The summation theorem - Summation of chord diagrams and Dodgson polynomials

5.2 Summation of chord diagrams and Dodgson polynomials

5.2.3 The summation theorem

The statement of the main theorem is now that the polynomial Z_Γ⁰, defined like this, is in fact equal to the expression that we abbreviated with this notation in section 4.4. In order to prove this we first need some auxiliary results. First we attempt to study the summation by essentially working backwards and looking at sums

Pχ^(u_Γ^id^|v^id⁾ for (u_id,v_id) ∈ P₂, which appear in the partition polynomial for the single part partition E ={E_D¹}.

Lemma 5.2.5. Let P_j be as above and c^j₂(D) the number of two-coloured cycles consisting of chords and j-coloured base edges (such that c₂(D) =c¹₂(D) +c²₂(D)).

5. Structure of the integrand II: Polynomial identities

Proof. Quick computations show that the claim holds for all n with N =^Pni = 1,2, and even N = 3 is only mildly tedious, as shown below in example 5.2.6. We now reduce the l.h.s. of 5.36 to a sum over expressions corresponding N −1, in order to prove by induction.

Consider a word pair (x11· · ·x_1N,x₂₁· · ·x_2N) with all x_ij ∈ A different letters.

Assuming this word is a representative (u_id,v_id) ∈ P_j, each pair (x_1k,x_2k) of k-th letters corresponds to a base edge ofE_D^j₀, for a chord diagram D0 ∈ D0ⁿ. With eq.

(5.11) its Dodgson polynomial can be written as ΨΓχ^(x_Γ¹¹^···x^1N^|x²¹^···x^2N⁾

=^X^N

k=1

(−1)^1+kχ^(x_Γ¹¹^|x^2k⁾χ^(x_Γ¹²^···x^1N^|x²¹^···ˆ^x^2k^···x^2N⁾

=χ^(x_Γ¹¹^|x²¹⁾χ^(x_Γ¹²^···x^1N^|x²²^···x^2N⁾−

k=2

χ^(x_Γ¹¹^|x^2k⁾χ^(x_Γ^1k^x¹²^···ˆ^x^1k^···x^1N^|x²¹^···ˆ^x^2k^···x^2N⁾. (5.37) Moving the letter x_1k in the second line guarantees that the k-th letter pairs are still paired up in the expansion. In fact, the word pairs

(x_1kx₁₂· · ·ˆx_1k· · ·x_1N,x₂₁· · ·ˆx_2k· · ·x_2N) (5.38) are the representatives (u⁰_id,v⁰_id) of an equivalence class of word pairs associated to the diagram π₀(D), where D is D₀ together with the chord corresponding to the letter pair (x11,x_2k). The sum over all equivalence classes in P_j can be realised by summing word pairs of the form

(x_(1+t₁₎₁· · ·x_(1+t_N_)N, x_(2−t₁₎₁· · ·x_(2−t_N_)N) (5.39) over all N-tuples in T ={t∈ {0,1}^N | t₁ = 0}. One finds

ΨΓX

t∈T

χ^(x_Γ^(1+t¹⁾¹^···x⁽¹⁺^tN^)N^|x^(2−t¹⁾¹^···x⁽²⁻^tN^)N⁾

=χ^(x_Γ¹¹^|x²¹⁾^X

t∈T

χ^(x_Γ^(1+t²⁾²^···x⁽¹⁺^tN^)N^|x^(2−t²⁾²^···x⁽²⁻^tN^)N⁾

−

k=2

t∈T

χ^(x_Γ¹¹^|x⁽²⁻^tk^)k⁾χ^(x_Γ⁽¹⁺^tk^)k^x^(1+t²⁾²^···ˆ^x⁽¹⁺^tk^)k^···x⁽¹⁺^tN^)N^|x²¹^···ˆ^x⁽²⁻^tk^)k^···x⁽²⁻^tN^)N⁾. (5.40) Now we want to translate this back to vertices of a chord diagram. Letu, v ∈VD0

