the endpoints of the selected arcs, and therefore the new chord can be viewed as belonging to the complement share. We need these operators to construct the decomposition of a bunch of $n$ chords into trivial shares. Proposition 2. The following recurrence relations hold for any integer $n \geqslant 1$: Before we prove Proposition 2, note the following relationship between the operators $\widetilde{T}$ and $\widetilde{S}$. Proof. This expression follows by applying the 4-term relation to the left-hand side of the equality $k$ times and pulling out successively a blue chord from under the black one:
The lemma is proved. Lemma 3 can be regarded as a 4-term relation for the trivial share: if in (3.5) we replace all blue chords by one, then we obtain the usual 4-term relation. Due to the linearity of $w_{\mathfrak{sl}_2}$, the 4-term relation can be generalized to any decomposable share (that is, as we will see from Theorem 6, to any share). This generalization is presented in [9], Ch. 4, exercise (14). Proof of Proposition 2. We apply the first 6-term relation (2.2) to $\widetilde{S}(x^{n+1})$: Let us express the difference between the last two shares $\widetilde{S}^2(x^{n-1})- \widetilde{T}^2(x^{n-1})$ using the linear operators $\widetilde{T}$ and $\widetilde{S}$ taken to the first power. To do this, we apply $\widetilde{T}$ and $\widetilde{S}$ to both sides of (3.4) for $k=n-1$:
To get rid of the sequential application of the operators $\widetilde{T}\widetilde{S}$ and $\widetilde{S}\widetilde{T}$ we use the 4-term relation Then we obtain
$$
\begin{equation}
\widetilde{S}\widetilde{T}(x^{n-1})-\widetilde{T}\widetilde{S}(x^{n-1})= \widetilde{S}(x^n) -x\widetilde{S}(x^{n-1}).
\end{equation}
\tag{3.9}
$$
We add the expressions (3.7) and (3.8) by taking (3.9) into account:
$$
\begin{equation*}
\widetilde{T}^2(x^{n-1})-\widetilde{S}^2(x^{n-1})= x\widetilde{S}(x^{n-1})+ \widetilde{T}(x^{n})-c\widetilde{S}(x^{n-1})- c\widetilde{T}(x^{n-1}).
\end{equation*}
\notag
$$
We substitute the linear combination of shares on the right-hand side into (3.6) and, recalling that $\widetilde{T}$ and $\widetilde{S}$ are connected by Lemma 3, we prove both recurrence formulae.
The proposition is proved. Corollary 1. The share $\widetilde{T}(x^n)$ has a decomposition into trivial shares such that the share with the largest number of chords consists of exactly $n+1$ chords and is included in this decomposition with coefficient $1$. Proof. For $n=0$ we have $\widetilde{T}(1)=x$, for $n=1$ the statement follows from the $3$-term relation (3.1). Induction on $n$ and Proposition 2 ensure the decomposability of $\widetilde{T}(x^{n+1})$, and from the form of the recurrence formula (3.3) in this proposition we obtain that the coefficient at the leading monomial of the decomposition polynomial of the share $\widetilde{T}(x^{n+1})$ is $1$. By the induction assumption the leading monomials of the decomposition polynomials for $\widetilde{T}(x^{n})$ and $\widetilde{T}(x^{n-1})$ are equal to $x^{n+1}$ and $ x^{n}$, respectively. By substituting these monomials into the recurrence formula we obtain the required leading monomial $x^{n+2}$: This monomial corresponds to the trivial share with $n+2$ chords.
The corollary is proved. Proof of Proposition 1 (bunch decomposition algorithm). We carry out the proof using induction on the number of chords in the bunch. Suppose that a bunch with $n$ chords is decomposed into trivial parts. In a bunch with $n+1$ chords we select a subbunch with $n$ chords. By Remark 3 we can decompose this subbunch into trivial parts without violating the equality. Decomposing it, we obtain a decomposition of the bunch of $n+1$ chords into shares of the form $\widetilde{T}(x^{k})$, the maximum index $k$ in this decomposition is equal to $n$, and $\widetilde{ T}(x^{n})$ participates in this decomposition with coefficient $1$. However, every share $\widetilde{T}(x^{k})$ is, in its turn, decomposable into trivial shares, hence the bunch is decomposable. The trivial share with the greatest number of chords in the decomposition of this bunch comes from the decomposition of $\widetilde{T}(x^{n})$, consists of $n+1$ chords, and its coefficient in the decomposition is equal to $1$.
