Values of the \mathfrak{sl}_2 weight system on complete bipartite graphs
aa r X i v : . [ m a t h . G T ] F e b Values of the sl weight system on complete bipartitegraphs. P. Filippova ∗ February 9, 2021
Abstract
A weight system is a function on chord diagrams that satisfies the so-called four-termrelations. Vassiliev’s theory of finite-order knot invariants describes these invariants interms of weight systems. In particular, there is a weight system corresponding to thecolored Jones polynomial. This weight system can be easily defined in terms of the Liealgebra sl , but this definition is too cumbersome from the computational point of view,so that the values of this weight system are known only for some limited classes of chorddiagrams.In the present paper we give a formula for the values of the sl weight system for aclass of chord diagrams whose intersection graphs are complete bipartite graphs with nomore than three vertices in one of the parts.Our main computational tool is the Chmutov–Varchenko reccurence relation. Fur-thermore, complete bipartite graphs with no more than three vertices in one of the partsgenerate Hopf subalgebras of the Hopf algebra of graphs, and we deduce formulas for theprojection onto the subspace of primitive elements along the subspace of decomposableelements in these subalgebras. We compute the values of the sl weight system for theprojections of chord diagrams with such intersection graphs. Our results confirm certainconjectures due to S.K.Lando on the values of the weight system sl at the projections ofchord diagrams on the space of primitive elements. Key words: chord diagram, intersection graph, weight system, complete bipartite graph, Hopfalgebra.
Contents ∗ National Research University Higher School of Economics, e-mail: [email protected]. The publicationwas prepared within the framework of the Academic Fund Program at HSE University in 2020–2021 (grant № sl Weight System 12 sl weight system . . . . . . . . . . . . . . . . . . . . . . . . . 133.2 The values of the sl weight system at the graphs K ,n , K ,n , and K ,n . . . . . . 153.3 The values of the sl weight system at the projections π ( K l,n ), l = 1 , ,
3. . . . . 18
Finite-order knot invariants, which were introduced in [16] by Vassiliev about 1990, can beexpressed in terms of weight systems, that is, functions on chord diagrams satisfying the so-called Vassiliev four-term relations. In the paper [8], Kontsevitch proved that over a field ofcharacteristic zero every weight system corresponds to some finite-order invariant.There are multiple approaches to constructing weight systems. In particular, Bar-Natan [1]and Kontsevitch [8] suggested a construction of a weight system from a finite-dimensional Liealgebra endowed with an invariant nondegenerate bilinear form. The simplest case of thisconstruction is the sl weight system, which is constructed from the Lie algebra sl . Its valueslie in the center of the universal enveloping algebra of sl . The center is isomorphic to thering of polynomials in one variable (the Casimir element). In Vassiliev’s approach this weightsystem corresponds to a well-known knot ivariant, the colored Jones polynomial. The value ofthe sl weight system at a chord diagram of order n is a monic polynomial of degree n in theCasimir element.The sl weight system was studied in many papers. Despite the simplicity of the defini-tion of this weight system, it is difficult to compute its value at a chord diagram by usingthis definition, because doing this involves computations in a noncommutative algebra. TheChmutov–Varchenko recurrence relations [5] significantly simplify these computations; however,using them in computations is laborious as well, and the explicit values of the sl weight systemare known only for chord diagrams of low order and for a small number of simple series of chorddiagrams. In particular, the values of the sl weight system at chord diagrams with completeintersection graph are unknown. The conjecture of Lando about the form of the correspondingpolynomials has been proved only for the linear terms of these polynomials [2].The Chmutov–Lando theorem [4] states that the value of the sl weight system at a chorddiagram depends only on the intersection graph of this chord diagram; i.e., if two chord dia-grams have isomorphic intersecton graphs, then the values of the weight system at these chorddiagrams coincide. This raises the following natural question, which was asked by Lando in [4]:Is it possible to extend this weight system to a polynomial graph invariant satisfying the four-term relations for graphs? E. Krasil’nikov showed that such an extension exists and is uniquefor all graphs with n ≤ define a polynomial invariant for arbitrary graphs and then show that this invariant satisfiesthe four-term relations for graphs and coincides with the sl weight system on the intersectiongraphs. To make this possible, it is necessary to have enough examples of the explicit values ofthe sl weight system for various graph families. In our present paper we compute such valuesat the complete bipartite graphs such that the number of vertices in one of the parts is at most2hree.The quotient space of the vector space of chord diagrams by the four-term relations carriesthe structure of a connected graded commutative cocommutative Hopf algebra [8]. The sameis true for the vector space of graphs. Further, the complete bipartite graphs generate a Hopfsubalgebra in the Hopf algebra of graphs, and for any l = 0 , , , . . . , this Hopf subalgebracontains the Hopf subalgebra generated by complete bipartite graphs such that one of the partscontains no more than l vertices.According to the Milnor–Moore theorem [14], each of these Hopf algebras is generated byits primitive elements and is a polynomial Hopf algebra in its primitive elements. As a con-sequence, in each of these Hopf algebras there is a well-defined projection onto the subspaceof primitive elements along the space of decomposable elements. There is a universal formulawhich expresses this projection as a logarithm of the identity homomorphism ([10, 15]). How-ever, this formula is cumbersome, and we construct its compact forms for the Hopf algebras ofcomplete bipartite graphs with sizes of one of the parts at most 1 , , and 3. Then we use theseforms to explicitly compute of the values of the sl weight system at the projections on thesubspace of primitive elements of the complete bipartite graphs with size of one of the parts atmost l = 1 , ,
