The Number of

Independent Vassiliev Invariants in

the Homfly and Kauffman Polynomials

Jens Lieberum

Received: February 17, 2000

Communicated by Günter M. Ziegler

Abstract. We consider vector spaces H_{n,ℓ} and F_{n,ℓ} spanned by the degree-n coefficients in power series forms of the Homfly and Kauffman polynomials of links with ℓ components. Generalizing previously known formulas, we determine the dimensions of the spaces H_{n,ℓ}, F_{n,ℓ} and H_{n,ℓ} + F_{n,ℓ} for all values of n and ℓ. Furthermore, we show that for knots the algebra generated by ⨁_{n} H_{n,1} + F_{n,1} is a polynomial algebra with dim(H_{n,1} + F_{n,1}) − 1 = n + [n/2] − 4 generators in degree n ≥ 4 and one generator in degrees 2 and 3.

1991 Mathematics Subject Classification: 57M25.

Keywords and Phrases: Vassiliev invariants, link polynomials, Brauer algebra, Vogels algebra, dimensions.

1 Introduction

Soon after the discovery of the Jones polynomial V ([Jon]), two 2-parameter generalizations of it were introduced: the Homfly polynomial H ([HOM]) and the Kauffman polynomial F ([Ka2]) of oriented links. Let V_{n,ℓ} be the vector space of ℚ-valued Vassiliev invariants of degree n of links with ℓ components. After a substitution of parameters, the polynomial H (resp. F) can be written as a power series in an indeterminate h, such that the coefficient of h^{n} is a polynomial-valued Vassiliev invariant p_{n} (resp. q_{n}) of degree n. Let H_{n,ℓ} (resp. F_{n,ℓ}) be the vector space generated by the coefficients of p_{n} (resp. q_{n}) regarded as a subspace of V_{n,ℓ}. The dimensions of H_{n,ℓ} and F_{n,ℓ} have been determined in [Men] for n ≥ 0 and ℓ = 1 and partial results were also known for ℓ > 1. We complete these formulas by calculating dim H_{n,ℓ}, dim F_{n,ℓ} and dim(H_{n,ℓ} + F_{n,ℓ}) for n ≥ 0 and all pairs (n, ℓ).

Theorem 1. (1) For all n, ℓ ≥ 1 we have

dim H_{n,ℓ} = min \{n, [\frac{n − 1 + ℓ}{2}]\} = \{ \begin{matrix}n & if n < ℓ, \\ [\frac{n−1+ℓ}{2}] & if n ≥ ℓ. \end{matrix}

(2) If n ≥ 4, then

dim F_{n,ℓ} = \{ \begin{matrix}n − 1 & if ℓ = 1, \\ 2n − 1 & if ℓ ≥ 2 and n ≤ ℓ, \\ n + ℓ − 1 & if ℓ ≥ 2 and n ≥ ℓ. \end{matrix}

The values of dim F_{n,ℓ} for n ≤ 3 are given in the following table

\begin{matrix}(n, ℓ) & (1, 1) & (1, ≥ 2) & (2, 1) & (2, 2) & (2, ≥ 3) & (3, 1) & (3, 2) & (3, ≥ 3) \\ dim F_{n,ℓ} & 0 & 1 & 1 & 2 & 3 & 1 & 4 & 5 \end{matrix}

(3) For all n, ℓ ≥ 1 we have

dim(H_{n,ℓ} ∩ F_{n,ℓ}) = min{dim H_{n,ℓ}, 2}.

In the framework of Vassiliev invariants it is natural to consider the elements of ⨁_{n,ℓ}(H_{n,ℓ} ∩ F_{n,ℓ}) as the common specializations of H and F. It is known that a one-variable polynomial Y ([CoG], [Kn1], [Lik], [Lie], [Sul]) appears as a lowest coefficient in H and F. This is used in the proof of Theorem 1 to derive lower bounds for dim(H_{n,ℓ} ∩ F_{n,ℓ}). Let r_{n}^{ℓ} be the coefficient of h^{n} in the Jones polynomial V (e^{h/2}) and let y_{n}^{ℓ} be the coefficient of h^{n} in Y (e^{h/2}). Then we have r_{n}^{ℓ}, y_{n}^{ℓ} ∈ H_{n,ℓ} ∩ F_{n,ℓ}. The following corollary to the proof of Theorem 1 says that the Jones polynomial V and the polynomial Y are the only common specializations of H and F in the sense above (compare [Lam] for common specializations in a different sense).

Corollary 2. For all n ≥ 0, ℓ ≥ 1 we have H_{n,ℓ} ∩ F_{n,ℓ} = span{r_{n}^{ℓ}, y_{n}^{ℓ}}.

The main part of the proofs of Theorem 1 and Corollary 2 will not be given on the level of link invariants, but on the level of weight systems. A weight system of degree n is a linear form on a space Ā_{n,ℓ} generated by certain trivalent graphs with ℓ distinguished oriented circles and 2n vertices called trivalent diagrams. There exists a surjective map W from V_{n,ℓ} to the space Ā_{n,ℓ}^{∗} = Hom(Ā_{n,ℓ}, ℚ) of weight systems. The restriction of W to H_{n,ℓ} + F_{n,ℓ} is injective. So we may study the spaces H′_{n,ℓ} = W (H_{n,ℓ}) and F′_{n,ℓ} = W (F_{n,ℓ}) ⊆ Ā_{n,ℓ}^{∗} instead of H_{n,ℓ} and F_{n,ℓ}. Using an explicit description of weight systems in H′_{n,ℓ} and F′_{n,ℓ} we derive upper bounds for dim H′_{n,ℓ} and dim F′_{n,ℓ}. We obtain an upper bound for dim(H′_{n,ℓ} + F′_{n,ℓ}) from a lower bound for dim(H′_{n,ℓ} ∩ F′_{n,ℓ}). We evaluate the weight systems in H′_{n,ℓ} and F′_{n,ℓ} on many trivalent diagrams which gives us lower bounds for dim H′_{n,ℓ}, dim F′_{n,ℓ} and dim(H′_{n,ℓ} +F′_{n,ℓ}). These lower bounds always coincide with the upper bounds. The resulting dimension formulas will imply Theorem 1.

Number of Independent Vassiliev Invariants in H and F 277

For simplicity of notation we will drop the index ℓ when ℓ = 1. The fact that the Jones polynomial and the square of the Jones polynomial appear by choosing special values of parameters of the Kauffman polynomial gives us quadratic relations between elements of ⨁_{n=0}^{∞} F_{n,ℓ}. We will use the Hopf algebra structure of Ā = ⨁_{n=0}^{∞} Ā_{n} to show that we know all algebraic relations between elements of ⨁_{n=0}^{∞} H_{n} + F_{n}:

Theorem 3. The algebra generated by ⨁_{n=0}^{∞} H_{n} + F_{n} is a polynomial algebra with

max{dim(H_{n} + F_{n}) − 1, 1} = max{n + [n/2] − 4, 1}

generators in degree n ≥ 2.

If knot invariants v_{i} satisfy v_{i}(K_{1}) = v_{i}(K_{2}), then polynomials in the invariants v_{i} also cannot distinguish the knots K_{1} and K_{2}. By Theorem 3 there is only one algebraic relation between elements v_{i} ∈ ⨁_{n=1}^{m−1}(H_{n} + F_{n}) and elements of H_{m} + F_{m} in each degree m ≥ 4. This gives us a hint why it is possible to distinguish many knots by comparing their Homfly and Kauffman polynomials. The plan of the paper is the following. In Section 2 we recall the definitions of the link polynomials H, F, V, Y, and we give the exact definitions of H_{n,ℓ} and F_{n,ℓ}. Then we express relations between these polynomials in terms of Vassiliev invariants. In Section 3 we define Ā_{n,ℓ} and recall the connection between the Vassiliev invariants in H_{n,ℓ} + F_{n,ℓ} and their weight systems in H′_{n,ℓ} + F′_{n,ℓ}. In Section 4 we use a direct combinatorial description of the weight systems in H′_{n,ℓ} and F′_{n,ℓ} to derive upper bounds for dim H′_{n,ℓ} and dim F′_{n,ℓ}. For the proof of lower bounds we state formulas for values of weight systems in H′_{n,ℓ} and F′_{n,ℓ} on certain trivalent diagrams in Section 5. We prove these formulas by making calculations in the Brauer algebra Br_{k}. In Section 6 we complete the proofs of Theorem 1, Corollary 2 and Theorem 3 by using a module structure on the space of primitive elements P of Ā over Vogel’s algebra Λ ([Vog]).