The proposition is proved. In combination, Lemma 2 and Proposition 1 present an algorithm for the computation of $K_n(c)$. § 4. Proof of Theorem 4Let us introduce a linear operator $T$, on the space of polynomials in $x$ with coefficients that are polynomials in $c$. On the basis vectors $x^{n}$ it is defined using the recurrence relation from Proposition 2:
$$
\begin{equation}
\begin{gathered} \, T(x^{n+1})=(2x-1) T(x^{n})+(2c-x-x^{2}) T(x^{n-1})+x^{n- 1}(c-x)^{2}, \\ T(1)=x, \qquad T(x)=x^2-x. \end{gathered}
\end{equation}
\tag{4.1}
$$
In a similar way we introduce an operator $S$ on this space of polynomials:
$$
\begin{equation*}
\begin{gathered} \, S(x^{n+1})=(2x-1) S(x^{n})+(2c-x-x^{2}) S(x^{n-1})-x^{n- 1}(c-x)^{2}, \\ S(1)=c, \qquad S(x)=(c-1)x. \end{gathered}
\end{equation*}
\notag
$$
It follows from Lemma 2 that the value of the $ {\mathfrak{sl}_2}$ weight system at a diagram with a complete intersection graph with $n$ vertices can be defined using the operator $T$ in the following form: Let us introduce a new variable $y:=x-c$. In terms of this new variable, Changing the variable makes it easier to calculate $K_n(c)$, since it is easier to substitute zero into the polynomial: to do this, we need only know the free term $T^{n}(1)$ of the polynomial in $y$. Therefore, we consider the operator $T$ itself in the basis $y^n$. When the basis changes, the recurrence formula (4.1) for $T(x^n)$ turns to a new recurrence formula for $T(y^n)$, which is generalized in the following proposition. Proposition 3. The following relations hold for every polynomial $p(y)$: Proof. We prove the recurrence formula for the operator $T$; for $S$ the proof is similar. By the linearity of $T$ it is sufficient to prove the recurrence relation only for the monomials $p(y)=y^{n-1}$, $n \geqslant 1$.
The first initial condition is given by the definition of the operator, and the second follows from its linearity: We prove the recurrence relation. To this end we write out $T(y^2 p(y))$ using the linearity of $T$ and the equality $y^2=x^2-2cx+ c^2$:
$$
\begin{equation}
T(y^2p(y))=T(y^{n+1})=T(x^{2}y^{n-1})-2cT(xy^{n-1})+ c^ 2T(y^{n-1}).
\end{equation}
\tag{4.3}
$$
Each term on the right-hand side can be expressed in terms of $T(y^{n})$ and $T(y^{n-1})$: To simplify $T(x^{2}y^{n-1})$ we use the equality $y^{n-1}=(x-c)^{n-1}=\sum_{k=0}^{n -1} \eta_{kn} x^{k}$ (where $\eta_{kn}=\binom{n-1}{k}(-c)^{n-k-1}$, although the particular values are not important in this case):
$$
\begin{equation}
\begin{aligned} \, \notag &T(x^{2}y^{n-1}) =\sum_{k=0}^{n-1} \eta_{kn} T(x^{k+2}) \\ \notag &\qquad= (2x-1)\sum_{k=0}^{n-1} \eta_{kn} T(x^{k+1})+(2c-x-x^2)\sum_{k= 0}^{n-1} \eta_{kn} T(x^{k})+(x-c)^2\sum_{k=0}^{n-1} \eta_{kn} x^{k} \\ &\qquad= (2x-1)T(xy^{n-1})+(2c-x-x^2) T(y^{n-1})+y^{n+1}. \end{aligned}
\end{equation}
\tag{4.5}
$$
All that remains is to substitute (4.4) and (4.5) into (4.3).
The proposition is proved. To connect the operator $T$ with the generating function presented in Theorem 4 in the form of a continued fraction, we need Stieltjes’s theorem. Theorem 5 (see [10]). The coefficients of the continued fraction It turns out that in some basis the operator $T$ can be written in the form of a tridiagonal matrix of the same form as a matrix on the right-hand side of the equality in Theorem 5, with elements specified in Theorem 4. Proposition 4. In the basis Proof. Proposition 4 is equivalent to the following relation for the elements of the basis $y_i$ and their images under the action $T$:
Every $y_i$, as a polynomial, is divisible by the $y_j$ with indices $j\leqslant i$. Therefore, the above equality can be rewritten as follows:
$$
\begin{equation}
T(y_{k})=\biggl(y^2+(c-k) y-c\frac{k(k+1)}{2}\biggr)y_{k-1}.