5.2. Summation of chord diagrams and Dodgson polynomials such that x₁₁=a_u, x₂₁ =a_v and (u, v)∈E_D^j₀. Then eq. (5.40) becomes

ΨΓ X

(uid,vid)∈Pj

χ^(u_Γ^id^|v^id⁾ = 2χ^(a_Γ^u^|a^v⁾ ^X

(u⁰_id,v_id⁰ )∈P^u,v_j

χ^(u_Γ⁰^id^|v^id⁰ ⁾− ^X

w∈VD0

w6=u,v

χ^(a_Γ^u^|a^w⁾ ^X

(u⁰_id,v⁰_id)∈P^u,w_j

χ^(u_Γ⁰^id^|v⁰^id⁾, (5.41) whereP^u,v_j andP^u,w_j are the classes of word pairs after addition of the chords (u, v) or (u, w) respectively. Replacing these sums with the corresponding r.h.s. of eq.

(5.36) finishes the proof, where the factor of −(−2) in the first term corresponds to the addition of the cycle that consists of thej-coloured base edge (u, v) and the chord between those same vertices. All other chords (u, w) added to D₀ do not add two-coloured cycles but only split, twist or merge base cycles when projected out with π0.

Example 5.2.6. Consider as an example N = 3 with a single base cycle. Label vertices consecutively from 1 to 6and choose j to be the colour of (1,2). Then the sum over word pairs in P_j on the l.h.s. of eq. (5.36) is

χ^(a_Γ¹^a³^a⁵^|a²^a⁴^a⁶⁾+χ^(a_Γ¹^a⁴^a⁵^|a²^a³^a⁶⁾+χ^(a_Γ¹^a³^a⁶^|a²^a⁴^a⁵⁾+χ^(a_Γ¹^a⁴^a⁶^|a²^a³^a⁵⁾. (5.42) Expanding each term as defined in eq. (5.11) yields Ψ⁻_Γ² times 24 terms, 15 of which are distinct, such that one finds

The obvious next questions is now: Can we find such an identity for all parti-tions? Indeed, we can.

5. Structure of the integrand II: Polynomial identities

Lemma 5.2.7. Let E ∈ P(E_D¹) be any partition of 1-coloured base edges. Then

(u,v)∈P2

sgnE(u,v) ^Y

(u⁰,v⁰)∈λE(u,v)

χ^(u_Γ⁰^|v⁰⁾ = (−1)¹^−|E|(−ΨΓ)^|E|−N^X

D∈D|E

(−2)^c²²^(D)⁻¹ ^Y

(u,v)∈E_D⁰

χ^(a_Γ^u^|a^v⁾, (5.43) where D|^E ⊂ DNⁿ ' DΓ⁰ is the subset of complete chord diagrams with base cycles given by n (and vertices labelled by edges ofΓ) that is restricted by demanding that all chords of a diagram can only connect vertices that lie within the same part of E.

Proof. Begin with partitions with exactly two parts E = {E1, E2} in which one part consists of only one edge, say E₁ ={(u, v)}. Then for all (u,v)∈P₂ the pair (a_u,a_v) is contained in λE(u,v) such that χ^(a_Γ^u^|a^v⁾ simply factors out. Proposition 5.2.5 can be applied to the remaining terms to finish the proof for partitions of this type, and in particular one notes that the definition ofλE(u,v) precisely yields the restriction of chord diagrams to this partition. Moreover, repetition of this argument proves the claim for all partitions with at most one part that contains more than one edge.