Acknowledgements

I would like to thank C.-F. Bödigheimer, C. Kassel, J. Kneissler, T. Mennel, H. R. Morton, and A. Stoimenov for helpful remarks and discussions that influenced an old version of this article entitled ”The common specializations of the Homfly and Kauffman polynomials”. I thank the Graduiertenkolleg for mathematics of the University of Bonn, the German Academic Exchange Service, and the Schweizerischer Nationalfonds for financial support.

2 Vassiliev invariants and link polynomials

A singular link is an immersion of a finite number of oriented circles into ℝ^{3} whose only singularities are transversal double points. A singular link without double points is called a link. We consider singular links up to orientation preserving diffeomorphisms of ℝ^{3}. The equivalence classes of this equivalence relation are called singular link types or by abuse of language simply singular links. A link invariant is a map from link types into a set. If v is a link invariant with values in an abelian group, then it can be extended recursively to an invariant of singular links by the local replacement rule v(L_{×}) = v(L_{+})−v(L_{−}) (see Figure 1). A link invariant is called a Vassiliev invariant of degree n if it vanishes on all singular links with n + 1 double points. Let V_{n,ℓ} be the vector space of ℚ-valued Vassiliev invariants of degree n of links with ℓ components.

[figure: L_{×}; L_{+}; L_{−}; L_{||}; L_{=}]

Figure 1: Local modifications (of a diagram) of a (singular) link

Let us recall the definitions of the link invariants H,F,V, and Y (see [HOM], [Ka2], [Jon], and Proposition 4.7 of [Lik]; the normalizations of H and V we will use are equivalent to the original definitions). For a link L, the Homfly polynomial H_{L}(x, y) ∈ ℤ[x^{±1}, y^{±1}] is given by

xH_{L_{+}}(x, y) − x^{−1}H_{L_{−}}(x, y) = yH_{L_{||}}(x, y),  (1)
H_{O^{k}}(x, y) = (\frac{x − x^{−1}}{y} )^{k} .  (2)

The links in Equation (1) are the same outside of a small ball and differ inside this ball as shown in Figure 1. The symbol O^{k} denotes the trivial link with k ≥ 1 components. A link diagram L ⊂ ℝ^{2} is a generic projection of a link together with the information which strand is the overpassing strand at each double point of the projection. Call a crossing of a link diagram as in L_{+} (see Figure 1) positive and a crossing as in L_{−} negative. Define the writhe w(L) of a link diagram L as the number of positive crossings minus the number of negative crossings. Similar to the Homfly polynomial, the Dubrovnik version of the Kauffman polynomial F_{L}(x, y) ∈ ℤ[x^{±1}, y^{±1}] of a link diagram L is given by

xF_{L_{+}}(x, y) − x^{−1}F_{L_{−}}(x, y) = y (F_{L_{||}}(x, y) − x^{w(L_{=,or})−w(L_{||})}F_{L_{=,or}}(x, y)) , (3)
F_{O^{k}}(x, y) = (\frac{x − x^{−1} + y}{y} )^{k} .  (4)

Here the link diagrams L_{+}, L_{−}, L_{||}, L_{=} differ inside of a disk as shown in Figure 1 and coincide on the outside of this disk, and L_{=,or} is the link diagram L_{=} equipped with an arbitrary orientation of the components of the corresponding link. The symbol O^{k} denotes an arbitrary diagram of the trivial link with k ≥ 1 components. The Homfly and the Kauffman polynomials are invariants of links. Let |L| denote the number of components of a link L. For the links in Equation (1) we have |L_{+}| = |L_{−}| = |L_{||}| ± 1. Since Equations (1) and (2) are sufficient to calculate H this implies H_{L}(x, y) = (−1)^{|L|}H_{L}(x, −y) for every link L. The Jones polynomial V can be expressed in terms of the Homfly polynomial as

V_{L}(x) := H_{L} (x^{2}, x^{−1} − x) = (−1)^{|L|}H_{L} (x^{2}, x − x^{−1}) ∈ ℤ[x^{±1}].

It is easy to see that for every link L we have

H̃_{L}(x, y) := y^{|L|}H_{L}(x, y) ∈ ℤ[x^{±1}, y] , F̃_{L}(x, y) := y^{|L|}F_{L}(x, y) ∈ ℤ[x^{±1}, y].(5)

The link invariant Y is defined by

Y_{L}(x) = H̃_{L}(x, 0) ∈ ℤ[x^{±1}].

After substitutions of parameters we can express H and F as

H_{L} (e^{ch/2}, e^{h/2} − e^{−h/2}) = ∑_{j=0}^{∞} ∑_{i=1}^{j+|L|} p_{i,j}^{|L|}(L)c^{i}h^{j} ∈ ℚ[c][[h]], (6)

F_{L} (e^{(c−1)h/2}, e^{h/2} − e^{−h/2}) = ∑_{j=0}^{∞} ∑_{i=1}^{j+|L|} q_{i,j}^{|L|}(L)c^{i}h^{j} ∈ ℚ[c][[h]], (7)

for the following reasons: Equation (5) implies that the sum over i is limited by j+|L| in these expressions and one sees that no negative powers in h appear and that the sum over i starts with i = 1 directly by using the defining equations of H and F with the new parameters. For j = 0 we have p_{i,0}^{|L|} = q_{i,0}^{|L|} = δ_{i,|L|}, where δ_{i,j} is 1 for i = j and is 0 otherwise. It follows from Equations (1) and (3) that the link invariants p_{i,n}^{ℓ} and q_{i,n}^{ℓ} are in V_{n,ℓ}. Define

H_{n,ℓ} = span{p_{1,n}^{ℓ}, p_{2,n}^{ℓ}, …, p_{n+ℓ,n}^{ℓ}} ⊆ V_{n,ℓ},  (8)
F_{n,ℓ} = span{q_{1,n}^{ℓ}, q_{2,n}^{ℓ}, …, q_{n+ℓ,n}^{ℓ}} ⊆ V_{n,ℓ}.  (9)

Define the invariants y_{n}^{ℓ}, r_{n}^{ℓ} of links with ℓ components by

Y_{L} (e^{h/2}) = ∑_{n=0}^{∞} y_{n}^{|L|}(L)h^{n} ∈ ℚ[[h]],  (10)

V_{L} (e^{h/2}) = ∑_{n=0}^{∞} r_{n}^{|L|}(L)h^{n} ∈ ℚ[[h]].  (11)

In the following proposition we state the consequences of Propositions 4.7, 4.2, 4.5 of [Lik] for the versions of the Homfly and Kauffman polynomials from Equations (6) and (7).

Proposition 4. For all n ≥ 0, ℓ ≥ 1 we have

(1) y_{n}^{ℓ} = p_{n+ℓ,n}^{ℓ} = q_{n+ℓ,n}^{ℓ},
(2) r_{n}^{ℓ} = (−1)^{ℓ} ∑_{i=1}^{n+ℓ} 2^{i}p_{i,n}^{ℓ} = (−1/2)^{n} ∑_{i=1}^{n+ℓ}(−2)^{i}q_{i,n}^{ℓ},

(3) (−2)^{n} ∑_{i=1}^{n+ℓ} 4^{i}q_{i,n}^{ℓ} = ∑_{m=0}^{n} ∑_{i=1}^{m+ℓ} ∑_{j=1}^{n−m+ℓ}(−2)^{i+j}q_{i,m}^{ℓ}q_{j,n−m}^{ℓ}.

Sketch of Proof. (1) The following formulas for Y can directly be derived from its definition:

(a) xY_{L_{+}}(x) − x^{−1}Y_{L_{−}}(x) = Y_{L_{||}}(x) if |L_{+}| < |L_{||}|,
(b)  xY_{L_{+}}(x) = x^{−1}Y_{L_{−}}(x)  if |L_{+}| > |L_{||}|,
(c)  Y_{O^{k}}(x) = (x − x^{−1})^{k}.