3. It turns out that these values are polynomials of degree at most l . Thus, ourcomputations confirm another conjecture of Lando, which states that if C is a chord diagramsuch that the circumference (that is, the length of the longest cycle) of its intersection graph isat most 2 l, l ≥
1, then the value of the sl weight system at its projection on the subspace ofprimitive elements is a polynomial of degree at most l .Note that the value of a multiplicative graph invariant can be uniquely reconstructed fromthe values of this invariant at the projections of graphs on the subspace of primitive elements.What is more, this invariant is often significantly simplified under the projection. For instance,it is known that the value of the sl weight system at the projection of a chord diagram oforder n on the subspace of primitive elements is a polynomial of degree less than or equal to[ n ], i.e., the degree of the polynomial is at least halved. In [9] the coefficient of the term ofdegree n was extended to arbitrary graphs satisfying the four-term relations for graphs. Thisgives the hope for the existence of such an extension for the other coefficients of the polynomial.(In [12], an extension of this coefficient for the more general case of binary delta-matroids wasgiven.)This paper is organized as follows. In Section 2 we give definitions of the Hopf algebras ofchord diagrams and graphs and of their subalgebras that we are interested in. Our main newresult here is explicit formulas for the projections of complete bipartite graphs with size of oneof the parts at most three on the subspace of primitive elements. These formulas may be usefulfor computing such projections for other graph invariants.In Section 3 we compute explicitly the values of the sl weight system at the chord diagramswhose intersection graph is a complete bipartite graph with size of one of the parts at mostthree. Using this result and the projection formula from Section 2, we compute the values ofthe sl weight system at the projections of such complete bipartite graphs on the subspace ofprimitive elements.We follow the approach of [13]; see also [3].3 Hopf algebras of graphs and chord diagrams
In this section we define the Hopf algebra of graphs and the Hopf algebra of chord diagramsmodulo the four-term relations. We also define Hopf subalgebras generated by complete bipar-tite graphs in the Hopf algebra of graphs. Using the universal formula for the projection ontothe subspace of primitive elements, we derive formulas for the projection onto the subspaceof primitive elements in the Hopf algebras of complete bipartite graphs with no more than l , l = 1 , , counital coassociative coalgebra over a field K is a vector space C over K together with K -linear maps µ : C → C ⊗ Cε : C → K such that ( id C ⊗ µ ) ◦ µ = ( µ ⊗ id C ) ◦ µ, ( id C ⊗ ε ) ◦ µ = id C = ( ε ⊗ id C ) ◦ µ. A bialgebra over a field K is a vector space B over K endowed with two structures, that ofa unital associative algebra over K (with multiplication m and unit η ) and that of a counitalcoassociative coalgebra (with comultiplication µ and counit ε ) such that µ ◦ m = ( m ⊗ µ ) ◦ ( id ⊗ τ ⊗ id ) ◦ ( µ ⊗ µ ) ,ε ⊗ ε = ε ◦ m,η ⊗ η = µ ◦ η,id = ε ◦ ν. Here by τ : B ⊗ B → B we denote the linear map defined by τ ( x ⊗ y ) = y ⊗ x, x, y ∈ B. A Hopf algebra over a field K is an associative, unital, coassociative, and counital bialgebra H together with an antipode , that is, a K -linear map S : H → H such that (in the abovenotation) m ◦ ( S ⊗ id ) ◦ µ = η ◦ ε = m ◦ ( id ⊗ S ) ◦ µ. Further on we assume that the characteristic of the ground field K is zero. In this paper by a graph we mean an isomorphism class of finite simple graphs (i.e. finite graphswith no loops and multiple edges). Formal linear combinations of graphs form a vector space,which is graded by the number of graph verticesWe define the product of two graphs G and G as their disjoint union: G G := G ⊔ G .This multiplication is extended to the vector space of graphs by linearity. It preserves thegrading. Thus, this vector space is endowed with the structure of a graded algebra.By V ( G ) we denote the vertex set of a graph G .4ny subset U ⊂ V ( G ) induces a subraph of G . We denote such a subgraph by G | U . Wedefine the comultiplication µ which acts on a graph G as µ ( G ) := X U ⊂ V ( G ) G | U ⊗ G | V ( G ) \ U . Both multiplication and comultiplication are extended to the vector space of graphs bylinearity and preserve the grading. Thus, this vector space is endowed with both the structureof a graded algebra and that of a graded coalgebra. Moreover, the following assertion holds.
Claim 1.
The multiplication and comultiplication defined above, together with the naturallydefined unit, counit, and antipode, turn the vector space of graphs into a Hopf algebra.
