On a weight system conjecturally related to sl 2
OOn a weight system conjecturally related to sl ∗ , T. Mukhutdinova † , G. RybnikovNational Research University Higher School of Economics Abstract
We introduce a new series R k , k = 2 , , , . . . , of integer valued weightsystems. The value of the weight system R k on a chord diagram is a signednumber of cycles of even length 2 k in the intersection graph of the diagram.We show that this value depends on the intersection graph only. We checkthat for small orders of the diagrams, the value of the weight system R k on adiagram of order exactly 2 k coincides with the coefficient of c k in the value ofthe sl -weight system on the projection of the diagram to primitive elements. Below, we use standard notions from the theory of finite order knot invariants; see,e.g. [4] or [10].A chord diagram of order n is an oriented circle endowed with 2 n pairwise distinctpoints split into n disjoint pairs, considered up to orientation-preserving diffeomor-phisms of the circle. A weight system is a function on chord diagrams satisfying the4-term relation; see Fig. 1. For a chord diagram d with two chords A and B hav-ing neighboring ends, as in Fig. 1, we will write this relation as f ( d ) − f ( d (cid:48) AB ) = f ( ˜ d AB ) − f ( ˜ d (cid:48) AB ).In figures, the outer circle of the chord diagram is always assumed to be orientedcounterclockwise. Dashed arcs may contain ends of arbitrary sets of chords, same forall the four terms in the picture.To each weight system, a finite order knot invariant can be associated in a canon-ical way, which makes studying weight systems an important part of knot theory. ∗ Partly supported by the Russian Foundation for Basic Research Grant no. 13-01-00383a † Partly supported by the Russian Foundation for Basic Research Grant no. 13-01-00383a a r X i v : . [ m a t h . G T ] M a y igure 1: 4-term relationThere is a number of approaches to constructing weight systems. In particular, weightsystems can be constructed from semisimple Lie algebras, although the result is com-plicated. The present paper has been motivated by an aspiration for understandingthe weight system corresponding to the simplest nontrivial case of the Lie algebra sl .The intersection graph γ ( d ) of a chord diagram d is the simple graph whose verticesare in one-to-one correspondence with the chords in d , and two vertices are connectedby an edge iff the corresponding chords intersect one another. The 4 -term relationfor graphs , introduced in [9], is the graph counterpart of the 4-term relation for chorddiagrams. It is defined as follows.Denote by V (Γ) the set of vertices of a graph Γ and by E (Γ) the set of its edges.Let us associate to each ordered pair of (distinct) vertices A, B ∈ V (Γ) of a graph Γtwo other graphs Γ (cid:48) AB and (cid:101) Γ AB .The graph Γ (cid:48) AB is obtained from Γ by erasing the edge AB ∈ E (Γ) in the case thatthis edge exists, and by adding the edge otherwise. In other words, we simply changethe adjacency between the vertices A and B in Γ. This operation is an analogue ofedge deletion, but we prefer to formulate it in a slightly more symmetric way.The graph (cid:101) Γ AB is obtained from Γ in the following way. For any vertex C ∈ V (Γ) \ { A, B } we switch its adjacency with A to the opposite one if C is joined with B , and we do nothing otherwise. All other edges do not change. Note that the graph (cid:101) Γ AB depends not only on the pair ( A, B ), but on the order of vertices in the pair aswell.
Definition 1.1 (4-invariant)
A graph invariant f is a 4 -invariant if it satisfies the4 -term relation f (Γ) − f (Γ (cid:48) AB ) = f ( (cid:101) Γ AB ) − f ( (cid:101) Γ (cid:48) AB ) (1)for each graph Γ and for any pair A, B ∈ V (Γ) of its vertices.Figure 2 shows the 4-term relation for intersection graphs corresponding to asample 4-term relation for chord diagrams.Any graph invariant satisfying the 4-term relation (that is, a 4-invariant) deter-mines a weight system [9]: the value of this weight system on a chord diagram is set2 (cid:37)(cid:39)(cid:36) (cid:38)(cid:37)(cid:39)(cid:36) (cid:38)(cid:37)(cid:39)(cid:36) (cid:38)(cid:37)(cid:39)(cid:36) − − = (cid:2)(cid:2)(cid:2)(cid:2) (cid:2)(cid:2)(cid:2)(cid:2) (cid:2)(cid:2)(cid:2)(cid:2) (cid:2)(cid:2)(cid:2)(cid:2) E E E ED D D D (cid:80)(cid:80)(cid:80)(cid:80) (cid:80)(cid:80)(cid:80)(cid:80) (cid:80)(cid:80)(cid:80)(cid:80) (cid:80)(cid:80)(cid:80)(cid:80)
F F F F (cid:33)(cid:33)(cid:33)(cid:33) (cid:33)(cid:33)(cid:33)(cid:33) (cid:33)(cid:33)(cid:33)(cid:33) (cid:33)(cid:33)(cid:33)(cid:33)
B B B B (cid:68)(cid:68)(cid:68)(cid:68) (cid:68)(cid:68)(cid:68)(cid:68) (cid:68)(cid:68)(cid:68)(cid:68) (cid:68)(cid:68)(cid:68)(cid:68)
C C C C (cid:4)(cid:4)(cid:4)(cid:4)
A A A A (cid:10)(cid:10)(cid:10)(cid:10) (cid:65)(cid:65)(cid:65) (cid:64)(cid:64) (cid:114) (cid:114) (cid:114) (cid:114)(cid:114) (cid:114) (cid:114) (cid:114)(cid:114) (cid:114) (cid:114) (cid:114)(cid:114) (cid:114) (cid:114) (cid:114)(cid:114) (cid:114) (cid:114) (cid:114)(cid:114) (cid:114) (cid:114) (cid:114) − − = A A A AB B B BE E E ED D D DF F F FC C C C (cid:1)(cid:1) (cid:1)(cid:1) (cid:1)(cid:1) (cid:1)(cid:1)(cid:65)(cid:65) (cid:65)(cid:65)(cid:81)(cid:81)(cid:81) (cid:81)(cid:81)(cid:81) (cid:81)(cid:81)(cid:81) (cid:81)(cid:81)(cid:81)(cid:17)(cid:17)(cid:17) (cid:17)(cid:17)(cid:17)(cid:65)(cid:65) (cid:65)(cid:65) (cid:65)(cid:65) (cid:65)(cid:65)(cid:1)(cid:1) (cid:1)(cid:1) (cid:1)(cid:1) (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1)(cid:1)