\end{equation}
\tag{4.7}
$$
We prove this identity below, using induction. It can readily be seen that for $k=1$ we have $T(y_{1})=T(y)= (y-1)(y+c)=y^2+(c-1) y-c$ (see Proposition 3). Assume that (4.7) holds for all $k \leqslant n$; we claim that it also holds for $k=n+1$. To prove this we express $T(y_{n+1})$ in terms of $y_{n}$ and $y_{n-1}$ in a way similar to the proof of Proposition 3:
$$
\begin{equation*}
T(y_{n+1})=T(y^2 y_{n-1})+n^2T(y y_{n-1})+ \frac{n^2(n^2-1)} {4}T(y_{n-1}).
\end{equation*}
\notag
$$
Next we express each summand on the right-hand side in terms of $T(y_{n})$ and $T(y_{n-1})$.
The first term on the right-hand side is simplified using Proposition 3:
$$
\begin{equation*}
T(y^{2} y_{n-1})= (2y-1)T(y y_{n-1})+(-y^2-y)T(y_{n-1})+y^2 y_{n-1}.
\end{equation*}
\notag
$$
For the second term we have Substituting the last two equalities into the expression for $T(y_{n+1})$ we obtain
$$
\begin{equation*}
\begin{aligned} \, T(y_{n+1}) &=(2y+n^2-1)T(y_{n}) \\ &\qquad-\biggl(y^2+(n^2-n+1)y+\frac{(n^2-2n)(n^2-1)}{4}\biggr)T(y_{n -1})+y^2 y_{n-1}. \end{aligned}
\end{equation*}
\notag
$$
We apply the induction assumption to the first and second terms, namely, we use the equality (4.7) for $k=n$ and $k=n-1$. Note that the coefficient of $T(y_{n-1})$ can be factorized:
$$
\begin{equation*}
\begin{aligned} \, &y^2+(n^2-n+1)y+\frac{(n-2)(n-1)n(n+1)}{4} \\ &\qquad= \biggl(y+\frac{n(n+1)}{2}\biggr)\biggl(y+\frac{(n-2)(n-1)}{2}\biggr). \end{aligned}
\end{equation*}
\notag
$$
The second term $(y+(n-2)(n-1)/2)$ in this factorization is equal to the ratio $y_{n-1}/y_{n-2}$. Then $T(y_{n+1})$ is the product of $y_{n-1}$ and a third-degree polynomial in $y$:
$$
\begin{equation}
\begin{aligned} \, \notag \frac{T(y_{n+1})}{y_{n-1}} &=y^3+\biggl(c+\frac{n(n-3)}{2}-1\biggr)y^2 \\ &\qquad- \biggl(2nc+c+\frac{n(n^2-1)}{2}\biggr)y-\frac{c(n^2-1)(n^2+2n)}{ 4}. \end{aligned}
\end{equation}
\tag{4.8}
$$
The coefficient of $y_{n-1}$ in (4.8) can also be factorized:
$$
\begin{equation*}
T(y_{n+1})=\biggl(y^2+(c-n-1) y- \frac{c(n+1)(n+2)}{2}\biggr)\biggl(y+\ \frac{n(n-1)}{2}\biggr)y_{n-1}.
\end{equation*}
\notag
$$
All that remains is to combine the second and third factors into $y_{n}$ to prove identity (4.7).
Proposition 4 is proved. Proposition 5. Let $\sum_{n=0}^{\infty} d_n(c) t^n$ be the formal power series equal to the continued fraction in Theorem 4. Then $K_{n}(c)=d_{n}(c)$ for all $n$. Proof. In accordance with Proposition 4, we write the operator $T$ in the basis $y_{n}$ in the form of a tridiagonal matrix $A$. It follows from Theorem 5 that because multiplication by $A^{n}$ transforms the $j$th column of the matrix $(k_{ij})_{i,j=0}^{\infty}$ into the $(j+n)$th one. Since $y_0=1$ and $y_k|_{y=0}=0$ for all integers $k \geqslant 1$, it follows that By (4.2) we have $T^{n}(1)|_{y=0}=K_n(c)$.
The proposition is proved. Thus, we have justified the form of the generating function for the values of the $\mathfrak{sl}_2$ weight system at chord diagrams with complete intersection graphs, and Theorem 4 is proven. § 5. Existence and uniqueness of the decomposition of an arbitrary share into trivial sharesThe proof of the main theorem is based on the decomposition of a bunch into trivial shares. We have presented an explicit algorithm for decomposing this type of share. It turns out that in the general case one can find such a decomposition for an arbitrary share, and it is unique and can be constructed using combinatorial relations. The existence of this decomposition follows directly from combinatorial relations, and to prove uniqueness we use the above form of the generating function for the sequence $K_n(c)$. Let us proceed to the decomposition of an arbitrary share into trivial parts. Proposition 6 (existence of decomposition). Every share can be decomposed into trivial parts, and this decomposition can be obtained by applying combinatorial relations to the chords of the share. Definition 10. By an arch we mean a chord of a share whose endpoints are located on a single selected arc of the share, and by a bridge we mean a chord of a share with endpoints on different arcs. Given an arch, we also define its length to be the number of endpoints of chords positioned between the endpoints of the arch on the same distinguished arc of the share as the endpoints of the arch. Proof of Proposition 6. We consider only shares with connected intersection graphs. We carry out induction on the number of chords in a share and their intersections.