This construction was introduced in [7].By G we denote the Hopf algebra of graphs. The set of all graphs with n vertices generatea vector subspace G n in G . Thus, G = G ⊕ G ⊕ G ⊕ . . . Let A and B be two vertices of a graph G . By G ′ AB we denote the graph obtained from G by changing the adjacency between the vertices A and B in Γ, i.e., by deleting the edge AB if it exists and adding the edge AB otherwise. By ˜ G AB we denote the graph obtained from G by changing the adjacency with A of each vertex in V ( G ) \ { A, B } joined with B . A four-termelement in the space of graphs is a linear combination G − G ′ AB − ˜ G AB + ˜ G ′ AB , Note that all graphs in a four-term element have the same number of vertices.Let F n denote the quotient space of G n by the subspace spanned by the four-term elementscontaining n -vertex graphs. The graded Hopf algebra structure on G induces a graded Hopfalgebra structure on the space F : F = F ⊕ F ⊕ F ⊕ . . . Our description of the Hopf algebra structure on graphs follows [11].
Definition 1. A chord diagram of order n (a chord diagram with n chords) is an orientedcircle together with n pairwise distinct points splitted into n disjoint pairs considered up toorientation-preserving diffeomorphisms of the circle. We connect the points belonging to the same pair by a segment of a line or of a curve, calleda chord . (The shape of a chord is irrelevant, but it must have no common points with the circleexcept its endpoints.)The vector space spanned by the chord diagrams is graded. Each componet is spanned bydiagrams of the same order.A four-term element in the space of chord diagrams is the linear combination of diagramswhich is shown in Fig. 1. The four diagrams in this combinations contain the same set of chordsin addition to those shown in the figure, but the endpoints of all these chords must belong tothe dashed arcs.Equating four-term elements in spaces of graphs and chord diagrams to zero, we obtain four-term relations in the corresponding spaces.5 + − Figure 1: 4-term element in the space of chord diagramsFigure 2: An example of an arc representation of a chord diagram.
Definition 2. An arc diagram of order n is an oriented line together with n pairwise distinctpoints splitted into n disjoint pairs, which is considered up to orientation-preserving diffeomor-phisms of the line. Each of these n pairs of points is shown as an arc joining these points and lying in the upperhalf-plane.Choosing a point on a chord diagram (different from the endpoints of all chords) and cuttingthe chord diagram at this point, we obtain an arc diagram, or an arc representation, of thischord diagram (see Fig. 2). A chord diagram may have up to 2 n different arc representations,while an arc diagram uniquely determines the corresponding chord diagram.The product of two chord diagrams C and C is the chord diagram corresponding to thearc diagram obtained by the concatenation of two arc representations of C and C (see Fig. 3)The multiplication of chord diagrams is well-defined (i.e., the result does not depend on the arcrepresentations) modulo the four-term relations.Let C be a chord diagram. We denote its set of chords by V ( C ). Let C | U denote the chorddiagram consisting of all chords in the subset U ⊂ V ( C ).The comultiplication of chord diagrams is defined as µ ( C ) := X U ⊂ V ( C ) C | U ⊗ C | V ( C ) \ U . · = =Figure 3: Multiplication of chord diagrams.6ultiplication and comultiplication are extended to the vector space of chord diagrams bylinearity and preserve the grading. Claim 2.
The multiplication and comultiplication defined above turn the quotient space of thevector space of chord diagrams by the four-term relations into a Hopf algebra.
We denote this Hopf algebra by C . In the sequel we will call it the Hopf algebra of chorddiagrams , assuming that we consider chord diagrams up to four-term relations.To each chord diagram C we associate its intersection graph γ ( C ). The vertices of γ ( C )correspond to the chords of the diagram, and two vertices v a and v b in V ( γ ( C )) are connected byan edge if and only if the corresponding chords in V ( C ) intersect (i.e., their endpoints a , a , b , and b are arranged on the circle in the order a , b , a , b ). Claim 3 ([11]) . The map taking each chord diagram to its intersection graph extends to agraded homomorphism C → G of Hopf algebras. This homomorphism descends to a gradedhomomorphism C → F of Hopf algebras. Note that not every graph can be realized as the intersection graph of a chord diagram.Besides, two distinct chord diagrams may have the same intersection graph.
Definition 3.
A graph G is a complete bipartite graph if the set V ( G ) of its vertices can bepartitioned into two subsets ( parts ) U and W so that any two vertices v ∈ V ( G ) and v ∈ V ( G )
1. are not connected by an edge if they belong to the same part ( v , v ∈ U or v , v ∈ W ) ,2. are connected by an edge if they belong to different parts ( v ∈ U and v ∈ W or v ∈ W and v ∈ U ).The complete bipartite graph with parts of sizes n and m is denoted by K n,m . One ofthe parts of such a graph may be empty. Note that K ,n = K n , . By B we denote the Hopfsubalgebra generated by all connected complete bipartite graphs in the Hopf algebra of graphs.Note that any subgraph of a complete bipartite graph is also a complete bipartite graph. Thus,the vector space of complete bipartite graphs is closed under comultiplication.By B ( n ) we denote the Hopf subalgebra in B generated by complete bipartite graphs withno more than n vertices in one of the parts, n = 0 , , , , . . . : B ( l ) = h K , , K , , K , , K , , . . . , K , , K , , . . . K l,l , K l,l +1 , . . . i . These Hopf subalgebras are nested: B (0) ⊂ B (1) ⊂ B (2) ⊂ . . . ⊂ B .In this paper we study the structure of the Hopf algebras B (0) , B (1) , B (2) , and B (3) . TheHopf algebra B (0) is generated by the graph K , . The Hopf algebra B (1) is generated by thegraphs of the form K ,n (the so-called star graphs S n ). The Hopf algebra B (2) is generated bythe graphs of the forms K ,n and K ,n . The Hopf algebra B (3) is generated by the graphs ofthe forms K ,n , K ,n , and K ,n . 7igure 4: The complete bipartite graph K , and a chord diagram with intersection graphisomorphic to K , . Definition 4.
