Algebraic intersection of noodles and arcs

Homological Representations of the Braid Groups

3.6 Noodles vs. spanning arcs

3.6.2 Algebraic intersection of noodles and arcs

The intersection of a noodle N and a spanning arc α can be measured in terms of a so-calledalgebraic intersection N, α. This is an element of the ringZ[q^±1, t^±1] defined up to multiplication by monomialsq^wt^u with w∈Z and u ∈ 2Z⊂ Z. The algebraic intersection N, α depends on a choice of orientation on α, which we fix from now on. As above, we endow Σ with counterclockwise orientation. The orientations ofαand Σ allow us to speak of the “right” and “left” sides ofαinΣ. Pushingαslightly to the left (keeping the endpoints), we obtain a “parallel” oriented spanning arcα⁻on (D, Q) with the same starting and terminal endpoints asαand disjoint fromαotherwise.

Slightly deformingN, we can assume thatNintersectsαtransversely inm≥0 pointsz1, . . . , zm(the numeration is arbitrary). We choose the parallel arcα⁻ very closely to α so that α⁻ meets N transversely in m points z₁⁻, . . . , z_m⁻, where each pairz_i⁻, zi is joined by a short subarc of N lying in the narrow strip onΣ bounded byα⁻∪α; cf. Figure 3.9 below (the strip in question is shaded). Fori∈ {1, . . . , m}, letεi =±1 be the intersection sign ofN andα atzi (recall that both N andαare oriented). Thus, εi = +1 ifN crossesα atz_i from left to right andε_i =−1 otherwise. Denote the starting endpoint and the terminal endpoint ofNbyd⁻andd, respectively. Fix arbitrary points

z⁻∈α⁻−∂α⁻, z∈α−∂α

and fix pathsθ⁻, θin Σ leading respectively fromd⁻ to z⁻ and fromdto z and having disjoint images (these paths are allowed to meet N, α⁻, and α elsewhere).

Recall the spaceC of nonordered pairs of distinct points of Σ. For every pair i, j ∈ {1, . . . , m}, we define a loop ξi,j in C as follows. Let β_i⁻ be an oriented embedded arc on α⁻ leading from z⁻ to z_i⁻ (the orientation ofβ_i⁻ may be opposite to that of α⁻). Let β_j be an oriented embedded arc on α leading fromz to zj. Let γ_i,j⁻ and γi,j be disjoint oriented arcs inN leading from the points z_i⁻, zj ∈ N to the endpoints ofN. These oriented arcs are determined only by the position of the pointsz⁻_i , zj onN and do not depend on the orientation of N. Recall the notation for paths in C introduced in Section 3.5.1. Consider the paths{θ⁻, θ},{β_i⁻, βj}, and{γ_i,j⁻, γi,j}inC. They lead from{d⁻, d} ∈ C to{z⁻, z} ∈ C, from{z⁻, z}to{z_i⁻, z_j} ∈ C, and from {z_i⁻, zj}to{d⁻, d}, respectively. The product of these three paths

ξi,j={θ⁻, θ} {β_i⁻, βj} {γ_i,j⁻, γi,j} (3.19) is a loop inC beginning and ending in{d⁻, d}. Set

N, α= m i=1

m j=1

ε_iε_jq^w(ξi,j)t^u(ξi,j)∈Z[q^±1, t^±1],

where w and u are the integral invariants of loops in C introduced in Sec- tion 3.5.1. The expression on the right-hand side does not depend on the numeration of points in N ∩α. Under a different choice of z⁻, z, θ⁻, θ, all loopsξ_i,j are multiplied on the left by one and the same loop inC of the form {ξ1, ξ2}, whereξ1, ξ2 are loops inΣ. ThenN, αis multiplied by a monomial inq^±1, t^±2.

For example, if N is disjoint from α, then N, α = 0. IfN crossesαin only one point, thenm= 1 and N, α=q^kt for somek, ∈Z.

We state two fundamental properties of the algebraic intersections of noo- dles and arcs.

Lemma 3.20.The algebraic intersection N, αis invariant under isotopies ofN andαin Σconstant on the endpoints.

Proof. It suffices to fixN and to prove thatN, αis invariant under isotopies of the spanning arcα. A generic isotopy of αin Σ can be split into a finite sequence of local moves of three types:

(i) an isotopy ofαinΣ keepingαtransversal toN,

(ii) a move pushing a small subarc of αacross a subarc ofN, (iii) an inverse to (ii).