These relations are sufficient to calculate Y_{L}(x) for every link L. The link invariant Y′_{L}(x) := F̃_{L}(x, 0) satisfies the same Relations (a), (b), (c) as Y, hence we have H̃_{L}(x, 0) = Y_{L}(x) = Y′_{L}(x) = F̃_{L}(x, 0). Now the formulas

H̃_{L} (e^{h/2}, 0) = ∑_{n=0}^{∞} p_{n+|L|,n}^{|L|}(L)h^{n} and F̃_{L} (e^{h/2}, 0) = ∑_{n=0}^{∞} q_{n+|L|,n}^{|L|}(L)h^{n}

imply Part (1) of the proposition.

(2) Let < L > (A) be the Kauffman bracket (see [Ka1], [Ka3]) defined by

< [figure] > = A <  > + A^{−1} <  > , < O^{k} >= (−A^{2} − A^{−2})^{k}.

For a link diagram L define the link invariant f_{L}(A) with values in ℤ[A^{2}, A^{−2}] by f_{L}(A) = (−A^{3})^{−w(L)} < L > (A), where w(L) denotes the writhe of L. Then one can show that

F_{L} (e^{−3h/2}, e^{h/2} − e^{−h/2}) = f_{L} (−e^{−h/2}) = f_{L} (e^{−h/2})  and
(−1)^{|L|}H_{L} (e^{h}, e^{h/2} − e^{−h/2}) = V_{L} (e^{h/2}) = f_{L} (e^{h/4}) .

This implies Part (2) of the proposition.

(3) With the notation of Part (2) of the proof we have

F_{L}(B^{3}, B − B^{−1}) = f_{L}(−A^{−1})^{2} = F_{L}(A^{−3}, A − A^{−1})^{2}, where B = A^{−2}.

Substituting A = e^{h/2} and B = e^{ℏ/2} with ℏ = −2h and comparing with Equation (7) gives us Part (3) of the proposition.

Parts (1) and (2) of Proposition 4 imply that r_{n}^{ℓ}, y_{n}^{ℓ} ∈ H_{n,ℓ} ∩ F_{n,ℓ}. In other words, the polynomials V and Y are common specializations of H and F. This was the easy part of the proofs of Theorem 1 and Corollary 2. Part (3) of Proposition 4 will be used in the proof of Theorem 3.

3 Spaces of weight systems

We recall the following from [BN1]. A trivalent diagram is an unoriented graph with ℓ ≥ 1 disjointly embedded oriented circles such that every connected component of this graph contains at least one oriented circle, every vertex has valency three, and the vertices that do not lie on an oriented circle have a cyclic orientation. We consider trivalent diagrams up to homeomorphisms of graphs that respect the additional data. The degree of a trivalent diagram is defined as half of the number of its vertices. An example of a diagram on two circles of degree 8 is shown in Figure 2.

[figure]

Figure 2: A trivalent diagram

In the picture the distinguished circles are drawn with thicker lines than the remaining part of the diagrams. Orientation of circles and vertices are assumed to be counterclockwise. Crossings in the picture do not correspond to vertices of a trivalent diagram. Let A_{n,ℓ} be the ℚ-vector space generated by trivalent diagrams of degree n on ℓ oriented circles together with the relations (STU), (IHX) and (AS) shown in Figure 3. The diagrams in a relation are assumed to coincide everywhere except for the parts we have shown. Let Ā_{n,ℓ} be the quotient of A_{n,ℓ} by the relation (FI), also shown in Figure 3. A weight system is a linear map from Ā_{n,ℓ} to a ℚ-vector space. A chord diagram is a trivalent diagram where every trivalent vertex lies on an oriented circle. It is easy to see that A_{n,ℓ} is spanned by chord diagrams. If D is a chord diagram of degree n on ℓ oriented circles, then one can construct a singular link L_{D} with ℓ components such that the preimages of double points

(STU)-relation:  [figure] =  \overline{}− [figure]

(IHX)-relation:  [figure]  =  − [figure]
(AS)-relation (anti-symmetry):  [figure] = −
(FI)-rel. (framing-independence):  \overline{}= 0

Figure 3: (STU), (IHX), (AS) and (FI)-relation

of L_{D} correspond to the points of D connected by a chord. The singular link L_{D} described above is not uniquely determined by D, but, if v ∈ V_{n,ℓ}, then the linear map W (v) : Ā_{n,ℓ} ⟶ ℚ which sends D to v(L_{D}) is well-defined. This defines a linear map W : V_{n,ℓ} ⟶ Hom(Ā_{n,ℓ}, ℚ) = Ā_{n,ℓ}^{∗}. Let us define the spaces

H′_{n,ℓ} = W (H_{n,ℓ})  and  F′_{n,ℓ} = W (F_{n,ℓ}) ⊆ Ā_{n,ℓ}^{∗}.

If v_{1} ∈ V_{n,ℓ} and v_{2} ∈ V_{m,ℓ}, then the link invariant v_{1}v_{2} defined by (v_{1}v_{2})(L) = v_{1}(L)v_{2}(L) is in V_{n+m,ℓ}. Weight systems are multiplied by using the algebra structure dual to the coalgebra structure of ⨁_{n=0}^{∞} Ā_{n,ℓ} (see [BN1]). The following proposition is a well-known consequence of a theorem of Kontsevich (see Proposition 2.9 of [BNG] and Theorem 7.2 of [KaT], Theorem 10 of [LM3] or [LM1], [LM2]).

Proposition 5. For all ℓ ≥ 1 there exists an isomorphism of algebras

Z^{∗} : ⨁_{n=0}^{∞} Ā_{n,ℓ}^{∗} ⟶ ⋃_{n=0}^{∞} V_{n,ℓ}

such that for all n ≥ 0 we have

Z^{∗} ∘ W_{|(H_{n,ℓ}+F_{n,ℓ})} = id_{(H_{n,ℓ}+F_{n,ℓ})}.

This proposition reduces the study of H_{n,ℓ} and F_{n,ℓ} to that of H′_{n,ℓ} and F′_{n,ℓ}: we have the following corollary.

Corollary 6. For all n ≥ 0 and ℓ ≥ 1 we have

dim H′_{n,ℓ} = dim H_{n,ℓ},
dim F′_{n,ℓ} = dim F_{n,ℓ},
dim(H′_{n,ℓ} + F′_{n,ℓ}) = dim(H_{n,ℓ} + F_{n,ℓ}).

We will often use Corollary 6 without referring to it.

4 Upper bounds for dim H′_{n,ℓ} and dim F′_{n,ℓ}

Let us recall the explicit descriptions of W (p_{i,j}^{ℓ}) and W (q_{i,j}^{ℓ}) from [BN1]. Let D be a trivalent diagram. Cut it into pieces along small circles around each vertex. Then replace the simple parts as shown in Figure 4.

⇝  ⇝  [figure] ⇝  [figure]− [figure]

Figure 4: The map W_{𝔤𝔩}

Glue the substituted parts together. Sums of parts of diagrams are glued together after multilinear expansion. The result is a linear combination of unions of circles. Replace each circle by a formal parameter c and call the resulting polynomial W_{𝔤𝔩}(D). It is well-known that this procedure determines a linear map W_{𝔤𝔩} : A_{n,ℓ} ⟶ ℚ[c] (see [BN1], Exercise 6.36). Proceeding with the replacement patterns shown in Figure 5, we get the linear map W_{𝔰𝔬} : A_{n,ℓ} ⟶ ℚ[c].

⇝  ⇝  − [figure]  [figure] ⇝  [figure]

Figure 5: The map W_{𝔰𝔬}

For a trivalent diagram D, define the linear combination of trivalent diagrams ι(D) by replacing each chord as shown in Figure 6. Connected components of D \ S^{1}^{⨿ℓ} with an internal trivalent vertex stay as they are.

[figure] ⇝ [figure]− \frac{1}{2} ( [figure]+ [figure])

Figure 6: The deframing map ι

This definition determines a linear map ι : Ā_{n,ℓ} ⟶ A_{n,ℓ}, such that π ∘ ι = id where π : A_{n,ℓ} ⟶ Ā_{n,ℓ} denotes the canonical projection (compare [BN1], Exercise 3.16). By the following proposition ([BN1], Chapter 6.3) the weight systems \overline{W}_{𝔤𝔩} = W_{𝔤𝔩} ∘ ι and \overline{W}_{𝔰𝔬} = W_{𝔰𝔬} ∘ ι belong to the Homfly and Kauffman polynomials.