An element p of a bialgebra is called primitive if µ ( p ) = 1 ⊗ p + p ⊗ . It is easy to show that primitive elements form a vector subspace in a bialgebra. Since anyhomogeneous component of a primitive element is primitive, such a vector subpace of a gradedbialgebra is also graded.
Claim 4 (Milnor–Moore theorem [14]) . Over a field of characteristic zero, each connected commutative cocommutative graded bialgebra isisomorphic to the polynomial bialgebra generated by its primitive elements. (A graded bialgebra A ⊕ A ⊕ A ⊕ . . . over a field K is connected if A ∼ = K .)Decomposable elements (i.e., products of homogeneous elements of lower degree) span avector subspace in each homogeneous subspace of a graded bialgebra. It follows from the Milnor-Moore theorem that every homogeneous subspace is the direct sum of a subspace generated bydecomposable elements and a subspace of primitive elements. Therefore, a projection π of eachhomogeneous subspace to the subspace of primitive elements along the subspace generated bydecomposable elements is well defined. Claim 5 ([11, 15]) . The projection π ( G ) of any graph G on the subspace of primitive elementsalong the subspace generated by decomposable elements in the Hopf algebra G is given by theformula π ( G ) := G − X V ⊔ V = V ( G ) G | V · G | V + 2! X V ⊔ V ⊔ V = V ( G ) G | V · G | V · G | V − . . . , (1) where V ( G ) is the vertex set of G and V , V , V , . . . are nonempty nonintersecting subsets of V ( G ) . The general formula (1) for the projection onto the subspace of primitive elements in the Hopfalgebra of graphs is hard to use. But this formula can be significantly simplified in some special8ases. We will derive an explicit projection formula in Hopf algebras of complete bipartitegraphs in terms of the generating functions K l ( x ) := ∞ X n =0 K l,n x n + l n ! , (2) P l ( x ) := ∞ X n =0 π ( K l,n ) x n + l n !for l = 0 , , ,
3. Note that, for l = 0, we have K ( x ) = ∞ X n =0 K ,n x n n ! = ∞ X n =0 K n , x n n ! = exp( K , x ) . Remark 1.
Each graph K l,n has n ! automorphisms preserving the part of the graph that consistsof l vertices ( we call them selected vertices ) . In each summand of the generating functions (2) the denominator n ! equals the number of such automorphisms of the corresponding graph, andthe exponent of x equals the number of vertices of the corresponding graph. Our first main result is the following theorem.
Theorem 1.
In the case of the complete bipartite graphs K ,n , K ,n , K ,n , and K ,n , thegenerating functions for projections are expressed in terms of the generating functions for graphsas follows: P ( x ) = log K ( x ) = K , x P ( x ) = K ( x ) exp( − K , x ) P ( x ) = K ( x ) exp( − K , x ) − ( K ( x ) exp( − K , x )) = K ( x ) exp( − K , x ) − P ( x ) (3) P ( x ) = K ( x ) exp( − K , x ) − K ( x ) K ( x ) exp( − K , x ) + 2( K ( x ) exp( − K , x )) = K ( x ) exp( − K , x ) − P ( x ) P ( x ) − P ( x ) (4)The following definition is needed for the proof of Theorem 1.The Stirling number of the second kind is the number of ways to partition a set of n labeledobjects into m nonempty unlabeled subsets. It is denoted by (cid:8) nm (cid:9) , n, m ≥
0. Here are someexamples: (cid:26) n (cid:27) = 0 , n ∈ N ; (cid:26) nn (cid:27) = 1 , n ∈ N ∪ { } ; (cid:26) nn − (cid:27) = (cid:18) n (cid:19) , n ∈ N ; (cid:26) (cid:27) = 7 . Lemma 1.
For any positive integer a and N , N + a X m = a ( − m − ( m − (cid:26) Nm − a (cid:27) = ( − a − ( a − − a ) N . (5)The proof of this lemma uses the following well-known fact (see, e.g, [6]).9 laim 6. Let k, N ∈ N ∪ { } , and let x k denote the falling factorial: x := x = 1 ,x k := x ( x − x − . . . ( x − k + 1) . Then N X k =1 (cid:26) Nk (cid:27) x k = x N . (6) Proof of Lemma 1. If a = m , then the first summand in (5) is zero, since (cid:8) N (cid:9) = 0 if N ∈ N . Thus, we rewrite (6), replacing k by m − a , as N + a X m = a (cid:26) Nm − a (cid:27) x ( x − x − . . . ( x − ( m − a − x N . We substitute x = − a : N + a X m = a (cid:26) Nm − a (cid:27) ( − a )( − a − − a − . . . ( − m + 1) = ( − a ) N . To conclude the proof, it remains to multiply both sides by ( − a − ( a − Proof of Theorem 1.