It is clear from the definitions that the moves of type (i) do not changeN, α. Any move of type (ii) adds two new intersection pointszm+1, zm+2to the set N∩α={z1, . . . , zm}. Assume for concreteness that the subarc ofN connect- ing zm+1 with zm+2 lies on the right of the arc α; see Figure 3.9. Clearly, the sign εi = ±1 is preserved under this move for i = 1, . . . , m. For all

i, j= 1, . . . , m, the loopsξi,j computed before and after the move are homo- topic to each other. Therefore such pairs (i, j) contribute the same expression to N, αbefore and after the move. For i= 1, . . . , m+ 2, the loopsξ_i,m+1 and ξ_i,m+2 are homotopic and the obvious equality ε_m+1 = −ε_m+2 implies that the contributions of the pairs (i, m+ 1), (i, m+ 2) cancel each other.

Similarly, for anyi = 1, . . . , m, the loops ξm+1,i and ξm+2,i are homotopic and the contributions of the pairs (m+ 1, i), (m+ 2, i) cancel each other.

ThereforeN, αis preserved under the move.

α α

N

α⁻ α⁻

zm+1 zm+2

z

z⁻_m+1 z⁻_m+2

Fig. 3.9.Additional crossings

We say that a spanning arcαon (D, Q) can be isotopped off a noodleN if there is a continuous family of spanning arcs {α_s}s∈[0,1] on (D, Q) such thatα₀=αandα₁is disjoint fromN. Such a family{α_s}sis called anisotopy ofα. Note that the spanning arcsα_s necessarily have the same endpoints.

Lemma 3.21.A spanning arc αcan be isotopped off a noodleN if and only ifN, α= 0.

Proof. If there is an isotopy {αs}s of α =α0 in Σ such that α1 is disjoint from N, then N, α = N, α1 = 0. The hard part of the lemma is the opposite implication. Applying a preliminary isotopy to α, we can assume that α intersects N transversely at a minimal number of points z1, . . . , zm

withm≥0. We assume thatm≥1 and show thatN, α = 0.

We keep the notation introduced above in the definition of N, α. For anyi, j∈ {1, . . . , m}, setwi,j=w(ξi,j)∈Zandui,j=u(ξi,j)∈Z. Then

N, α= m

i=1

m j=1

εiεjq^wi,jt^ui,j. (3.20) Observe that

ε_i= (−1)^ui,i

for alli. Indeed, ifεi= +1, thenN crosses the arc αatzi from left to right and therefore the pathsγ⁻_i,i, γi,i end respectively in d⁻, d. Then ξi,i has the form{ξ1, ξ2}, whereξ1, ξ2are loops inΣ. In this case

ui,i=u(ξi,i) = 0 (mod 2).

Similarly, ifεi=−1, then ui,i= 1 (mod 2). In both cases εi= (−1)^ui,i.

We shall use the lexicographic order on monomials q^wt^u with w, u ∈ Z.

More precisely, we writeq^wt^u≥q^wt^u with w, u, w, u ∈Zif eitherw > w orw=wandu≥u. We say that an ordered pair (i, j) withi, j∈ {1, . . . , m} ismaximal (for givenN, α) if q^wi,jt^ui,j ≥ q^wk,lt^uk,l for all k, l ∈ {1, . . . , m}. A maximal pair necessarily exists because the lexicographic order on the monomials is total. A maximal pair may be nonunique. We claim that

if (i, j)is maximal, thenui,i=uj,j. (3.21) This claim implies that every maximal pair (i, j) contributes the monomial

εiεjq^wi,jt^ui,j = (−1)^ui,i(−1)^uj,jq^wi,jt^ui,j =q^wi,jt^ui,j

toN, α. All maximal pairs necessarily contribute the same monomial, which then occurs inN, αwith a positive coefficient. ThereforeN, α = 0.