Figure 2: A 4-term relation for chord diagrams and the corresponding intersectiongraphsto be the value of the 4-invariant on the intersection graph of the diagram.Let d be a chord diagram. Denote by E l ( d ) the number of circuits of length l (= edge l -gons) in the intersection graph of d . It is well known (see [10], Exer-cise 6.4.10) that, for l ≥
4, the parity of the number E l ( d ) is a weight system withvalues in Z / Z . However, no integer-valued weight system with the same parity hasbeen known.In this paper, we construct, for each k = 2 , , , . . . , an integer-valued weight sys-tem R k whose value on each chord diagram has the same parity as E k , R k ( d ) ≡ E k ( d )mod 2 for any chord diagram d . This weight system counts circuits of length 2 k inthe intersection graph of the diagram with signs depending on the mutual position ofthe corresponding chords.For completeness, let us recall the proof of the fact mentioned above. It is worthto be compared with the proof for the integer-valued analogue below. Proposition 1.2
The value E l mod 2 is a weight system for each l ≥ . Proof.
The value E l ( d ) obviously depends on the intersection graph of a chorddiagram rather than on the diagram itself. Let us extend the function E l to arbitrarygraphs (that are not necessarily intersection graphs) in the obvious way: let E l (Γ)be the number of edge l -gons in Γ having l pairwise distinct vertices. We are goingto prove that E l (Γ) mod 2 satisfies the 4-term relation for graphs; the propositionthen follows.Suppose Γ contains an edge AB . Then for the two terms on the left-hand side ofthe 4-term relation we have that E l (Γ) − E l (Γ (cid:48) AB ) is the number of edge l -gons in Γpassing through the edge AB . Similarly, for the right-hand side, E l ( (cid:101) Γ) − E l ( (cid:101) Γ (cid:48) AB ) isthe number of edge l -gons in (cid:101) Γ AB passing through AB . All the l -gons in Γ passing3hrough AB contain a chain CABD and split into three disjoint classes according tothe adjacency of the vertices C and D to A and B :1. the vertex C is adjacent to B and D is adjacent to A ;2. the vertex C is adjacent to B , but D is not adjacent to A ;3. the vertex C is not adjacent to B.All edge l -gons in (cid:101) Γ AB passing through the four points A, B, C, D admit a similarclassification.
Example 1.3
The three edge 4-gons in the leftmost graph in Fig. 2 passing throughthe edge AB split in the three classes in the following way: • the edge quadrangles F ABC and
CABF belong to the first class; • there are no edge quadrangles belonging to the second class; • the edge quadrangle DABC belongs to the third class.Now, the l -gons in Γ belonging to the second class are in one-to-one correspondencewith the l -gons in (cid:101) Γ AB containing the path CBAD . The edge l -gons of the third kindare the same in both graphs Γ and (cid:101) Γ AB . And, finally, the edge l -gons of the first kindin each of the two graphs come in pairs: the chain CABD can be replaced with thechain
CBAD . Hence, the number of edge l -gons of the first type is even for each ofthe two graphs, and the required assertion follows. (cid:3) The authors are grateful to the participants of the seminar “Combinatorics offinite order knot invariants” at the Department of mathematics NRU HSE for usefuldiscussions. The authors would also like to express their gratitude to the referees forcareful proofreading and valuable suggestions. R k k -gons In order to define the weight system R k , let us take a chord diagram d and choosean arbitrary orientation of its chords. This orientation induces an orientation of theedges of the intersection graph γ ( d ) in the following way. We orient an edge AB from A to B if the beginning of the chord B belongs to the arc of the outer circle4f d which starts at the beginning of A and goes in the positive direction to the endof A ; see Fig. 3. We denote the directed intersection graph of an oriented chorddiagram d by γ ( d ), like in the case of ordinary intersection graph of a non-orientedchord diagram, since this convention causes no misunderstanding. Oriented edges ina directed graph will also be called arrows .Figure 3: An oriented chord diagram and the corresponding directed intersectiongraph; the 4-circuit ADBC is oriented positivelyWe say that a circuit of even length l = 2 k in a directed graph is positively oriented (or its sign is +) if the number of arrows in this circuit oriented in either directionis even; otherwise the circuit is negatively oriented (or has the sign − ). Since thetotal number of arrows is even, the sign is independent of the choice of the directionof the circuit. For example, all the arrows in the 4-circuit ADBC in the directedintersection graph in Fig. 3 are oriented in the same direction, meaning it is positive.