For l = 0, there is nothing to prove. Indeed, for n = 1, the graph K , is aprimitive element of the Hopf algebra of graphs; therefore, π ( K , ) = K , . Further, for n = 1,the graph K ,n is a decomposable element (since it is a disconnected graph); thus, π ( K ,n ) = 0.Now consider l > l = 1, then, by Claim 5, π ( K ,n ) = n +1 X m =1 ( − m − ( m − X V ⊔ V ⊔ ... ⊔ V m = V ( K ,n ) K ,n | V · K ,n | V · . . . · K ,n | V m . (By V ⊔ V ⊔ . . . ⊔ V m we denote a partition of V ( K ,n ) into m disjoint nonempty subsets.)Given m = 1 , , . . . , n + 1, consider all possible partitions of V ( K ,n ) into m nonemptysubsets. One of these subsets contains the selected vertex, and the remaining m − K ,n corresponding to the partitions. Foreach partition, by i (0 ≤ i ≤ n − m + 1) we denote the number of vertices belonging to thesame subset as the selected vertex. The vertices of this subset induce a subgraph K ,i of K ,n . There are (cid:0) ni (cid:1) ways to choose these vertices. Furthermore, there are (cid:8) n − im − (cid:9) partitionsof the remaining n − i vertices into m − K ,n connectingthese vertices. Therefore, for any such partition, the product of the corresponding graphsis just the disjoint union of n − i vertices, which is equal to the product of n − i copies of K , . It now follows that π ( K ,n ) = n +1 X m =1 ( − m − ( m − n − m +1 X i =0 (cid:18) ni (cid:19)(cid:26) n − im − (cid:27) K ,i K n − i , . (7)10f i > n − m + 1, then (cid:8) n − im − (cid:9) = 0, so we can change the summation limits in the secondsum (without changing the value of (7)) as π ( K ,n ) = n +1 X m =1 ( − m − ( m − n X i =0 (cid:18) ni (cid:19)(cid:26) n − im − (cid:27) K ,i K n − i , . Changing the order of summation, we obtain π ( K ,n ) = n X i =0 K ,i K n − i , (cid:18) ni (cid:19) n +1 X m =1 ( − m − ( m − (cid:26) n − im − (cid:27) . The application of Lemma 1 for a = 1 yields π ( K ,n ) = n X i =0 K ,i K n − i , (cid:18) ni (cid:19) ( − n − i . (8)On the other hand, K ( x ) exp( − K , x ) = ∞ X n =0 K ,n x n +1 n ! ! ∞ X j =0 ( − K , x ) j j ! ! . The coefficient of x n +1 in this expression equals n X i =0 K ,i ( − K , ) n − i ( n − i )! i ! . Therefore, the coefficient of x n +1 n ! equals (8), as required.2. Suppose that l = 2. Now we consider the projections of K ,n and partitions of V ( K ,n ).There are two cases: in the first case, both selected vertices belong to the same part,and in the second, they belong to two different parts. Thus, P is the sum of the twoexpressions corresponding to these cases. For the first case, the reasoning is the same asfor K ,n . Therefore, the first summand in P equals K ( x ) exp( − K , x ). Given a graph K ,n and m = 1 , . . . , n + 1, the second summand equals n +2 X m =2 ( − m − ( m − n X k =0 k X i =0 (cid:18) ni, k − i, n − k (cid:19)(cid:26) n − km − (cid:27) K ,i K ,k − i K n − k , , where k is the total number of vertices in the two parts containing selected vertices, and i and k − i are the numbers of vertices in each of these parts.Now we argue as in case 1. We change the summation limits, replacing 0 ≤ k ≤ n − m + 2by 0 ≤ k ≤ n , and the order of summation. Then, in view of the fact that (cid:8) n − km − (cid:9) = 0for m > n − k , we again change the summation limits, replacing 2 ≤ m ≤ n + 2 by2 ≤ m ≤ n − k + 2. We obtain n X k =0 k X i =0 (cid:18) ni, k − i, n − k (cid:19) K n − k , K ,i K ,k − i n − k +2 X m =2 ( − m − ( m − (cid:26) n − km − (cid:27) . a = 2 yields − n X k =0 k X i =0 (cid:18) ni, k − i, n − k (cid:19) K ,i K ,k − i K n − k , ( − n − k . (9)On the other hand, let us consider the coefficient of x n +2 n ! in the second summand on theright-hand side of (3). This summand equals − ( K ( x ) exp( − K , x )) = − ( K ( x ) exp( − K , x )) = − ∞ X i =0 K ,i x i +1 i ! ∞ X j =0 K ,j x j +1 j ! ∞ X a =0 ( − a K a , x a a ! , and the coefficient of x n +2 n ! in this expression equals − n ! n X k =0 k X i =0 K ,i K ,k − i i !( k − i )!( n − k )! ( − n − k K n − k , . This is equal to (9), which concludes the proof of (3).3. Let l = 3. The projection of K ,n equals π ( K ,n ) = n +1 X m =1 ( − m − ( m − n − m +1 X k =0 (cid:18) nk (cid:19)(cid:26) n − km − (cid:27) K ,k K n − k , + n +2 X m =2 ( − m − ( m − (cid:18) (cid:19) n − m +2 X k =0 k X i =0 (cid:18) ni, k − i, n − k (cid:19)(cid:26) n − km − (cid:27) K ,i K ,k − i K n − k , + n +3 X m =3 ( − m − ( m − n − m +3 X k =0 k X i =0 k − i X j =0 (cid:18) ni, j, k − ( i + j ) , n − k (cid:19)(cid:26) n − km − (cid:27) K ,i K ,j K ,k − ( i + j ) K n − k , . Here the second summand corresponds to the partitions of V ( K ,n ) such that two ofthe selected vertices belong to one part and the third, to another part. The binomialcoefficient (cid:0) (cid:1) in the second summand is the number of ways to choose these two vertices.The further argument is the same as for l = 1 , sl Weight System
In this section we compute the values of the sl weight system at the chord diagrams withintersection graph K l,n , l ≤