Proposition 7. For all n ≥ 0, i, ℓ ≥ 1 the weight system W (p_{i,n}^{ℓ}) (resp. W (q_{i,n}^{ℓ})) is equal to the coefficient of c^{i} in \overline{W}_{𝔤𝔩}_{|Ā_{n,ℓ}} (resp. \overline{W}_{𝔰𝔬}_{|Ā_{n,ℓ}}).

The direct description of W (p_{i,n}^{ℓ}) and W (q_{i,n}^{ℓ}) from the proposition above will simplify the computation of dimensions.

Lemma 8. (1) For all n, ℓ ≥ 1 we have

dim H′_{n,ℓ} ≤ \{ \begin{matrix}n & if n < ℓ, \\ [\frac{n−1+ℓ}{2}] & if n ≥ ℓ. \end{matrix}

(2) For all n, ℓ ≥ 1 we have

dim F′_{n,ℓ} ≤ \{ \begin{matrix}n − 1 & if ℓ = 1, \\ 2n − 1 & if ℓ ≥ 2 and n ≤ ℓ, \\ n + ℓ − 1 & if ℓ ≥ 2 and n ≥ ℓ. \end{matrix}

Proof. In the proof D will denote a chord diagram of degree n ≥ 1 on ℓ circles.

(1) If n ≥ ℓ, then we get [(n − 1 + ℓ)/2] as an upper bound for dim H′_{n,ℓ} by the following observations:

(a) The polynomial \overline{W}_{𝔤𝔩}(D) has degree ≤ n + ℓ and vanishing constant term because the number of circles can at most increase by one with each replacement of a chord as shown in Figure 4, and there remains always at least one circle.

(b) The coefficients of c^{n+ℓ−1−2i} (i = 0, 1, …) vanish because the number of circles changes by ±1 with each replacement of a chord as shown in Figure 4.

(c) We have \overline{W}_{𝔤𝔩}(D)(1) = 0 because W_{𝔤𝔩}(D′)(1) = 1 for each chord diagram D′ and ι(D) is a linear combination of chord diagrams D′ having 0 as sum of their coefficients. If D is a chord diagram of degree n < ℓ, then by similar arguments \overline{W}_{𝔤𝔩}(D) is a linear combination of c^{ℓ−n}, c^{ℓ−n+2}, …, c^{ℓ+n} with \overline{W}_{𝔤𝔩}(D)(1) = 0. This implies the upper bound for dim H′_{n,ℓ}.

(2) If n ≥ ℓ, then by the same arguments as above \overline{W}_{𝔰𝔬}(D) is a polynomial of degree ≤ n + ℓ with vanishing constant term and \overline{W}_{𝔰𝔬}(D)(1) = 0. This implies dim F′_{n,ℓ} ≤ n + ℓ − 1 in this case. If ℓ = 1, then for chord diagrams D′ of degree n the value W_{𝔰𝔬}(D′)(2) is constant because 𝔰𝔬_{2} is an abelian Lie algebra (see [BN1]). This implies \overline{W}_{𝔰𝔬}(D)(2) = 0 and hence dim F′_{n,ℓ} ≤ n − 1 in this case. If ℓ ≥ 2 and n < ℓ, then the coefficient of c^{ℓ−n} in \overline{W}_{𝔰𝔬}(D) is 0 by the following argument: Assume that a chord diagram D′ has the minimal possible number of ℓ − n connected components (in other words, if we contract the oriented circles of D′ to points, then the resulting graph is a forest). Then we see that W_{𝔰𝔬}(D′) = 0 by using Figure 5. Hence \overline{W}_{𝔰𝔬}(D) is a linear combination of c^{ℓ−n+1}, c^{ℓ−n+2}, …, c^{ℓ+n} with \overline{W}_{𝔰𝔬}(D)(1) = 0. This completes the proof of the upper bounds for dim F′_{n,ℓ}.

5 The Brauer algebra and values of \overline{W}_{𝔤𝔩} and \overline{W}_{𝔰𝔬}

In order to find lower bounds for dim H′_{n,ℓ}, dim F′_{n,ℓ} and dim(H′_{n,ℓ} + F′_{n,ℓ}), we shall evaluate the weight systems \overline{W}_{𝔤𝔩} and \overline{W}_{𝔰𝔬} on sufficiently many trivalent diagrams. Let ω_{k}, L_{k}, C_{k}, T_{k} be the diagrams of degree k shown in Figure 7.

\begin{matrix}ω_{k} = [figure: ...] & L_{k} = [figure: ⋯] \\ T_{k} = [figure: ⋮] & C_{k} = [figure: ⋯] \end{matrix}

Figure 7: The diagrams ω_{k}, L_{k}, C_{k}, T_{k}

For technical reasons we extend this definition by setting L_{0} = C_{0} = T_{0} = S^{1} and C_{1} = L_{1}. An important ingredient in the proofs of Theorems 1 and 3 is the following lemma.

Lemma 9. (1) For all k ≥ 2 we have

\overline{W}_{𝔤𝔩}(ω_{k}) = \{ \begin{matrix}c^{k+1} + c^{3} − 2c & if k is even, \\ c^{k+1} − c^{2} & if k is odd, and \end{matrix}
\overline{W}_{𝔰𝔬}(ω_{k}) = c(c − 1)(c − 2)R_{k}(c),

where R_{k} is a polynomial with R_{k}(0) ≠ 0. If k = 2, then R_{2} = 2, and if k ≠ 3, then R_{k}(2) ≠ 0.

(2) For all k ≥ 1 we have

\overline{W}_{𝔤𝔩}(L_{k}) = c(1 − c^{2})^{k},  \overline{W}_{𝔰𝔬}(L_{k}) = c^{k+1}(1 − c)^{k},
\overline{W}_{𝔤𝔩}(T_{k}) = (−c)^{k}(c^{2} − 1), \overline{W}_{𝔰𝔬}(T_{k}) = c(c − 1)Q_{k}(c),
\overline{W}_{𝔰𝔬}(C_{k}) = c(c − 1)P_{k}(c),

where P_{k} and Q_{k} are polynomials in c such that for k ≥ 2 we have P_{k}(0) ≠ 0, Q_{k}(0) = 2^{k−1}, and Q_{k}(2) = (−2)^{k}.

In the proof of the lemma we will determine the polynomials P_{k}, Q_{k}, and R_{k} explicitly, which will be helpful to us for calculations in low degrees. For the main parts of the proofs of Theorems 1 and 3 it will be sufficient to know the properties of these polynomials stated in the lemma. We do not need to know the value of \overline{W}_{𝔤𝔩}(C_{k}). In the proof of Lemma 9 we use the Brauer algebra ([Bra]) on k strands Br_{k}. As a ℚ[c]-module Br_{k} has a basis in one-to-one correspondence with involutions without fixed-points of the set {1, …, k}×{0, 1}. We represent a basis element corresponding to an involution f graphically by connecting the points (i, j) and f(i, j) by a curve in ℝ × [0, 1]. Examples are the diagrams u_{−}, x_{+}, x_{−}, u_{+} = d, e, f, g, h in Figures 8 and 9.

H  ⇝  u_{+}  [figure: u_{−}]  [figure: x_{+}]  [figure: x_{−}]

Figure 8: Elements of Br_{3} needed to calculate W_{𝔰𝔬}(ω_{k})

[figure: d; e; f; g; h]

Figure 9: Diagrams needed to calculate W_{𝔤𝔩}(ω_{k})

The product of basis vectors a and b is defined graphically by placing a onto the top of b, by gluing the lower points (i, 0) of a to the upper points (i, 1) of b, and by introducing the relation that a circle is equal to the formal parameter c of the ground ring ℚ[c]. We have a map tr : Br_{k} ⟶ ℚ[c], called trace, that is defined graphically by connecting the vertices (i, 0) and (i, 1) of a diagram by curves, and by replacing each circle by the indeterminate c. As an example, the trace of the diagram x_{+}u_{−} is shown in Figure 10.

tr [figure: (; )] = [figure] = c

Figure 10: The trace of a diagram The elements u_{+}, u_{−}, x_{+}, x_{−} arise among others when the replacement rules belonging to W_{𝔰𝔬} (see Figure 5) are applied to the part H (see Figure 8) of a trivalent diagram. Similarly, the elements d and h arise when we apply the replacement rules belonging to W_{𝔤𝔩} (see Figure 4) to the part H of a trivalent diagram. We have ι(ω_{k}) = ω_{k} (see Figure 6) because the diagram ω_{k} contains no chords. The proof of the following lemma is now straightforward.