Definition 2.1
The function R k takes a chord diagram d to the difference betweenthe number of positively and negatively oriented 2 k -gons in the directed intersectiongraph γ ( d ) for arbitrarily chosen orientation of the chords in d .Now we are going to show that this definition makes sense. Proposition 2.2
The function R k is well defined, that is, its value on a chord dia-gram does not depend on the chosen orientation of the chords. Proof.
It suffices to check that if we switch the direction of a single chord in achord diagram d to the opposite one, then the sign of each circuit remains the same.Changing the direction of a chord A in d leads to switching directions of all the arrowsin γ ( d ) incident to the vertex A , preserving the directions of all other arrows. Hence,the sign of any circuit not containing the vertex A remains the same. Any circuitpassing through A contains exactly two arrows incident to A . After switching thedirection of both of them, the sign also remains the same. (cid:3) xample 2.3 Let us compute the value of R on the chord diagram d shown in Fig. 4.Due to the Proposition above, an arbitrary orientation of the chords can be chosen.For the one in the figure, the directed intersection graph γ ( d ) is shown in Fig. 4.Figure 4 also shows all the oriented 4-gons in the intersection graph. The differencebetween the number of positively and negatively oriented 4-gons is 2 − R ( d ). (cid:38)(cid:37)(cid:39)(cid:36) (cid:55)→ (cid:64)(cid:64)(cid:64)(cid:82)(cid:0)(cid:0)(cid:0)(cid:18)(cid:63)(cid:45) ABDC (cid:114)(cid:114) (cid:114)(cid:114) (cid:0)(cid:0)(cid:0)(cid:18)(cid:45)(cid:54) (cid:54)(cid:64)(cid:64)(cid:64)(cid:73)(cid:27)
A BD C (cid:114)(cid:114) (cid:114)(cid:114) (cid:45)(cid:54) (cid:54)(cid:27) (cid:114)(cid:114) (cid:114)(cid:114) (cid:0)(cid:0)(cid:0)(cid:18)(cid:45)(cid:64)(cid:64)(cid:64)(cid:73)(cid:27) (cid:114)(cid:114) (cid:114)(cid:114) (cid:0)(cid:0)(cid:0)(cid:18)(cid:54) (cid:54)(cid:64)(cid:64)(cid:64)(cid:73) − + +Figure 4: An oriented chord diagram with labeled chords, its directed intersectiongraph, and all the 4-gons in it, with signs indicated R k is a weight system Theorem 2.4
The function R k on chord diagrams is indeed a weight system, thatis, it satisfies the -term relation. Proof.
The proof is similar to that of Proposition 1.2, but is slightly more com-plicated. Let d be a chord diagram, and let A and B be a pair of chords in d havingneighboring ends and intersecting one another. Pick an orientation of the chords in d such that the arrow AB in the directed graph γ ( d ) of d is oriented from A to B .The difference R k ( d ) − R k ( d (cid:48) AB ) counts the number of signed circuits of length 2 k in γ ( d ) containing the edge AB .Each such circuit contains a sequence of vertices CABD . Let us split the set ofsuch circuits into the following three groups:1. the vertex C is adjacent to B and D is adjacent to A ;2. the vertex C is adjacent to B , but D is not adjacent to A ;3. the vertex C is not adjacent to B.All edge 2 k -gons in (cid:103) γ ( d ) AB passing through the four points A, B, C, D admit asimilar classification.Now, the 2 k -gons in γ ( d ) belonging to the second class are in one-to-one corre-spondence with the 2 k -gons in (cid:103) γ ( d ) AB containing the path CBAD : just preserve the6ther edges in each 2 k -gon. In order to show that the signs of the correspondingcircuits in both directed graphs coincide, it suffices to consider all possible mutualpositions of the four chords A, B, C, D in the chord diagram d . There are essentiallytwo different such positions, depending on whether the chords C and D intersectone another. In both cases, for a chosen orientation of the chords, we easily checkcoincidence of the signs of the corresponding 2 k -gons.The edge 2 k -gons of the third kind are the same in both graphs γ ( d ) and (cid:101) γ ( d ) AB .Finally, the edge 2 k -gons of the first kind in each of the two directed intersectiongraphs come in pairs: the chain CABD can be replaced with the chain
CBAD . Inorder to prove that the signs of the two 2 k -gons in a pair are opposite, it sufficesto consider all possible mutual positions of the four chords A, B, C, D in the chorddiagram d .Hence, the required assertion follows. (cid:3) R k and intersection graphs Different chord diagrams can have one and the same intersection graph. Below, weshow that the value of the weight system R k depends on the intersection graph ofthe chord diagram rather than on the diagram itself. A natural question then arises,whether R k can be extended to a 4-invariant of graphs. We show that this is true for k = 2 and k = 3. The case of arbitrary k is discussed in Sec. 5. R k depends on the intersectiongraph only Theorem 3.1
The weight system R k acquires the same value on any two chord dia-grams with coinciding intersection graphs. Since weight systems taking the same values on chord diagrams with coincidingintersection graphs are in one-to one correspondence with finite order knot invariantsnot distinguishing mutant knots [5], we conclude the following:
Corollary 3.2
The canonical knot invariant associated to the weight system R k doesnot distinguish mutant knots. The proof of the theorem is based on a statement in [5] giving a complete descrip-tion of the situations where two chord diagrams have the same intersection graph.7e start with the definition of a share.