3. Then we use Theorem 1 to compute the values of the sl weightsystem at the projections of such complete bipartite graphs on the space of primitive elements.The results which we obtain confirm a conjecture due to Lando, which states that if G isthe intersection graph of a chord diagram such that the length of the longest cycle of G (thecircumference of G ) is at most 2 l, l ≥
1, then the value of the sl weight system at the projectionof this chord diagram on the subspace of primitive elements is a polynomial of degree at most l .12 i x i x i x i x i x i x i x i x i x i Figure 5: Computation of the value of the weight system that corresponds to a Lie algebrawith orthonormal basis x , . . . , x m at an arc representation of a chord diagram. sl weight system Let R be a ring,and let A be an algebra over R . A linear function w : C → A vanishing at anyfour-term element is called a weight system on C . We consider only the case where R = C and A = C [ c ].Let g be a Lie algebra of finite dimension over C endowed with a nondegenerate bilinearinvariant form ( · , · ). (A form is invariant if ([ x, y ] , z ) = ( x, [ y, z ]) for any x, y, z ∈ g .) Let X = { x , x , . . . , x m } be an orthonormal basis of g with respect to this form. We use U ( g ) todenote the universal enveloping algebra of the Lie algebra g . Consider the map w g : C → U ( g )defined as follows.Suppose given a chord diagram C and an arc representation a of C . Let V ( a ) be the setof all arcs of a , and let ν be a map ν : V ( a ) → { , , . . . , m } . With the diagram a and themap ν we associate the element w X ( a, ν ) ∈ U ( g ) obtained as follows: at both ends of each arc v ∈ V ( a ) we write the element x ν ( v ) ∈ X and multiply all written elements from left to right.We denote this product by w X ( a, ν ) and the sum of such products over all possible maps by w X ( a ): w X ( a ) := X ν w X ( a, ν ) . (10)For example, the value of the weight system corresponding to a Lie algebra with orthonormalbasis x , . . . , x m at the arc diagram shown in Fig. 5 equals m X i =1 m X i =1 m X i =1 m X i =1 m X i =1 x i x i x i x i x i x i x i x i x i x i . Claim 7.
1. For any C ∈ C the result of this operation is determined uniquely and does notdepend on the choice of an arc representation of C .2. For any a , w X ( a ) ∈ Z ( U ( g )) , where Z ( U ( g )) is the center of the universal envelopingalgebra.3. The element w X ( a ) does not depend on the choice of an orthonormal basis.4. The map from chord diagrams to Z ( U ( g )) thus defined satisfies the four-term relations.Therefore, it extends to a homomorphism of commutative algebras. Since we define the multiplication of chord diagrams as the concatenation of its arc repre-sentations, the weight system corresponding to any Lie algebra is multiplicative.13e consider this construction for the simplest case of a noncommutative Lie algebra, namely,for the Lie algebra sl . This Lie algebra is generated by three elements x, y, and z with relations[ x, y ] = z, [ y, z ] = x, [ z, x ] = y. The bilinear form is given by the relations( x, x ) = ( y, y ) = ( z, z ) = 1 , ( x, y ) = ( y, z ) = ( z, x ) = 0 . The center of the universal enveloping algebra Z ( U ( sl )) is isomorphic to the algebra of poly-nomials in the Casimir element c = x + y + z ∈ U ( sl ). Hence the formula (10) defines a map w sl : C → C [ c ]. This map is a homomorphism of algebras and is called the sl weight systemon C . This definition immediately implies the following assertion.
Corollary 1.
The value of the weight system sl on a chord diagram with only one chord equals c . In [4] the following nontrivial statement, which links the sl weight system with polynomialgraph invariants, was proved. Note that its analogue for more complicated Lie algebras, e.g,for sl , turns out to be wrong. Claim 8.
The value of the sl weight system at a chord diagram depends only on the intersectiongraph of this diagram. To compute the values of the sl weight system at chord diagrams, we use the multiplicativityof this weight system and the Chmutov–Varchenko recurrence relations . For simplicity, we willidentify the value of the sl weight system at a chord diagram with the diagram itself. As inFig. 1, diagrams may contain other chords with endpoints on the dashed arcs; the sets of theseadditional chords must be the same for all terms of each equation. Claim 9 (Chmutov–Varchenko reccurence relations, [5]) . Let D be a chord diagram. Assumethat its intersection graph is connected, i.e, D is not a product of two chord diagrams of lowerorder. Then the following assertions hold.1. If D contains a leaf , i.e., a chord intersecting only one chord, then w sl ( D ) = ( c − w sl ( D ′ ) , (11) where D ′ is the chord diagram obtained from D by deleting the leaf: = ( c − · . . The following equations hold:= − + + − ,= − + + − . Now we will compute the values of the sl weight system on complete bipartite graphs withrestricted size of one of the parts, using these reccurence relations. sl weight system at the graphs K ,n , K ,n , and K ,n . Relation (11) implies the following assertion
Corollary 2.