Lemma 10. The following two formulas hold:

\overline{W}_{𝔤𝔩}(ω_{k}) = tr ((d − h)^{k})  and  \overline{W}_{𝔰𝔬}(ω_{k}) = tr ((u_{+} − u_{−} + x_{+} − x_{−})^{k}) .

Now we can prove Lemma 9 by making calculations in the Brauer algebra.

Proof of Lemma 9. (1) With the elements u_{±}, x_{±} ∈ Br_{3} shown in Figure 8 we define u = u_{+} − u_{−} and x = x_{+} − x_{−}. It is easy to verify that

(u + x)u = (c − 2)u and x^{3} = x^{2} + 2x.  (12)

In view of the expression for x^{3} it is clear that x^{k} can be expressed as a linear combination of x and x^{2}:

d_{k}x^{2} + e_{k}x = x^{k}.  (13)

It can be shown by induction that the sequence of pairs (d_{k}, e_{k})_{k≥1} is given by (d_{1}, e_{1}) = (0, 1) and (d_{k+1}, e_{k+1}) = (d_{k} + e_{k}, 2d_{k}). We deduce

d_{1} − e_{1} = −1, d_{k+1} − e_{k+1} = d_{k} + e_{k} − 2d_{k} = e_{k} − d_{k}
⇒ d_{k} − e_{k} = (−1)^{k},  (14)
d_{k+1} + (−1)^{k} = (d_{k} + e_{k}) + (d_{k} − e_{k}) = 2d_{k}.  (15)

By Equations (12) and (13) we have

(u + x)^{k} = x^{k} + ∑_{i=0}^{k−1}(u + x)^{i}ux^{k−i−1}

= d_{k}x^{2} + e_{k}x + (c − 2)^{k−1}u + ∑_{i=0}^{k−2}(c − 2)^{i}(d_{k−i−1}ux^{2} + e_{k−i−1}ux).(16)

It is easy to see that

tr (x^{2}) /(c − 1) = −tr(x) = tr(u) = −tr(ux) = tr (ux^{2}) = c^{2} − c.  (17)

Applying the trace to Equation (16) yields by Lemma 10 and Equations (14) and (17):

\overline{W}_{𝔰𝔬}(ω_{k})
= (c^{2} − c) [d_{k}(c − 1) − e_{k} + (c − 2)^{k−1} + ∑_{i=0}^{k−2}(c − 2)^{i}(d_{k−i−1} − e_{k−i−1})]

= (c^{2} − c) [d_{k}(c − 2) + (−1)^{k} − ∑_{i=0}^{k−1}(−1)^{k−i}(c − 2)^{i}]

= (c^{2} − c)(c − 2) [(d_{k} + (−1)^{k}) + ∑_{i=1}^{k−2}(−1)^{k−i}(c − 2)^{i}] .

Define the sequence (a_{k})_{k≥2} inductively by a_{2} = 2 and a_{k+1} = 2a_{k} − 4(−1)^{k}. We have a_{2} = d_{2} + (−1)^{2} and by definition of a_{k}, induction and Equation (15) also

a_{k+1} = 2a_{k} − 4(−1)^{k} = 2(d_{k} + (−1)^{k}) − 4(−1)^{k} = d_{k+1} + (−1)^{k+1}.

This implies \overline{W}_{𝔰𝔬}(ω_{k}) = c(c − 1)(c − 2)R_{k}(c) with

R_{k}(c) = a_{k} + ∑_{i=1}^{k−2}(−1)^{k−i}(c − 2)^{i}.

The properties of R_{k} stated in the lemma are satisfied because by a simple computation we have R_{k}(2) = a_{k} > 0 for k ≠ 3 and

R_{k}(0) = a_{k} + (−1)^{k}(2^{k−1} − 2) ≡ 2 mod 4.

We only give a sketch of the proof of the formula for \overline{W}_{𝔤𝔩}(ω_{k}). Let d, e, f, g, h be the elements of Br_{3} shown in Figure 9. Then one can prove by induction on k that

(d − h)^{2k+1} = c^{2k}d − h + ∑_{i=0}^{k−1} c^{2i}(d + e) − c^{2i+1}(f + g).

Using Lemma 10 this formula allows to conclude by distinguishing whether k is even or odd.

(2) Let a, b, 1 be the elements of Br_{2} shown in Figure 11. Then we have ab = ba = a, a^{2} = ca, b^{2} = 1, tr(a) = tr(b) = c, tr(1) = c^{2}, and by convention (a − b)^{0} = 1. This implies for k ≥ 1 that

a =  b = [figure]  1 =

Figure 11: Diagrams in Br_{2}

\overline{W}_{𝔰𝔬}(T_{k}) = tr ((a − b + 1 − c1)^{k})
= tr [∑_{i=0}^{k} \binom{k}{i}(1 − c)^{k−i}(a − b)^{i}]

= tr \{∑_{i=0}^{k} \binom{k}{i}(1 − c)^{k−i} [(−b)^{i} + ∑_{j=1}^{i} \binom{i}{j}c^{j−1}(−1)^{i−j}a]\}

= ∑_{i=0}^{k} \binom{k}{i}(1 − c)^{k−i} [tr ((−b)^{i}) + (c − 1)^{i} − (−1)^{i}]

= ∑_{i=0}^{k} \binom{k}{i}(1 − c)^{k−i} [tr ((−b)^{i}) − (−1)^{i}]

= ∑_{0≤i≤k_{i even}} \binom{k}{i}(1 − c)^{k−i}(c^{2} − 1) + ∑_{1≤i≤k_{i odd}} \binom{k}{i}(1 − c)^{k−i}(1 − c)

= ∑_{i=0}^{k} \binom{k}{i}(1 − c)^{k−i}(c − 1)(−1)^{i} + c ∑_{0≤i≤k_{i even}} \binom{k}{i}(1 − c)^{k−i}(c − 1)

= c(c − 1) [−(−c)^{k−1} + ∑_{0≤i≤k_{i even}} \binom{k}{i}(1 − c)^{k−i}] .

Now one checks the properties of Q_{k} using the last expression for \overline{W}_{𝔰𝔬}(T_{k}). The remaining formulas follow by easy computations. For example, \overline{W}_{𝔰𝔬}(L_{k}) is given by the value of W_{𝔰𝔬} on the diagrams in ι(L_{k}) where no chord connects two different circles. Furthermore, one can show for k ≥ 2 that

\overline{W}_{𝔰𝔬}(C_{k}) = tr ((1 − b)^{k})+(1−c)\overline{W}_{𝔰𝔬}(L_{k−1}) = c(c−1) (2^{k−1} − c^{k−1}(1 − c)^{k−1}).

The property P_{k}(0) ≠ 0 from the lemma is obvious from the formula above.

6 Completion of proofs using Vogel’s algebra

In the case of diagrams on one oriented circle, the coalgebra structure of Ā = ⨁_{n=0}^{∞} Ā_{n} can be extended to a Hopf algebra structure (see [BN1]). The primitive elements P of Ā are spanned by diagrams D such that D \ S^{1} is connected, where S^{1} denotes the oriented circle of D. Vogel defined an algebra Λ which acts on primitive elements (see [Vog]). The diagrams t and x_{3} shown in Figure 12 represent elements of Λ.

t = x_{3} =

Figure 12: Elements of Λ

The space of primitive elements P of Ā becomes a Λ-module by inserting an element of Λ into a freely chosen trivalent vertex of a diagram of a primitive element. Multiplication by t increases the degree by 1 and multiplication by x_{3} increases the degree by 3. An example is shown in Figure 13.

x_{3}ω_{4} = x_{3} [figure] = [figure]

Figure 13: How P becomes a Λ–module

If D and D′ are classes of trivalent diagrams with a distinguished oriented circle modulo (STU)-relations (see Figure 3), then their connected sum D#D′ along these circles is well defined. We state in the following lemma how the weight systems \overline{W}_{𝔤𝔩} and \overline{W}_{𝔰𝔬} behave under the operations described above: Part (1) of the lemma is easy to prove; for Part (2), see Theorem 6.4 and Theorem 6.7 of [Vog].