Definition 3.3 A share is a part of a chord diagram consisting of two arcs of theouter circle possessing the following property: each chord one of whose ends belongsto these arcs has both ends on these arcs.The complement of a share also is a share. The whole chord diagram is its ownshare whose complement contains no chords. Definition 3.4 A mutation of a chord diagram is another chord diagram obtainedby a rotation/reflection of a share; see Fig. 5.Figure 5: Mutations of a chord diagramObviously, mutations preserve the intersection graphs of chord diagrams. We callthe subgraph of the intersection graph induced by the vertices corresponding to chordsforming a given share also a share in the intersection graph. Theorem 3.5 [5]
Two chord diagrams have the same intersection graph if and onlyif they are related by a sequence of mutations.
Proof of Theorem 3.1. We are going to prove that a mutation of a chord diagramdoes not change the value of the weight system R k , for arbitrary k = 2 , , , . . . . Itsuffices to consider only reflection and rotation of a share; for the composition ofreflection and rotation, the result will follow automatically.Pick an arbitrary orientation of chords in a chord diagram. A mutation of achord diagram with oriented chords produces a chord diagram with oriented chordsin a natural way. For a given share in the intersection graph, its reflection results ininverting the orientation of all the arrows belonging to the corresponding share in theintersection graph. Similarly, rotation of a share results in inverting all the arrowsbetween the two shares in the intersection graph.8enote the two sets of vertices belonging to the complementary shares in theintersection graph by U and W , respectively. The set U contains a subset u ⊂ U andthe set W contains a subset w ⊂ W such that • any vertex from u is connected to all vertices of w (and vice versa); • each edge connecting vertices from U and W connects, in fact, a vertex from u to a vertex from w .Denote the set of all edges in the intersection graph connecting a vertex from u to avertex from w by K ( u, w ). The graph with the set of vertices u (cid:116) w and the set ofedges K ( u, w ) is the complete bipartite graph with the parts u, w .Any circuit in the intersection graph contains an even number of edges from K ( u, w ).Indeed, any path starting in U switches between U and W after each passing throughan edge in K ( u, w ). Since a circuit returns to the original vertex, the number of suchpassings must be even.Now, the rotation mutation changes the orientation of all arrows in K ( u, w ),whence of an even number of arrows in any circuit. In particular, it changes orienta-tion of an even number of arrows in any 2 k -gon, hence preserving its sign.Pick an arbitrary orientation of the chord diagram such that all the arrows in K ( u, w )are oriented from U to W . Such an orientation always exists. Indeed, if the share U consists of two arcs U , U and the share W consists of two arcs W , W following alongthe positive direction of the outer circle in the alternating order, U , W , U , W , seeFig. 6, then it suffices to orient all the chords connecting U , U and W , W from U to U and from W to W , respectively.What happens if we reflect the share W ? If the 2 k -gon we consider has evennumber of arrows in the share W , then its sign remains the same. The circuits withodd number of arrows in W split into pairs in the following way. The circuit intersectsthe set K ( u, w ) by even number of arrows. Let us split these arrows into pairs: twoedges belong to one and the same pair iff their ends in the share W are connected bya path that is a part of the circuit totally contained in W . For such pair of arrows,let the first one be u w and the second one be u w , u i ∈ U , w i ∈ W , i = 1 , w = w or u = u , but not both).By switching pairs we mean replacing each pair of edges u w , u w , with the pair u w , u w ; see Fig. 7.Switching all pairs for a circuit produces a new circuit. This transformation isan involution: switching all pairs in this new circuit restores the original one. Thus,all circuits are split into pairs. We are going to show that any two circuits of evenlength 2 k having odd number of arrows in W and belonging to the same pair have9igure 6: Distinguished orientation of chords allowing for a specific orientation ofshare connecting arrows in the directed intersection graphopposite signs. Indeed, if we choose directions in both circuits that coincide insidethe share U , then they are opposite inside the share W . Both circuits have the samenumber of edges inside K ( u, w ) in either direction. Therefore, the only difference inthe orientation is inside the share W , where each arrow has opposite directions in thetwo circuits. Since the number of arrows in W is odd, the signs of the two circuitsare opposite.Thus, both kinds of mutation preserve the value of R k . (cid:3) R k to arbitrary graphs Theorem 3.1 above shows that each weight system R k defines a function on inter-section graphs; we denote this function by R k as well. In this section we prove thefollowing special case of a general theorem proved in Sec. 5. Proposition 3.6
The functions R and R can be extended to -invariants of graphs. Proof.