To prove (3.21), we first compute wi,j for anyi, j∈ {1, . . . , m}(not nec- essarily maximal). Let η_i⁻ be the loop inΣ obtained as the product of the pathθ⁻β_i⁻ with the path going fromz⁻_i to d⁻ along N. Letηj be the loop inΣobtained as the product ofθβ_jwith the path going fromz_j todalongN. We claim that

wi,j=w(η_i⁻) +w(ηj). (3.22) Indeed, if the pathγ_i,j⁻ appearing in (3.19) ends atd⁻, then the pathγi,j ends at d, the paths θ⁻β_i⁻γ_i,j⁻ and θβjγi,j are loops, andwi,j is the sum of their total winding numbers. Formula (3.22) follows in this case from the equalities η⁻_i =θ⁻β⁻_i γ_i,j⁻ andηj =θβjγi,j. Assume thatγ_i,j⁻ ends atd. Thenγi,j ends atd⁻,

η_i⁻=θ⁻β⁻_i γ_i,j⁻N⁻¹, ηj =θβjγi,jN ,

where N is viewed as a path from d⁻ to d. By definition, wi,j is the total winding number of the loopθ⁻β_i⁻γ_i,j⁻θβjγi,j. This loop is homotopic inΣto the loop

θ⁻β_i⁻γ_i,j⁻N⁻¹N θβ_jγ_i,jN N⁻¹=η⁻_i N η_jN⁻¹. The loopη_i⁻N ηjN⁻¹is homologous toη_i⁻ηj inΣ. Hence (3.22).

Inspecting the loops η⁻_i and ηi, we observe that the difference between their homology classes [η_i⁻],[ηi]∈H1(Σ;Z) is represented by the loop going from d to d⁻ alongN⁻¹, then from d⁻ to z⁻ alongθ⁻, then from z⁻ to z along a path lying in the strip between α⁻ and α, and finally from z to d alongθ⁻¹. Therefore the difference [η_i⁻]−[ηi] ∈ H1(Σ;Z) does not depend oni. This implies that the number

W =w(η_i⁻)−w(ηi)∈Z

does not depend oni. Formula (3.22) implies that for alli, j= 1, . . . , m, wi,j =w(ηi) +w(ηj) +W . (3.23)

Suppose that the pair (i, j) is maximal. Thenwi,j is maximal among all the integerswk,l. By (3.23), both numbersw(ηi) andw(ηj) must be maximal among all the integersw(η_k). Then

w(ηi) =w(ηj) and wi,i=wi,j.

The maximality of (i, j) implies that ui,i ≤ ui,j. We claim that ui,i = ui,j. Fori=j, this is obvious and we assume thati=j.

Suppose, seeking a contradiction, thatui,i< ui,j. Letμbe the (embedded) subarc of α connecting zi and zj. Let ν be the (embedded) subarc of N connecting zi and zj. We orient μ from zi to zj and ν from zj to zi. The productμν is a loop onΣ based atz_i. We distinguish two cases.

Case 1: The arcνapproachesαatz_ifrom the right (in other words,νdoes not pass through z⁻_i ). Then the loop μν does not pass through z⁻_i and we can consider its winding number,v∈Z, aroundz_i⁻. We claim thatv >0. To see this, we computevas follows. As was already observed, 2v=u({z⁻_i , μν}), whereuis the invariant of loops inCdefined in Section 3.5.1 andz_i⁻stands for the constant path in the pointz⁻_i . Observe that βj ∼βiμ, where ∼denotes the homotopy of paths inΣ−{z_i⁻}relative to the endpoints. The assumption thatν does not pass throughz⁻_i implies that

γ_i,i⁻ =γ_i,j⁻ and γi,i=ν⁻¹γi,j; see Figure 3.10. Then

ξi,j ={θ⁻, θ}{β_i⁻, βj}{γ_i,j⁻, γi,j} ∼ {θ⁻, θ}{β_i⁻, βi}{z_i⁻, μν}{γ⁻_i,i, γi,i}. The latter loop is homologous inCto the loop

{θ⁻, θ}{β⁻_i , βi}{γ_i,i⁻, γi,i}{z_i⁻, μν}=ξi,i{z_i⁻, μν}. Therefore,

2v=u({z_i⁻, μν}) =u(ξi,j)−u(ξi,i) =ui,j−ui,i. The assumptionui,i< ui,j implies thatv >0.

α

μ

zi ν z_j

N

γi,j

γ_i,j⁻ z⁻_i

Fig. 3.10. Case 1: the pathsγ_i,j⁻ andγi,j

We can now bring one more loop into the picture. Consider the short subarc ofN connectingzi toz⁻_i in the strip betweenαandα⁻. Pick a loopρ in a small neighborhood of this subarc such that

(i) ρbegins and ends inz_i;

(ii) ρdoes not meetz_i⁻ and winds clockwisev times aroundz_i⁻; (iii) ρhasv−1 transversal self-crossings;

(iv) ρmeetsμν only atzi (see Figure 3.11).

z_i μ ν

ρ z_j

zi−

Fig. 3.11. The loopρforv= 3

Note that the winding number of the loop μνρ aroundz_i⁻ is equal to 0.