Lemma 11. (1) Let D and D′ be chord diagrams each one having a distinguished oriented circle. Then the connected sum of D and D′ satisfies

\overline{W}_{𝔤𝔩}(D#D′) = \overline{W}_{𝔤𝔩}(D)\overline{W}_{𝔤𝔩}(D′)/c and \overline{W}_{𝔰𝔬}(D#D′) = \overline{W}_{𝔰𝔬}(D)\overline{W}_{𝔰𝔬}(D′)/c.

(2) For a primitive element p ∈ P we have:

\overline{W}_{𝔤𝔩}(tp) = c\overline{W}_{𝔤𝔩}(p),
\overline{W}_{𝔰𝔬}(tp) = c̃\overline{W}_{𝔰𝔬}(p),
\overline{W}_{𝔤𝔩}(x_{3}p) = (c^{3} + 12c)\overline{W}_{𝔤𝔩}(p),
\overline{W}_{𝔰𝔬}(x_{3}p) = (c̃^{3} − 3c̃^{2} + 30c̃ − 24)\overline{W}_{𝔰𝔬}(p),

where c̃ = c − 2.

We have the following formulas concerning spaces of weight systems restricted to primitive elements.

Proposition 12. For the restrictions of the weight systems to primitive elements of degree n ≥ 1 we have

(1)  dim H′_{n}_{|P_{n}} = dim H′_{n} = [n/2],
(2)  dim F′_{n}_{|P_{n}} = max(n − 2, [n/2]) = \{ \begin{matrix}[n/2] & if n ≤ 3, \\ n − 2 & if n ≥ 3, \end{matrix}
(3)  dim (H′_{n}_{|P_{n}} ∩ F′_{n}_{|P_{n}}) = min(2, [n/2]) = \{ \begin{matrix}[n/2] & if n ≤ 3, \\ 2 & if n ≥ 4. \end{matrix}

The proof of Proposition 12 will be given in this section together with a proof of Theorem 1. The proof is divided into several steps. If q is a polynomial, then we denote the degree of its lowest degree term by ord(q). Now we start to derive lower bounds for dimensions of spaces of weight systems.

Proof of Part (1) of Proposition 12. By Lemma 9 we have ord(\overline{W}_{𝔤𝔩}(ω_{k})) = 1 for even k. By Lemma 11 we have \overline{W}_{𝔤𝔩}(t^{k}p) = c^{k}\overline{W}_{𝔤𝔩}(p) for p ∈ P. This implies

dim (\overline{W}_{𝔤𝔩} (span\{t^{n−2}ω_{2}, t^{n−4}ω_{4}, …, t^{n−2[n/2]}ω_{2[n/2]}\})) = [n/2] ≤ dim H′_{n}_{|P_{n}}.

Since this lower bound coincides with the upper bound from Lemma 8 we have dim H′_{n}_{|P_{n}} = [n/2].

Let D_{ijk} = (L_{i}#C_{j})#T_{k} (in this definition we choose arbitrary distinguished circles of L_{i}, C_{j}, (L_{i}#C_{j}) and for further use also for D_{ijk}). Let d_{ijk} be the number of oriented circles in D_{ijk} and define D_{i,j,k}^{ℓ} = D_{ijk} ⨿ S^{1}^{⨿(ℓ−d_{ijk})} for ℓ ≥ d_{ijk}. We will make use of the formulas for \overline{W}_{𝔤𝔩}(D_{i,0,k}^{ℓ}#ω_{m}) and \overline{W}_{𝔰𝔬}(D_{i,j,k}^{ℓ}#ω_{m}) implied by Lemmas 9 and 11 throughout the rest of this section.

Proof of Part (1) of Theorem 1. For all n ≥ 1 we have [n/2] primitive elements p_{i} such that the polynomials g_{i} = \overline{W}_{𝔤𝔩}(p_{i} ⨿ S^{1}^{⨿(ℓ−1)}) are linearly independent and c^{ℓ}|g_{i} (see the proof of Part (1) of Proposition 12). Let n < ℓ. The diagrams

D_{n,0,0}^{ℓ}, D_{n−2,0,0}^{ℓ}#ω_{2}, …, D_{n−2[(n−1)/2],0,0}^{ℓ}#ω_{2[(n−1)/2]}  (18)

are mapped by \overline{W}_{𝔤𝔩} to the values

c^{ℓ−n}(1 − c^{2})^{n}, c^{ℓ−n+2}f_{2}(c), …, c^{ℓ−1}f_{[(n+1)/2]}(c)

with polynomials f_{i} satisfying f_{i}(0) = −2 (i = 2, …, [(n + 1)/2]). So in this case we have found [n/2] + [(n + 1)/2] = n linearly independent values, which is the maximal possible number (see Lemma 8). If n ≥ ℓ, then we conclude in the same way using the following list of k − n + 1 + [(n − 1)/2] = k − [n/2] elements where k = [(n + ℓ − 1)/2]:

D_{2k−n,0,0}^{ℓ}#ω_{2n−2k}, D_{2k−n−2,0,0}^{ℓ}#ω_{2n−2k+2}, …, D_{n−2[(n−1)/2],0,0}^{ℓ}#ω_{2[(n−1)/2]}. (19)

We will use the upper bounds for dim H′_{n,ℓ} and dim F′_{n,ℓ} together with the following lower bound for dim(H′_{n,ℓ}∩F′_{n,ℓ}) to get an upper bound for dim(H′_{n,ℓ}+ F′_{n,ℓ}). In the case ℓ = 1 we will argue in a similar way for the restriction of weight systems to primitive elements.

Lemma 13. For all n, ℓ ≥ 1 we have

dim (H′_{n,ℓ} ∩ F′_{n,ℓ}) ≥ min(dim H′_{n,ℓ}, 2)
= min(n, [(n − 1 + ℓ)/2], 2) = dim(span{W (r_{n}^{ℓ}), W (y_{n}^{ℓ})}).

For all n ≥ 1 we have

dim (H′_{n}_{|P_{n}} ∩ F′_{n}_{|P_{n}}) ≥ min(dim H′_{n}, 2) = min([n/2], 2).

Proof. Propositions 4 and 7 imply that the weight system W (r_{n}^{ℓ}) ∈ H′_{n,ℓ} ∩ F′_{n,ℓ} is equal to (−1)^{ℓ}\overline{W}_{𝔤𝔩}(.)(2)_{|Ā_{n,ℓ}} and the weight system W (y_{n}^{ℓ}) ∈ H′_{n,ℓ} ∩ F′_{n,ℓ} is equal to the coefficient of c^{ℓ+n} in \overline{W}_{𝔤𝔩}_{|Ā_{n,ℓ}}. By the proof of Lemma 8 we have \overline{W}_{𝔤𝔩}(D)(0) = \overline{W}_{𝔤𝔩}(D)(1) = 0 and in the weight system \overline{W}_{𝔤𝔩}_{|Ā_{n,ℓ}} the coefficients of c^{ℓ+n−1}, c^{ℓ+n−3}, … and the coefficients of c^{ℓ−n−1}, c^{ℓ−n−2}, … vanish. By Part (1) of Theorem 1 these are the only linear dependencies between the coefficients of c^{ℓ+n}, c^{ℓ+n−1}, … in the polynomial \overline{W}_{𝔤𝔩}_{|Ā_{n,ℓ}}. This implies for dim H′_{n,ℓ} = 1 that the coefficient of c^{ℓ+n} in \overline{W}_{𝔤𝔩}_{|Ā_{n,ℓ}} is not the trivial weight system and this implies for dim H′_{n,ℓ} ≥ 2 that \overline{W}_{𝔤𝔩}(.)(2)_{|Ā_{n,ℓ}} and the coefficient of c^{ℓ+n} in \overline{W}_{𝔤𝔩}_{|Ā_{n,ℓ}} are linearly independent. By Part (1) of Proposition 12 we can argue in the same way with \overline{W}_{𝔤𝔩}_{|P_{n}}. This completes the proof.