Assume now again a partition with two parts E = {E1, E2}, but also with

|E₁|= 2 and|E₂| ≥2. The Dodgson polynomialχ^(u_Γ⁰¹^|v⁰¹⁾, where (u⁰₁,v₁⁰)∈λE(u,v) is the word pair corresponding to partE1of the partition, can be expanded analogous to eq. (5.36), with the caveat that the restriction toE1 removes all terms from the sum that correspond to vertices not in that part (including the first term, which contributes the factor 2, if the fixed vertex u and the vertex v that it shares a 2-coloured base edge with are not in the same part of the 1-coloured base edge partition). Then the coefficient of each χ^(a_Γ^u^|a^v⁾ corresponds to a partition with part sizes{|E1| −1 = 1,|E2|}, which was discussed above, such that term by term application of the identity proves it for this type of partition. Further repetition of this argument finishes the proof for all partitions, independent of number or size of parts.

The final ingredient for the proof of this chapter’s main theorem is an iden-tity allowing summation of Stirling numbers of the second kind³. To prove it we need a certain identity relating Stirling numbers and the classical polylogarithm.

However, while the literature contains a number of well known identities that do so, they are all either similar but not obviously equivalent to the one we need, or appear without proof. Moreover, the commonly cited references (e.g. [2, 91, 122],

3James Stirling (1692-1770),Methodus Differentialis, 1730. See also [130]. The Stirling num-bers of the second kindS(k, l) count the ways to partition a set ofk elements intol non-empty sets.

5.2. Summation of chord diagrams and Dodgson polynomials among many others) all appear to cite each other or unavailable older literature, so it may actually be somewhat elucidating to derive everything we need ourselves.

Proposition 5.2.8. Let

Lis(z) =^X^∞

l=1

z^l

l^s |z|<1, s ∈Z (5.44) be the classical polylogarithm andS(k, l) be the Stirling number of the second kind.

Then and use partial fraction decomposition to find

S(k−1, l) z In the sum one now has a telescopic cancellation involving the second terms of eqs.

(5.48) and (5.49). The only remaining terms are S(k−1,0)

z−1 = 0 and S(k−1, k−1)(k−1)!

(z−1)^k =S(k, k)(k−1)!

(z−1)^k,

5. Structure of the integrand II: Polynomial identities

Lemma 5.2.9. Let S(k, l) be the Stirling number of the second kind. Then

l=1S(k, l)(−1)^l(l+ 1)! = (−2)^k ∀k ≥1.

Proof. For k = 1 the claim is checked directly. For k ≥ 2 we use the identity derived for the polylogarithm in proposition 5.2.8 and note that a change of the argument allows us to write

(−1)^kLi−k+1

Computing the derivative one finds L(z) = (−1)^k

Both polylogarithms start with terms linear in (z+1)/z, yielding divergences when evaluating at z=−1, but upon closer inspection we see that they precisely cancel each other. With z < −1 one has |1 + 1/z| < 1 such that we are able to employ the classical sum representation of the polylogarithm, of which only the first two terms are of interest to us:

L(z) = (−1)^k

5.2. Summation of chord diagrams and Dodgson polynomials

Proof of theorem 5.2.4. First, use lemma 5.2.7 to rewrite the partition poly-nomial as

The sum already contains c²₂(D), the number of 2-coloured cycles. Regarding cy-cles of the other colour we can make the following observation: In each diagram withc¹₂(D)≤N the 1-coloured cycles can themselves be interpreted as a partition of E_D¹ in which each part is given by the base edges connected to each other by chords. The diagrams in D|^E can only have chords connecting base edges within the same part of E, so each part in the partition given by the 1-coloured cycles has to be a subset of a part of E. Counting the number of ways of partitioning the c¹₂(D) cycles of a given diagram into partitions with |E| parts (i.e. counting the number of partitions E with a certain number of parts such that D|^E con-tains the given diagram D) one finds precisely the Stirling numbers of the second kind S(c¹₂(D),|E|). Using this, we can exchange summation over diagrams and partitions and find

5. Structure of the integrand II: Polynomial identities Application to the integrand

Let us return to eq. (4.74). When inserting our definition for Z_Γ⁰ from eq. (5.34) it becomes For the vertex integral from eq. (4.80) the simplification is similar, although there it is confined to the coefficient of γ^µ.

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