If the intersection graph of a chord diagram is a tree with n vertices, then thevalue of the sl weight system at this chord diagram equals c ( c − n − . In particular, w sl ( K ,n ) = c ( c − n for n = 0 , , , . . . .The next theorem is the our main result about values of the sl weight system. Since itsvalue at a chord diagram depends only on the intersection graph of this chord diagram, we usegraphs as the arguments of the weight system. We set k i,j = w sl ( K i,j ), i, j = 0 , , , , . . . . Theorem 2.
The following relations hold: k ,n = c n ,k ,n = c ( c − n , (12) k ,n =( c − k ,n − + c ( c n + ( c − n − ) , n ≥ , (13) k ,n = − c + 3) k ,n − + 3 c ( c − k ,n − +2 k ,n +2 + 11 k ,n +1 − (12 c − c − k ,n +(16 c − c + 32 c + 9) k ,n − − c (2 c − c + 16 c − k ,n − + c ( c − n − (4 c − c − c + 1) − c n +1 , n ≥ . (14) Proof of the Theorem 2.
1. The equation k ,n = c n follows from the multiplicativity of theweight system.2. The equation k ,n = c ( c − n follows from the fact that the complete bipartite graph K ,n is a tree.3. To prove the recurrence relation (13) for k ,n , we use the multiplicativity of the weightsystem and the Chmutov–Varchenko relations (see Claim 9).Consider the following chord diagram: 15 { z } n The intersection graph of this chord diagram is a complete tripartite graph with parts ofsizes 1 , , and n and hence we denote this chord diagram by K , ,n and the value of the sl weight system at it by k , ,n .The first of the Chmutov–Varchenko six-term relations for a chord diagram with inter-section graph K ,n is= − + + − | { z } n − | { z } n − | { z } n − | { z } n − | { z } n − | { z } n − K , ,n − The sum of the last three summands equals ( c − k ,n − . The first summand on theright-hand side equals c n +1 . Therefore, k ,n = c n +1 − k , ,n − + ( c − k ,n − . (15)Now we will use the following Chmutov–Varchenko relation to compute the value of k , ,n := − + + − n − z}|{ n − z}|{ n − z}|{ n − z}|{ n − z}|{ n − z}|{ We obtain the following relation between k , ,n − and k ,n − :( c − k , ,n − = ck ,n − − k ,n + ( c − k ,n − + ck , ,n − − ck ,n − ; (16)therefore, k , ,n − = k ,n − − c ( c − n − . To conclude the proof of 13, it remains to substitute this expression for k , ,n into (15).4. Now let us prove (14). The following equations follow from the Chmutov–Varchenkorecurrence relations:(a) = − + + − ; n − z}|{ n − z}|{ n − z}|{ n − z}|{ n − z}|{ n − z}|{ − + + − ; n − z}|{ n − z}|{ n − z}|{ n − z}|{ n − z}|{ n − z}|{ (c) = − + + − ; { n { n { n { n { n { n (d) = − + + − ; n − z}|{ n − z}|{ n − z}|{ n − z}|{ n − z}|{ n − z}|{ (e) = − + + − ; n − z}|{ n − z}|{ n − z}|{ n − z}|{ n − z}|{ n − z}|{ (f) = − + + − . { n { n { n { n { n { n We have six linear equations. Each chord diagram in these equations contains a collectionof n nonintersecting chords represented by the parallel line segments in the figure. If weadd Eqs. 4a–4f with n − n chords in each such collection and Eqs. 4c and 4fwith n − sl weight systemat the complete bipartite graphs K ,n and K ,n . Corollary 3.
The following relations hold: k ,n = 16 c ((4 c − c − n + 3( c − n + 2 c n +1 ) (17) k ,n = 130 c (cid:0) c − c + 6)( c − n + 10(4 c − c − n + 6(3 c − c + 2)( c − n + 5 c n +1 (cid:1) . (18)17et e K l denote the exponential generating functions for the values of the weight system w sl at the complete bipartite graphs K l,n ( l = 1 , , Corollary 4.
The following relations hold: e K ( x ) = xce ( c − x e K ( x ) = x c (cid:0) (4 c − e ( c − x + 3 e ( c − x + 2 ce cx (cid:1)e K ( x ) = x c (cid:0) c − c + 6) e ( c − x + 10(4 c − e ( c − x + 6(3 c − c + 2) e ( c − x + 5 ce cx (cid:1) . sl weight system at the projections π ( K l,n ) , l = 1 , , . Let e P l denote the exponential generating function for the values of the sl weight system atthe projections of the complete bipartite graphs K l,n on the subspace of primitive elements.Combining Theorem 1 and Corollary 4, we obtain expressions for e P l , l = 1 , , Corollary 5.