Define the weight system w = (−2)^{n}\overline{W}_{𝔰𝔬}(·)(4) − 2(−2)^{ℓ}\overline{W}_{𝔰𝔬}(·)(−2) ∈ F′_{n,ℓ}. For n ≥ 4 Lemmas 9 and 11 imply that

w (ω_{2}#(t^{n−4}ω_{2}) ⨿ S^{1}^{⨿ℓ−1}) = 18(−4)^{n}4^{ℓ−1} ≠ 0.  (20)

Part (3) of Proposition 4 together with Propositions 5 and 7 implies that

0 ≠ w ∈ ⨁_{i=1}^{n−1} F′_{i,ℓ}F′_{n−i,ℓ}.  (21)

For ℓ = 1 Equation (21) implies w(P_{n}) = 0. Therefore we have

dim F′_{n}_{|P_{n}} ≤ dim F′_{n} − 1 ≤ n − 2 for all n ≥ 4.  (22)

Since we have dim H′_{n} = dim H′_{n}_{|P_{n}} by Part (1) of Proposition 12 we know that w ∉ H′_{n} and therefore

dim(H′_{n} + F′_{n}) ≥ dim(H′_{n} + F′_{n})_{|P_{n}} + 1 for all n ≥ 4.  (23)

Let (\overline{W}_{𝔤𝔩}, \overline{W}_{𝔰𝔬}) : Ā_{n,ℓ} ⟶ ℚ[c] × ℚ[c] be defined by

(\overline{W}_{𝔤𝔩}, \overline{W}_{𝔰𝔬}) (D) = (\overline{W}_{𝔤𝔩}(D), \overline{W}_{𝔰𝔬}(D)) .

Then by Proposition 7 we have

dim(H′_{n,ℓ} + F′_{n,ℓ}) = dim ((\overline{W}_{𝔤𝔩}, \overline{W}_{𝔰𝔬})(Ā_{n,ℓ})) ,  (24)

dim (H′_{n}_{|P_{n}} + F′_{n}_{|P_{n}}) = dim ((H′_{n} + F′_{n})_{|P_{n}}) = dim ((\overline{W}_{𝔤𝔩}, \overline{W}_{𝔰𝔬})(P_{n})) . (25)

We will use Equations (24) and (25) to derive lower bounds for dim(H′_{n,ℓ}+F′_{n,ℓ}) and for dim(H′_{n} + F′_{n})_{|P_{n}}. Now we can complete the proofs of Theorem 1 and Proposition 12.

Proof of Parts (2) and (3) of Proposition 12 and Theorem 1 for ℓ = 1. Let

Σ_{7} = span{ω_{7}, tω_{6}, t^{2}ω_{5}, t^{3}ω_{4}, t^{5}ω_{2}, x_{3}ω_{4}} ⊂ P_{7}.

Define for n > 7:

Σ_{n} = \{ \begin{matrix}tΣ_{n−1} + ℚω_{n} & if n is odd, \\ tΣ_{n−1} + ℚω_{n} + ℚx_{3}ω_{n−3} & if n is even. \end{matrix}

By a calculation using Lemmas 9 and 11 we obtain

dim ((\overline{W}_{𝔤𝔩}, \overline{W}_{𝔰𝔬})(Σ_{7})) = 6.

In view of the proof of Lemma 8 we can define a polynomial-valued weight system by W̃_{𝔰𝔬}(.) = \overline{W}_{𝔰𝔬}(.)/(c(c − 1)). We used Lemma 9 and Lemma 11 to compute the degree 1 coefficients of the values of \overline{W}_{𝔤𝔩} and W̃_{𝔰𝔬} on elements of Σ_{n} stated in Table 1.

tΣ_{n−1}  \begin{matrix}ω_{n} \\ (n odd) \end{matrix}  \begin{matrix}ω_{n} \\ (n even) \end{matrix}  \begin{matrix}x_{3}ω_{n−3} \\ (n even) \end{matrix}
coeff. of c̃ in W̃_{𝔰𝔬}(·):  0  R^{n}(2)  R^{n}(2)  −24R^{n−3}(2)
coeff. of c in ^{\overline{W}_{𝔤𝔩}}(·):  0  0  −2  0

Table 1: Degree 1 coefficients of \overline{W}_{𝔤𝔩} and W̃_{𝔰𝔬} on Σ_{n}

By Lemma 9 we have R_{k}(2) ≠ 0 if k ≠ 3. Then, by Table 1 and induction, we see that dim(\overline{W}_{𝔤𝔩}, \overline{W}_{𝔰𝔬})(Σ_{n}) = [n/2] + n − 4 for n ≥ 7. By Equation (25), Lemmas 8 and 13, and Equation (22) we obtain

[n/2] + n − 4 + 2 ≤ dim (H′_{n}_{|P_{n}} + F′_{n}_{|P_{n}}) + dim (H′_{n}_{|P_{n}} ∩ F′_{n}_{|P_{n}}) =
= dim H′_{n}_{|P_{n}} + dim F′_{n}_{|P_{n}} ≤ [n/2] + n − 2.

Thus equality must hold. This implies Parts (2) and (3) of Proposition 12 for n ≥ 7. By Equation (23) we get dim(H′_{n} + F′_{n}) ≥ [n/2] + n − 3. Now we see by Lemmas 8 and 13 that

[n/2] + n − 3 + 2 ≤ dim (H′_{n} + F′_{n}) + dim (H′_{n} ∩ F′_{n}) =
= dim H′_{n} + dim F′_{n} ≤ [n/2] + n − 1

which implies Part (2) and Part (3) of Theorem 1 for n ≥ 7 and ℓ = 1. Let ψ be the element of degree 6 shown in Figure 14.

ψ = [figure]

Figure 14: A primitive element in degree 6

A calculation done by computer yields

\overline{W}_{𝔤𝔩}(ψ) = c^{7} + 13c^{5} − 14c^{3},
W̃_{𝔰𝔬}(ψ) = c̃^{5} − 3c̃^{4} + 34c̃^{3} − 36c̃^{2} + 16c̃.

Let Σ_{4} = span{ω_{4}, t^{2}ω_{2}}, Σ_{5} = tΣ_{4} + ℚω_{5}, and Σ_{6} = tΣ_{5} + ℚω_{6} + ℚψ. We obtain again dim(\overline{W}_{𝔤𝔩}, \overline{W}_{𝔰𝔬})(Σ_{n}) = [n/2]+n−4 which implies Parts (2) and (3) of Proposition 12 and Theorem 1 for ℓ = 1 and n ≥ 4 by the same argument as before. In degrees n = 1, 2, 3 we have dim P_{n} = dim Ā_{n} = dim H′_{n} = dim F′_{n} = [n/2]. This completes the proof.

Proof of Parts (2) and (3) of Theorem 1 for ℓ > 1. Let n ≥ 4 and ℓ > 1. By the previous proof we have n + [n/2] − 3 elements a_{i} ∈ Ā_{n} such that the values

(\overline{W}_{𝔤𝔩}(D_{i}), \overline{W}_{𝔰𝔬}(D_{i})) ∈ ℚ[c] × ℚ[c]

of D_{i} = a_{i} ⨿ S^{1}^{⨿ℓ−1} are linearly independent. Consider the following lists of elements:

If n ≤ ℓ, then we take the n elements

D_{0,n,0}^{ℓ}, D_{1,n−1,0}^{ℓ}, …, D_{n−3,3,0}^{ℓ}, D_{0,0,n}^{ℓ}, E_{n}^{ℓ} := D_{0,0,n}^{ℓ} − D_{0,0,n−2}^{ℓ}#ω_{2}.(26) If n ≥ ℓ + 1, then we take the ℓ elements

D_{0,ℓ−1,n−ℓ+1}^{ℓ}, D_{1,ℓ−2,n−ℓ+1}^{ℓ}, …, D_{ℓ−3,2,n−ℓ+1}^{ℓ}, D_{0,0,n}^{ℓ}, E_{n}^{ℓ}.  (27)

Let M_{n,ℓ} be the list of elements D_{i} together with the elements from Equation (18) (resp. (19)) and Equation (26) (resp. (27)). We have

card (M_{n,ℓ}) = \{ \begin{matrix}3n − 3 & if n < ℓ, \\ n + ℓ − 3 + [(n + ℓ − 1)/2] & if n ≥ ℓ. \end{matrix}  (28)

The values of \overline{W}_{𝔤𝔩} and \overline{W}_{𝔰𝔬} on elements of M_{n,ℓ} have the properties stated in Table 2.