In order to prove that R k can be extended to a 4-invariant of graphs,it suffices to prove this for graphs with exactly 2 k vertices. Indeed, denote by R k the function whose values on graphs with 2 k vertices coincides with the constructedextension, and which vanishes on all other graphs. Then the convolution R k ∗ U provides us with the desired extension to arbitrary graphs; here U is the functionwhose value on the arbitrary graph is equal to 1. It is obvious that when restrictedto intersection graphs, R k ∗ U coincides with R k . Here we make use of the fact that10igure 7: Two circuits obtained from one another by switching pairsthe vector space spanned by graphs carries a natural graded bialgebra structure: theproduct of two graphs is given by their disjoint union, and the coproduct µ (Γ) of agraph Γ is given by µ (Γ) = (cid:88) I ⊂ V (Γ) Γ | I ⊗ Γ | V (Γ) \ I , where Γ | I is the subgraph of Γ induced by a subset I of its vertices. The convolutionproduct, denoted by ∗ , is induced on the dual space of graph invariants from thecoproduct on the space of graphs [9]: f ∗ g (Γ) = f ⊗ g ( µ ( G )) , for arbitrary graph invariants f and g . These operations naturally descend to the4-bialgebra of graphs and 4-invariants, respectively.If k = 2, then each graph with 2 k = 4 vertices is an intersection graph, and any4-term relation for intersection graphs has a chord diagram counterpart, meaning weare done.For k = 3, there are two graphs with 6 vertices that are not intersection graphs,namely, the 5-wheel and the 3-prism; see Fig. 8. There are three ways to express the11-wheel as a linear combination of intersection graphs through the 4-term relation,and for the 3-prism there are two such ways; see Fig. 8. A direct computation showsthat in both cases all the different representations provide the same value, which onecan admit for the value of the extended graph invariant R : for the 5-wheel it is − −
1. Now, a direct verification, taking into account those4-term relations for graphs that do not have chord diagram counterparts, ensures thatthis extended invariant of graphs with 6 vertices is indeed a 4-invariant. (cid:3) R k and the sl -weight system It is well known that for an arbitrary Lie algebra g endowed with a nondegenerateinvariant scalar product ( · , · ) one can construct a weight system b g with values inthe center ZU ( g ) of the universal enveloping algebra U ( g ). Invariancy of the scalarproduct means that ( x, [ y, z ]) = ([ x, y ] , z ) for any three elements x, y, z ∈ g .The 3-dimensional Lie algebra sl (taken, for definiteness, over the field C ofcomplex numbers) provides the first nontrivial example of this construction. Sincethe center ZU ( sl ) of the universal enveloping algebra U ( sl ) is isomorphic to the ringof polynomials in the Casimir element c of sl , we obtain a weight system with valuesin the ring C [ c ] of polynomials in a single variable c ; see [1] and [7]. We denote thisweight system by b sl : M → C [ c ]. Here M denotes the graded vector space spannedby chord diagrams modulo the 4-term relations; M = M ⊕ M ⊕ M ⊕ . . . , where the finite dimensional vector space M n is spanned by chord diagrams with n chords, modulo the 4-term relations. The space M is a graded commutative cocom-mutative Hopf algebra, with a multiplication m : M ⊗ M → M , and a comultipli-cation µ : M → M ⊗ M . The weight system b sl is an algebra homomorphism. Ittakes an arbitrary chord diagram with n chords to a polynomial of degree n in c withthe leading coefficient 1.The invariant nondegenerate scalar product ( · , · ) on sl can be chosen in a uniqueway up to a nonzero multiplicative constant. The choice of the multiplicative constantaffects the non-leading coefficients of the value of b sl on a chord diagram. Below, weuse the normalization chosen in [10], which differs from that in [6] and [4].Being a graded commutative cocommutative Hopf algebra, M is generated by itsprimitive elements. Recall that an element p of a Hopf algebra is said to be primitive if µ ( p ) = 1 ⊗ p + p ⊗
1. Primitive elements form a graded vector subspace PM of M , PM = PM ⊕ PM ⊕ PM ⊕ . . . , PM n ⊂ M n . n = 1 , , , . . . , there exists a natural projection π n : M n → PM n of thespace of chord diagrams of order n to the subspace PM n of primitive elements alongthe subspace DM n spanned by decomposable chord diagrams. A chord diagramwith n chords is said to be decomposable if it can be represented as a product of twochord diagrams, each with less than n chords.The following statement is an immediate consequence of a result obtained in [6],Theorem 3, for an arbitrary simple Lie algebra. Proposition 4.1 [6]
The value of the weight system b sl on an arbitrary elementin PM n is a polynomial of degree at most n/ in c . Now we are able to formulate the conjecture mentioned in the introduction, whichrelates the weight systems R k and b sl . Conjecture 4.2
For any chord diagram d with k chords, the coefficient of c k in thevalue b sl ( π k ( d )) is R k ( d ) . We did not manage to prove the Conjecture, but the computer experiments showthat it is true up to k = 4. Below, we present Table 1, which contains the values ofthe weight systems R k ( · ) and b sl ( π k ( · )) for some chord diagrams with 2 k chords.The original definition of a weight system b g is too cumbersome from the computa-tional point of view, which makes it difficult to be computed even for chord diagramswith very few chords. The computations we made used the recurrence relation for b sl obtained in [6], Theorem 2: Proposition 4.3
If a chord diagram contains a leaf, that is, a chord intersectingonly one other chord, then the value of the sl universal weight system on the diagramis ( c − times its value on the result of deleting the leaf. In addition, the value ofthe weight system b sl on a chord diagram satisfies the recurrence relations shown inFig. 9. The projection to the subspace of primitive elements along the subspace of de-composable elements has the following form.