Every chord of a share is either a bridge or an arch. Let there be at least one arch in the share. We choose an arch of minimum length. If its length is 1, then this chord is a leaf, and the share is simplified by removing this leaf and multiplying the fraction by $c-1$. Otherwise the length of the shortest arc is at least 2 (it cannot be equal to zero due to the connectivity of the intersection graph of the share), and an arch of minimum length with two adjacent chords of the share is located in one of the two configurations shown on the left-hand sides of the 6-term relations (see (5.1) and (5.2)). Applying one of these relations to the triple of chords thus found, we obtain five shares, and in two of them there are one chord fewer than in the original share, and in three of them there are fewer intersections: By the induction assumption each of the resulting shares is decomposable, and thus the original share is decomposable too.A share can also consist of bridges alone. If such a share is not trivial, then there are two adjacent endpoints of intersecting bridges. Applying the 4-term relation (5.3) to these two chords we obtain three decomposable shares instead of one. There are fewer chord intersections in the first share, and therefore it is decomposable by the induction assumption, while the other two shares contain an arch, that is, they are decomposable according to the above reasoning: This completes the proof of the proposition. Proposition 7 (uniqueness of the decomposition). Every share has at most one decomposition into trivial shares. Proof. Suppose that there exists a linear combination of trivial shares such that the value of the ${\mathfrak{sl}_2}$ weight system at it is equal to zero for any complement share, the same for all diagrams. Let us show that in this case there is a linear combination of bunches with the same property. To do this we decompose every trivial share into bunches. This decomposition is inverse to the decomposition of bunches into trivial shares, and it exists by Proposition 1. The coefficient of the bunch with the largest number of chords in the decomposition into bunches is equal to the coefficient of the trivial share with the maximum number of chords in the original decomposition into trivial shares, that is, it is not equal to zero.
Thus, there is a linear combination of bunches such that the value of the weight system $w_{\mathfrak{sl}_2}$ at it is equal to zero for any common complement share:
$$
\begin{equation*}
w_{\mathfrak{sl}_2}\bigl(p_n(c) B_n(D)+p_{n-1}(c) B_{n-1}(D)+\dots+ p_0(c) B_0(D)\bigr)=0,
\end{equation*}
\notag
$$
where $B_k(D)$ is the diagram consisting of the bunch $B_{k}$ with $k$ chords and an complement share $D$, $p_k(c)$ is a polynomial in $c$, and the equation holds for any share $D$. We substitute the complement share $D=B_m$ into the linear combination of bunches. Every resulting chord diagram $B_k(B_m)$ has a complete intersection graph with $n+m$ vertices. Then for any positive integer $m$ we have the equation
$$
\begin{equation*}
p_n(c) K_{n+m}(c)+p_{n-1}(c) K_{n+m-1}(c)+\dots+p_0(c) K_{m}(c)=0,
\end{equation*}
\notag
$$
so that the sequence $K_j(c)$ is given by a linear recurrence formula. The generating function of a sequence given by a linear recurrence formula is rational. In turn, the rational function expands in a finite continued fraction of the form (4.6) whose coefficients $\alpha_i(c)$ and $\beta_i(c)$ are rational functions of $c$ and vanish for sufficiently large values of $i$. However, the generating function of the sequence $K_j(c)$ has the form of an infinite continued fraction by Theorem 4, which leads to a contradiction.
This completes the proof of the proposition. Thus, we have proved the following general theorem about the structure of the space of shares. Theorem 6. Let $J$ be the ${\mathbb C}[c]$-module of shares whose elements are linear combinations of shares. Let two elements of $J$ be said to be equivalent if one of them can be obtained from the other by applying combinatorial relations from Theorem 2 to its chords. Then the trivial shares and bunches form two free bases in the quotient space of shares $J/{\sim}$ with respect to this equivalence relation. Trivial shares form a basis according to Theorems 6 and 7; Proposition 1 enables one to go over from a basis of trivial shares to a basis of bunches. 
Citation in format AMSBIB
\Bibitem{Zak23} |
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