The following relations hold: e P ( x ) = cxe − x , (19) e P ( x ) = 16 cx (cid:0) (4 c − e − x − ce − x + 3 e − x + 2 c (cid:1) , (20) e P ( x ) = 130 cx ((12 c − c + 18) e − x − c (60 c − e − x + (60 c + 40 c − e − x − ce − x − (12 c + 12 c − e − x + 5 c ) . (21)For each e P l ( x ), the coefficient of each exponent is a polynomial in c of degree at most l .This implies the following statement. Claim 10.
The following assertions hold:1. The value w sl ( π ( K ,n )) equals ( − n c
2. The value w sl ( π ( K ,n )) is a polynomial of degree less than or equal to . If n ≥ , thenthe degree of this polynomial equals 2.3. The value w sl ( π ( K ,n )) is a polynomial of degree less than or equal to . If n ≥ , thenthe degree of this polynomial equals 3. This proves the following conjecture in the particular case of the graphs K ,n , K ,n , and K ,n . Conjecture 1 (S.K.Lando) . Suppose given a chord diagram C . Let π ( C ) be its projection onthe space of primitive elements, and let γ ( C ) be its intersection graph. Then w sl ( π ( C )) is apolynomial of degree less than or equal to one half of the circumference (i.e., the length of thelongest cycle) of γ ( C ) . For the case of complete bipartite graphs, this conjecture can be restated as follows.18 onjecture 2 (S.K. Lando) . The value of the sl weight system at the complete bipartite graph K l,n is a polynomial of degree min( l, n ) . We also give formulas for the ordinary generating functions for the values of w sl at theprojections of complete bipartite graphs onto the subspace of primitive elements. We denotesuch a generating function by e P l ; then e P l ( s ) = ∞ X n =0 w sl ( π ( K l,n )) s n . Using the fact that the ordinary generating function corresponding to the exponential gen-erating fuction e ax is − as , we obtain the following result. Theorem 3.
The following relations hold: e P ( s ) = cs s , e P ( s ) = cs ((1 + 2 s ) + 2 cs (1 + s ))(1 + s )(1 + 2 s )(1 + 3 s ) , e P ( s ) = cs ((3 s − s )(1 + 4 s ) − cs (5 + 21 s + 10 s − s ) − c s (1 + 2 s ))(1 + s )(1 + 2 s )(1 + 3 s )(1 + 4 s )(1 + 6 s ) . References [1] Bar-Natan, D.
On Vassiliev knot invariants , Topology, vol. 34, no. 2 (1995), 423–472.[2] Bigeni, Ange.
A generalization of the Kreweras triangle through the universal sl weightsystem , J. Combin. Theory Ser. A 161 (2019), 309–326; arXiv:1712.05475v3.[3] Chmutov, S.; Duzhin, S.; Mostovoy, J. Introduction to Vassiliev knot invariants , CambridgeUniversity Press, Cambridge, 2012; arXiv:1103.5628.[4] Chmutov, Sergei V.; Lando, Sergei K.
Mutant knots and intersection graphs , Algebr. Geom.Topol. 7 (2007), 1579–1598; arXiv: 0704.1313v1.[5] Chmutov, S.V., Varchenko, A.N.
Remarks on the Vassiliev knot invariants coming from sl , Topology, vol. 36, no.1 (1997) 153–178.[6] R. L. Graham, D. E. Knuth, O. Patashnik, Concrete Mathematics: A Foundation forComputer Science , Addison-Wesley, Boston, 1994[7] S. A. Joni, G.-C. Rota,
Coalgebras and bialgebras in combinatorics , Stud. Appl. Math. 61,no. 2 (1979), 93–139.[8] Kontsevich, M.
Vassiliev knot invariants , in: Adv. in Soviet Math., vol. 16 (1993), Part 2,137–150.[9] Kulakova, E.; Lando, S.; Mukhutdinova, T.; Rybnikov, G.
On a weight system conjecturallyrelated to sl , European J. Combin. 41 (2014), 266–277. arXiv:1307.4933v2.1910] S. Lando, On primitive elements in the bialgebra of chord diagrams , in: Amer. Math. Soc.Transl. Ser. 2, no. 180 Providence, RI: AMS 1997, 167–174.[11] S. Lando.
On a Hopf Algebra in Graph Theory , Journal of Combinatorial Theory, SeriesB 80 (2000), 104-121[12] S. Lando, V. Zhukov,
Delta-Matroids and Vassiliev Invariants , Moscow MathematicalJournal. 2017. Vol. 17. No. 4. P. 741–755.[13] S. Lando, A. Zvonkin.
Graphs on Surfaces and their Applications
Springer-Verlag, Berlin,2004[14] J. Milnor, J. Moore,
On the structure of Hopf algebras , Ann. of Math. (2), 1965, v. 81,211–264.[15] W. R. Schmitt,
Incidence Hopf algebras , J. Pure Appl. Algebra, 1994, v. 96, 299–330[16] V. Vassiliev,
Cohomology of knot spaces , in: Theory of singularities and its applications,Advance in Soviet Math., V. I. Arnold ed., AMS, 1990,This is a preprint of the Work accepted for publication in Functional Analysis and Its Applications ©©