Proposition 4.4 [8]
The projection π n takes a chord diagram d with n chords tothe linear combination π n : d (cid:55)→ d − (cid:88) I (cid:116) J = V ( d ) d | I d | J + 2! (cid:88) I (cid:116) J (cid:116) K = V ( d ) d | I d | J d | K − . . . , here the summations run over all unordered partitions of the set V ( d ) of the chordsin d into disjoint union of nonempty subsets, and d | I is the chord subdiagram inducedfrom d by the subset I ⊂ V ( d ) of chords. Here is more evidence supporting the conjecture:1. the weight system b sl , similarly to the weight system R k , depends on the inter-section graphs of the chord diagram rather than on the diagram itself—see [5];2. the value R k ( d d ) on a product of two nontrivial diagrams with 2 k chords intotal is 0, since there are no 2 k -gons in the intersection graph; on the otherhand, π k ( d d ) = 0, since the diagram d d is decomposable;3. for a chord diagram d with 2 k chords having a leaf (a chord intersecting onlyone other chord), R k ( d ) = 0; it can also be easily proved that the degree of thepolynomial b sl ( π k ( d )) is less than k . Remark 4.5
The same argument as in the proof of Proposition 3.6 allows one toextend the value of the b sl weight system to a 4-invariant of graphs with up to 6edges. Namely, we can set the value of this invariant on the 5-wheel to be equalto c − c + 50 c − c + 176 c − c , with the projection to the subspace ofprimitive elements − c + 70 c − c , and on the 3-prism to be equal to c − c +40 c − c + 146 c − c , with the projection to the subspace of primitive elements − c + 58 c − c . After the first version of the present text has been spread as a preprint and sub-mitted for publication, a paper by Bar-Natan and Vo [3] appeared. In this paper,Conjecture 4.2 is shown to be true, and now it can be stated as a theorem:
Theorem 5.1 [3]
For any chord diagram d with k chords, the coefficient of c k inthe value b sl ( π k ( d )) is R k ( d ) . In fact, Bar-Natan and Vo show that the proof essentially constitutes a part ofthe proof of the Melvin–Morton–Rozansky Conjecture in [2].One of the key ingredients in the proof is Proposition 3.13 in [2] which readsas follows. Denote by w C the Z -valued function on chord diagrams equal to 1 if theadjacency matrix of the intersection graph of the diagram is nondegenerate over Z / Z ,14 d ∈ M k b sl ( d ) γ ( d ) R k ( d ) b sl ( π k ( d ))2 (cid:22)(cid:21)(cid:23)(cid:20) (cid:0)(cid:0)(cid:64)(cid:64)(cid:74)(cid:74) c ( c − ( c − (cid:114) (cid:114)(cid:114) (cid:114) (cid:64)(cid:0) − c (cid:22)(cid:21)(cid:23)(cid:20) c − c + 8 c − c (cid:114) (cid:114)(cid:114) (cid:114) c − c (cid:22)(cid:21)(cid:23)(cid:20) (cid:0)(cid:0)(cid:64)(cid:64) c − c + 10 c − c (cid:114) (cid:114)(cid:114) (cid:114) (cid:0)(cid:0) c − c (cid:22)(cid:21)(cid:23)(cid:20) (cid:0)(cid:0)(cid:64)(cid:64) c − c + 13 c − c (cid:114) (cid:114)(cid:114) (cid:114) (cid:0)(cid:0)(cid:64)(cid:64) c − c (cid:22)(cid:21)(cid:23)(cid:20) (cid:10)(cid:10)(cid:74)(cid:74) (cid:74)(cid:74)(cid:10)(cid:10) c − c + 15 c − c + 18 c − c (cid:114) (cid:114)(cid:114) (cid:114)(cid:114) (cid:114) (cid:1)(cid:1)(cid:65)(cid:65) (cid:65)(cid:65)(cid:1)(cid:1) c + 3 c − c (cid:22)(cid:21)(cid:23)(cid:20) (cid:10)(cid:10)(cid:74)(cid:74) (cid:2)(cid:2)(cid:2)(cid:66)(cid:66)(cid:66) c − c + 28 c − c + 66 c − c (cid:114) (cid:114)(cid:114) (cid:114)(cid:114) (cid:114) (cid:1)(cid:1)(cid:65)(cid:65) (cid:65)(cid:65)(cid:1)(cid:1)(cid:33)(cid:33)(cid:33)(cid:97)(cid:97)(cid:97) c − c (cid:22)(cid:21)(cid:23)(cid:20) (cid:72)(cid:72)(cid:72)(cid:65)(cid:65)(cid:65)(cid:8)(cid:8)(cid:8) (cid:1)(cid:1)(cid:1) c − c + 115 c − c + 657 c − c (cid:114) (cid:114)(cid:114) (cid:114)(cid:114) (cid:114) (cid:1)(cid:1)(cid:65)(cid:65) (cid:65)(cid:65)(cid:1)(cid:1)(cid:33)(cid:33)(cid:33)(cid:97)(cid:97)(cid:97)(cid:28)(cid:28)(cid:28)(cid:33)(cid:33)(cid:33)(cid:92)(cid:92)(cid:92)(cid:97)(cid:97)(cid:97) c − c + 295 c Table 1: Values of the weight systems b sl ( · ), b sl ( π ( · )), R k on certain chord diagrams15nd equal to 0 otherwise. The function w C is extended to M by linearity. It is easyto show that w C satisfies the so-called 2-term relation, w C ( d ) = w C ( ˜ d AB ), for anychord diagram d and any pair of chords A and B with neighboring ends in it [2]; the4-term relation is an obvious corollary of the 2-term one. Proposition 5.2 [2] R k ( d ) = − w C ( π k ( d )) for any chord diagram d with k chords. This argument allows one to prove the following statement generalizing Proposi-tion 3.6 to arbitrary values of k . Theorem 5.3
For arbitrary k , the function R k extends to a -invariant of graphs. Indeed, the function w C is known to be extendable to a multiplicative Z -valued 4-invariant of graphs, see [5] or [11]: set w C (Γ) to be equal to 1 if the adjacency matrixof Γ is nondegenerate over Z / Z , and equal to 0 otherwise. It satisfies obviouslythe 2-term relation for graphs, w C (Γ) = w C ( (cid:101) Γ AB ), for any graph Γ and any pair ofvertices A, B in it. For intersection graphs with 2 k vertices, the projection of thisgraph invariant to primitive elements coincides with the invariant − R k . Hence, theprojection to primitive elements w C ( π k (Γ)) of this 4-invariant on arbitrary graph Γwith 2 k vertices is a 4-invariant that coincides with − R k (Γ) if Γ is an intersectiongraph. (cid:3) References [1] D. Bar-Natan,
On Vassiliev knot invariants , Topology, vol. , no. 2 (1995),423–472.[2] D. Bar-Natan, S. Garoufalidis, On the Melvin–Morton–Rozansky Conjecture , In-ventiones Mathematicae, vol. (1996), 103–133.[3] D. Bar-Natan, H. Vo,
Proof of a conjecture of Kulakova et al. related to the sl weight system , preprint, arXiv:1401.0754 [math.QA] (2014).[4] S. Chmutov, S. Duzhin, Y. Mostovoy, Introduction to Vassiliev KnotInvariants , Cambridge University Press, 2012, ISBN 978-1-107-02083-2.