Hence, this loop lifts to an appropriate covering of the complement of{z_i⁻}. We now describe this lift in more detail.

Let D_• = D− {z_i⁻} and p : D_• → D_• be the universal (infinite cyclic) covering. Letμ: [0,1]→D_• be an arbitrary lift ofμ(so thatpμ=μ). There is a unique liftν: [0,1]→D_• ofν such thatν(0) =μ(1). Consider also the unique liftρ: [0,1]→D_• of ρsuch thatρ(0) = ν(1). By abuse of notation, we shall denote the pathsμ, ν, ρ,μ,ν, ρand their images by the same letters.

Since the winding number ofμνρaround z⁻_i is zero, the pathμνρis a loop.

Our choice ofρensures thatρis an embedded arc inD_•meeting μνonly at the endpoints. However, the embedded arcsμandνinD_•may meet in several points besides their common endpointμ(1) = ν(0). Let a be the first point ofμthat lies also onν(possiblya=μ(1)). Letμa be the initial segment ofμ going fromμ(0) toa. Letνa be the final segment ofνgoing froma toν(1).

Set

δ=μaνaρ .

The construction of the loopδensures that it has no self-crossings. This loop parametrizes an embedded circle inD_• denoted by the same symbol δ. We identify D_• with the half-open strip R×[0,1) ⊂R² so that the orientation inD_•induced by the counterclockwise orientation inD_•is identified with the counterclockwise orientation inR². The Jordan curve theorem implies thatδ bounds an embedded diskB⊂D_•.

We verify now that the loopδencirclesB counterclockwise. LetC be the component ofD_•−ρsurroundingz⁻_i . We check first thatC∩p(B) =∅. Indeed, suppose that there is a pointb∈B such thatp(b)∈C. We can connect the pointp(b) to any other pointb ofC by an arc inC. This arc lifts to an arc inD_• beginning inb. The latter arc never meetsδ, since its projection to D_• never meetsμ, ν, orρ. Hence this lifted arc lies in the interiorB^◦=B−∂B of B, and its terminal endpoint projects tob. Thus, C ⊂p(B). SinceB is compact, so is p(B). On the other hand, it is clear that C is not contained in a compact subset of D_•. This contradiction shows that C ∩p(B) = ∅. Observe now thatClies on the right ofρ. IfB lies on the right ofρ⊂δ, then necessarilyC∩p(B)=∅, a contradiction. Thus,B lies on the left of ρand ofδ. Hence,δ goes counterclockwise aroundB.

We claim that B∩p⁻¹(Q) = ∅. Indeed, being a compact subset ofD_•, the disk B may contain only a finite number of points of the (discrete) set p⁻¹(Q)⊂ D_•. Observe that the pathsμ, ν, ρ lie in Σ =D−Qand do not meet Q. Therefore ∂B∩p⁻¹(Q) = ∅, so that B∩p⁻¹(Q) ⊂ B^◦. The loop δ=∂Bis homologous inB−p⁻¹(Q) to the sum of small loops encircling the points of B∩p⁻¹(Q) counterclockwise. The latter loops are projected by p homeomorphically onto small loops encircling certain points of Q counter- clockwise. Therefore,

card(B∩p⁻¹(Q)) =w(p◦δ),

wherew(p◦δ) is the total winding number of the loop p◦δin Σaround the points ofQ. We have

p◦δ=μaνaρ ,

where μ_a =p(μ_a) is the initial segment ofμ going from z_i to p(a) alongα, and ν_a = p(ν_a) is the final segment of ν going from p(a) to z_i along N. Then p(a) ∈ N ∩α, so that p(a) = z_k for some k = 1, . . . , n. Since ρ is contractible inΣ, the loopμaνaρis homotopic toμaνa inΣ and

w(μaνaρ) =w(μaνa).

Recall the loops ηk, ηi in Σ based at the terminal endpoint d of N. The difference between their homology classes [ηk],[ηi]∈H1(Σ;Z) depends neither on the choice of the pathθ nor on the choice of its terminal endpointz ∈α.