ord(\overline{W}_{𝔤𝔩}(D_{i})) ≥ ℓ  ord(\overline{W}_{𝔰𝔬}(D_{i})) ≥ ℓ, \overline{W}_{𝔰𝔬}(D_{i})(2) = 0
ord(\overline{W}_{𝔤𝔩}(E_{n}^{ℓ})) ≥ ℓ  ord(\overline{W}_{𝔰𝔬}(E_{n}^{ℓ})) ≥ ℓ, \overline{W}_{𝔰𝔬}(E_{n}^{ℓ})(2) ≠ 0
ord(\overline{W}_{𝔤𝔩}(D_{i,0,0}^{ℓ}#ω_{n−i}))
= ℓ − i
(i > 0, n − i even) ord(\overline{W}_{𝔰𝔬}(D_{i,0,0}^{ℓ}#ω_{n−i})) ≥ ℓ
ord(\overline{W}_{𝔰𝔬}(D_{n−i,i,0}^{ℓ})) = ℓ + 1 − i (i ≥ 3)
ord(\overline{W}_{𝔰𝔬}(D_{i,ℓ−1−i,n−ℓ+1}^{ℓ})) = i + 1 (i ≤ ℓ − 3)
ord(\overline{W}_{𝔰𝔬}(D_{0,0,n}^{ℓ})) = ℓ − 1

Table 2: Properties of \overline{W}_{𝔤𝔩}(e) and \overline{W}_{𝔰𝔬}(e) for e ∈ M_{n,ℓ}

The statements from this table are easily verified. For example, we have

\overline{W}_{𝔰𝔬}(E_{n}^{ℓ}) = c^{ℓ−1}(c − 1)h(c)

with h(c) = Q_{n}(c)−2(c−1)(c−2)Q_{n−2}(c). We have h(0) = Q_{n}(0)−4Q_{n−2}(0) = 0 which implies

ord (\overline{W}_{𝔰𝔬}(E_{n}^{ℓ})) ≥ ℓ,

and h(2) = Q_{n}(2) = (−2)^{n} which implies \overline{W}_{𝔰𝔬}(E_{n}^{ℓ})(2) ≠ 0. Now let

f = ∑_{e∈M_{n,ℓ}} λ(e) (\overline{W}_{𝔤𝔩}(e), \overline{W}_{𝔰𝔬}(e)) = (f_{1}, f_{2}) ∈ ℚ[c] × ℚ[c]

be a linear combination with λ(e) ∈ ℚ. We want to show that f = 0 implies that all scalars λ(e) are 0. For our arguments we will use the entries of Table 2 beginning at its bottom. The coefficients λ(D_{n−i,i,0}^{ℓ}) (resp. λ(D_{i,ℓ−1−i,n−ℓ+1}^{ℓ})) and λ(D_{0,0,n}^{ℓ}) are 0 because they are multiples of

\frac{d^{k}f_{2}}{dc^{k}} (0), …, \frac{d^{ℓ−1}f_{2}}{dc^{ℓ−1}} (0)

with k = max{1, ℓ − n + 1}. The coefficients λ(D_{i,0,0}^{ℓ}#ω_{n−i}) must be 0 by a similiar argument for f_{1}. We get λ(E_{n}^{ℓ}) = 0 because \overline{W}_{𝔰𝔬}(D_{i})(2) = 0 and \overline{W}_{𝔰𝔬}(E_{n}^{ℓ})(2) ≠ 0. The remaining coefficients λ(D_{i}) are 0 because the values (\overline{W}_{𝔤𝔩}(D_{i}), \overline{W}_{𝔰𝔬}(D_{i})) are linearly independent. This implies dim(H′_{n,ℓ} +F′_{n,ℓ}) ≥ card(M_{n,ℓ}). By Lemma 8 and Lemma 13 we have

card(M_{n,l}) + 2 ≤ dim (H′_{n,ℓ} + F′_{n,ℓ}) + dim (H′_{n,ℓ} ∩ F′_{n,ℓ}) =
= dim H′_{n,ℓ} + dim F′_{n,ℓ} ≤ \{ \begin{matrix}3n − 1 & if n < ℓ, \\ n + ℓ − 1 + [(n + ℓ − 1)/2] & if n ≥ ℓ. \end{matrix} (29)

Comparing with Equation (28) shows that equality must hold in Equation (29). This completes the proof of Parts (2) and (3) of the theorem for all n ≥ 4. In degrees n = 1, 2, 3 we used the diagrams shown in Table (3) (possibly together with some additional circles S^{1}) to determine dim F′_{n,ℓ}.

\begin{matrix}n = 1 & L_{1} \\ n = 2 & ω_{2}, C_{2}, L_{2} \\ n = 3 & ω_{3}, Ω_{3} := [figure] , ω_{2}#L_{1}, T_{3}, C_{3} \end{matrix}

Table 3: Diagrams used in low degrees

In the calculation we used the explicit formulas for the values of \overline{W}_{𝔰𝔬} from the proof of Lemma 9 together with \overline{W}_{𝔰𝔬}(Ω_{3}) = 2c(c − 1)(2 − c). The number of linearly independent values coincides in all of these cases with the upper bound for dim F′_{n,ℓ} from Lemma 8 or with dim Ā_{n,ℓ}. For ℓ ≥ 4 and a ∈ Ā_{3,3} we have

ord (\overline{W}_{𝔤𝔩} (L_{3} ⨿ S^{1}^{⨿ℓ−4})) = ℓ − 3 and ord (\overline{W}_{𝔤𝔩} (a ⨿ S^{1}^{⨿ℓ−3})) ≥ ℓ − 2.

Together with Lemmas 8 and 13 this implies

3 + 5 − 2 ≥ dim(H′_{3,ℓ} + F′_{3,ℓ}) ≥ dim F′_{3,3} + 1 = 6

and therefore dim(H′_{3,ℓ} + F′_{3,ℓ}) = 6 and dim(H′_{3,ℓ} ∩ F′_{3,ℓ}) = 2 for ℓ ≥ 4. In the cases n = 1, 2, and in the case n = 3 and ℓ < 4, we have dim H′_{n,ℓ} ≤ 2 and obtain dim(H′_{n,ℓ} ∩ F′_{n,ℓ}) = dim H′_{n,ℓ} by applying Lemma 13. This completes the proof.

Corollary 2 can now be proven easily.

Proof of Corollary 2. Proposition 5, Lemma 13, and Part (3) of Theorem 1 imply

dim(span{r_{n}^{ℓ}, y_{n}^{ℓ}}) = dim(span{W (r_{n}^{ℓ}), W (y_{n}^{ℓ})})
= min(dim H′_{n,ℓ}, 2) = min(dim H_{n,ℓ}, 2) = dim(H_{n,ℓ} ∩ F_{n,ℓ}).

By Proposition 4 we have r_{n}^{ℓ}, y_{n}^{ℓ} ∈ H_{n,ℓ} ∩ F_{n,ℓ}. This implies the statement span{r_{n}^{ℓ}, y_{n}^{ℓ}} = H_{n,ℓ} ∩ F_{n,ℓ} of the corollary.

Using Theorem 1 and Proposition 12 we can also prove Theorem 3.

Proof of Theorem 3. By Proposition 12 we have

dim(H′_{n} + F′_{n})_{|P_{n}} = b_{n} := \{ \begin{matrix}[n/2] & n ≤ 3 \\ n + [n/2] − 4 & n ≥ 4. \end{matrix}

This implies that in the graded algebra A generated by ⨁_{n=0}^{∞}(H′_{n} + F′_{n}) we find a subalgebra B ⊆ A which is a polynomial algebra with b_{n} generators in degree n. For n ≥ 4 we find by Equation (21) a nontrivial element w ∈ F′_{n} lying in the algebra generated by ⨁_{n=1}^{n−1} F′_{n}. This shows that A is generated by a_{n} elements in degree n with a_{n} := dim(H′_{n} + F′_{n}) − 1 for n ≥ 4 and a_{n} := dim(H′_{n} + F′_{n}) for n ≤ 3. By Theorem 1 we have a_{n} = b_{n}. Now B ⊆ A implies A = B. By Proposition 5 the isomorphism Z^{∗} maps A to the algebra generated by ⨁_{n=0}^{∞}(H_{n} + F_{n}). This completes the proof.

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Jens Lieberum MSRI 1000 Centennial Drive Berkeley, CA 94720-5070 USA lieberum@msri.org 300

Documenta Mathematica 5 (2000)