Mutant knots and intersection graphs , Algebraicand Geometric Topology, vol. (2007), 101–120.[6] S. V. Chmutov, A. N. Varchenko, Remarks on the Vassiliev knot invariants com-ing from sl , Topology, vol. (1997), 153–178.[7] M. Kontsevich, Vassiliev knot invariants , in: Adv. in Soviet Math., vol. (1993), part 2, 137–150.[8] S. K. Lando, On primitive elements in the bialgebra of chord diagrams , in: Amer.Math. Soc. Transl. Ser. 2, AMS, Providence RI, 1997, vol. , 167–174.[9] S. K. Lando,
On a Hopf algebra in graph theory , J. Comb. Theory, Ser. B, vol. (2000), 104–121.[10] S. Lando, A. Zvonkin, Graphs on surfaces and their applications (Chapter 6),Springer, 2004.[11] B. Mellor,
A few weight systems arising from intersection graphs , Michigan Math.J., vol. , no. 3 (2003), 509–536.National Research University Higher School of Economics7 Vavilova Moscow 117312 RussiaE. Kulakova [email protected]. Lando [email protected]. Mukhutdinova [email protected]. Rybnikov [email protected] (cid:115)(cid:115) (cid:115)(cid:115) (cid:115) (cid:8)(cid:8)(cid:8)(cid:72)(cid:72)(cid:72) (cid:64)(cid:64)(cid:0)(cid:0) (cid:72)(cid:72)(cid:72)(cid:8)(cid:8)(cid:8)(cid:67)(cid:67)(cid:67) (cid:3)(cid:3)(cid:3) − A B (cid:115)(cid:115)(cid:115) (cid:115)(cid:115) (cid:115) (cid:8)(cid:8)(cid:8)(cid:72)(cid:72)(cid:72) (cid:64)(cid:64)(cid:0)(cid:0) (cid:72)(cid:72)(cid:72)(cid:8)(cid:8)(cid:8)(cid:67)(cid:67)(cid:67) (cid:3)(cid:3)(cid:3) = A B (cid:115)(cid:115)(cid:115) (cid:115)(cid:115) (cid:115) (cid:8)(cid:8)(cid:8)(cid:72)(cid:72)(cid:72) (cid:64)(cid:64)(cid:17)(cid:17)(cid:17)(cid:17)(cid:72)(cid:72)(cid:72)(cid:8)(cid:8)(cid:8)(cid:67)(cid:67)(cid:67) (cid:3)(cid:3)(cid:3) − A B (cid:115)(cid:115)(cid:115) (cid:115)(cid:115) (cid:115) (cid:8)(cid:8)(cid:8)(cid:72)(cid:72)(cid:72) (cid:64)(cid:64)(cid:17)(cid:17)(cid:17)(cid:17)(cid:72)(cid:72)(cid:72)(cid:8)(cid:8)(cid:8)(cid:67)(cid:67)(cid:67) (cid:3)(cid:3)(cid:3)
A B − − − (cid:115)(cid:115)(cid:115) (cid:115)(cid:115) (cid:115) (cid:8)(cid:8)(cid:8)(cid:72)(cid:72)(cid:72) (cid:64)(cid:64)(cid:0)(cid:0) (cid:72)(cid:72)(cid:72)(cid:8)(cid:8)(cid:8)(cid:67)(cid:67)(cid:67) (cid:3)(cid:3)(cid:3) − A B (cid:115)(cid:115)(cid:115) (cid:115)(cid:115) (cid:115) (cid:8)(cid:8)(cid:8)(cid:72)(cid:72)(cid:72) (cid:64)(cid:64)(cid:72)(cid:72)(cid:72)(cid:8)(cid:8)(cid:8)(cid:67)(cid:67)(cid:67) (cid:3)(cid:3)(cid:3) = A B (cid:115)(cid:115)(cid:115) (cid:115)(cid:115) (cid:115) (cid:8)(cid:8)(cid:8)(cid:72)(cid:72)(cid:72) (cid:64)(cid:64)(cid:0)(cid:0)(cid:2)(cid:2)(cid:2)(cid:2)(cid:17)(cid:17)(cid:17)(cid:17)(cid:72)(cid:72)(cid:72)(cid:8)(cid:8)(cid:8) (cid:3)(cid:3)(cid:3) − A B (cid:115)(cid:115)(cid:115) (cid:115)(cid:115) (cid:115) (cid:8)(cid:8)(cid:8)(cid:72)(cid:72)(cid:72) (cid:64)(cid:64)(cid:17)(cid:17)(cid:17)(cid:17)(cid:2)(cid:2)(cid:2)(cid:2)(cid:72)(cid:72)(cid:72)(cid:8)(cid:8)(cid:8) (cid:3)(cid:3)(cid:3)
A B − − (cid:115)(cid:115)(cid:115) (cid:115)(cid:115) (cid:115) (cid:8)(cid:8)(cid:8)(cid:72)(cid:72)(cid:72) (cid:64)(cid:64)(cid:0)(cid:0) (cid:72)(cid:72)(cid:72)(cid:8)(cid:8)(cid:8)(cid:67)(cid:67)(cid:67) (cid:3)(cid:3)(cid:3) − B A (cid:115)(cid:115)(cid:115) (cid:115)(cid:115) (cid:115) (cid:8)(cid:8)(cid:8)(cid:72)(cid:72)(cid:72) (cid:64)(cid:64)(cid:72)(cid:72)(cid:72)(cid:8)(cid:8)(cid:8)(cid:67)(cid:67)(cid:67) (cid:3)(cid:3)(cid:3) = B A (cid:115)(cid:115)(cid:115) (cid:115)(cid:115) (cid:115) (cid:8)(cid:8)(cid:8)(cid:0)(cid:0) (cid:72)(cid:72)(cid:72)(cid:8)(cid:8)(cid:8)(cid:67)(cid:67)(cid:67) (cid:3)(cid:3)(cid:3) − B A (cid:115)(cid:115)(cid:115) (cid:115)(cid:115) (cid:115) (cid:8)(cid:8)(cid:8)(cid:72)(cid:72)(cid:72)(cid:8)(cid:8)(cid:8)(cid:67)(cid:67)(cid:67) (cid:3)(cid:3)(cid:3)
B A − − (cid:115)(cid:115)(cid:115) (cid:115)(cid:115) (cid:115) (cid:8)(cid:8)(cid:72)(cid:72) (cid:8)(cid:8)(cid:72)(cid:72) − A B (cid:115)(cid:115)(cid:115) (cid:115)(cid:115) (cid:115) (cid:8)(cid:8)(cid:72)(cid:72) (cid:8)(cid:8)(cid:72)(cid:72) = A B (cid:115)(cid:115)(cid:115) (cid:115)(cid:115) (cid:115) (cid:8)(cid:8)(cid:72)(cid:72) (cid:8)(cid:8)(cid:72)(cid:72)(cid:65)(cid:65)(cid:65)(cid:65) (cid:1)(cid:1)(cid:1)(cid:1) − A B (cid:115)(cid:115)(cid:115) (cid:115)(cid:115) (cid:115) (cid:8)(cid:8)(cid:72)(cid:72) (cid:8)(cid:8)(cid:72)(cid:72)(cid:65)(cid:65)(cid:65)(cid:65) (cid:1)(cid:1)(cid:1)(cid:1)
A B − − (cid:115)(cid:115)(cid:115) (cid:115)(cid:115) (cid:115) (cid:8)(cid:8)(cid:72)(cid:72) (cid:8)(cid:8)(cid:72)(cid:72) − A B (cid:115)(cid:115)(cid:115) (cid:115)(cid:115) (cid:115) (cid:8)(cid:8)(cid:72)(cid:72) (cid:8)(cid:8) = A B (cid:115)(cid:115)(cid:115) (cid:115)(cid:115) (cid:115) (cid:72)(cid:72) (cid:8)(cid:8)(cid:72)(cid:72) (cid:1)(cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1)(cid:1) − A B (cid:115)(cid:115)(cid:115) (cid:115)(cid:115) (cid:115) (cid:72)(cid:72) (cid:8)(cid:8) (cid:1)(cid:1)(cid:1)(cid:1)
A B − − − R is indicated18 sl (2) (cid:18) (cid:114)(cid:114)(cid:114) (cid:114)(cid:114) (cid:114) (cid:19) − b sl (2) (cid:18) (cid:114)(cid:114)(cid:114) (cid:114)(cid:114) (cid:114) (cid:19) − b sl (2) (cid:18) (cid:114)(cid:114)(cid:114) (cid:114)(cid:114) (cid:114) (cid:19) + b sl (2) (cid:18) (cid:114)(cid:114)(cid:114) (cid:114)(cid:114) (cid:114) (cid:19) = b sl (2) (cid:18) (cid:114)(cid:114) (cid:114)(cid:114) (cid:19) − b sl (2) (cid:18) (cid:114) (cid:114)(cid:114) (cid:114) (cid:19) ; b sl (2) (cid:18) (cid:114)(cid:114)(cid:114) (cid:114)(cid:114) (cid:114) (cid:19) − b sl (2) (cid:18) (cid:114)(cid:114)(cid:114) (cid:114)(cid:114) (cid:114) (cid:19) − b sl (2) (cid:18) (cid:114)(cid:114)(cid:114) (cid:114)(cid:114) (cid:114) (cid:19) + b sl (2) (cid:18) (cid:114)(cid:114)(cid:114) (cid:114)(cid:114) (cid:114) (cid:19) = b sl (2) (cid:18) (cid:114)(cid:114) (cid:114)(cid:114) (cid:19) − b sl (2) (cid:18) (cid:114) (cid:114)(cid:114) (cid:114) (cid:19) . Figure 9: The recurrence relation for the complete slsl