Taking z = zi, one immediately deduces from the definition of ηk, ηi that [ηk]−[ηi] = [μaνa]. Therefore,

w(μaνa) =w(ηk)−w(ηi). To sum up, we have

card(B∩p⁻¹(Q)) =w(p◦δ) =w(μaνaρ) =w(μaνa) =w(ηk)−w(ηi). Sincew(ηi) is maximal, card(B∩p⁻¹(Q))≤0. HenceB∩p⁻¹(Q) =∅.

We shall need a few simple facts concerning the covering p: D_• → D_•. The group of covering transformations ofpis an infinite cyclic group gener- ated by the covering transformationg :D_• →D_• corresponding to the loop encircling z_i counterclockwise. The set p⁻¹(N) consists of an infinite num- ber of disjoint closed intervals inD_• with boundary on∂D_•. These intervals can be numerated by integers so that the action ofg shifts the index by 1.

This implies that any nontrivial covering transformationD_•→D_•maps each component of p⁻¹(N) to a different component of p⁻¹(N). The same facts hold for the setp⁻¹(α)⊂D_•with the only difference that its components are closed intervals lying in the interior ofD_•.

We claim that under our assumptions the pairN, αhas a digon. This would imply that the intersectionN∩αis not minimal. The latter contradicts our choice of α in its isotopy class. Therefore, the assumption u_i,i < u_i,j must have been false, so thatu_i,i=u_i,j.

We now construct a digon forN, α. Suppose first that B^◦∩p⁻¹(N)=∅ or B^◦∩p⁻¹(α)=∅

(or both). Observe that the circleδ=∂B is formed by three embedded arcs:

the arcμalying onp⁻¹(α), the arcνalying onp⁻¹(N), and the arcρmeeting the setp⁻¹(N)∪p⁻¹(α) only in its two endpoints. Note that the boundary of the one-manifoldp⁻¹(N) is contained in∂D_•and lies therefore outside ofB.

IfB^◦∩p⁻¹(N)=∅, thenB^◦∩p⁻¹(N) is a finite set of disjoint embedded arcs with endpoints onμa. At least one of these arcs bounds together with a subarc ofμ_a a diskD₁⊂Bwhose interior does not meetp⁻¹(N). IfB^◦∩p⁻¹(N) =∅, then we set D1 =B. Similarly, the boundary of p⁻¹(α) ⊂ D_• is contained inp⁻¹(Q) and lies outside of B. If the interiorD^◦₁ of D1 meetsp⁻¹(α), then they meet along a finite number of disjoint embedded arcs with endpoints onp⁻¹(N)∩∂D1. At least one of these arcs bounds together with a subarc of p⁻¹(N)∩∂D1 an embedded disk D2 ⊂ D1 whose interior does not meet p⁻¹(α). IfD^◦₁∩p⁻¹(α) =∅, then we setD2=D1. In any case, the boundary ofD2is formed by an arc onp⁻¹(N) and an arc onp⁻¹(α), while the interior D^◦₂ ofD2does not meetp⁻¹(N ∪α). Then

D₂^◦∩g(∂D2) =∅,

for any nontrivial covering transformation g : D_• → D_• of the coveringp : D_• → D_•. The properties of the sets p⁻¹(N) and p⁻¹(α) mentioned above imply that∂D₂∩g(∂D₂) =∅. This implies that eitherD₂∩g(D₂) =∅orD₂is contained in the interior of the diskg(D2). In the latter case,g⁻¹(D2)⊂D₂^◦, which contradicts the fact thatD₂^◦ does not meetp⁻¹(N∪α). We conclude that D2 ∩g(D2) = ∅. Thus, the disk D2 does not meet its images under nontrivial covering transformations of the coveringp:D_•→D_•. Hence, the restriction ofptoD2 is injective. This implies thatp(D2) is a digon forN, α inΣ.

It remains to construct a digon for the pairN, αwhenB^◦∩p⁻¹(N∪α) =∅. The setp⁻¹(ρ) consists ofv copies of the lineRembedded inD_•; these lines meet each other at an infinite number of points (see Figure 3.12, wherev= 3).

The arcsμa, νalie in the component ofD_•−ρadjacent to∂D_•≈S¹except for the pointsμa(0) =νa(1) =zi. Therefore the arcsμa,νa lie in the component ofD_•−p⁻¹(ρ) adjacent to∂D_•≈Rexcept for the points μa(0) =μ(0) and

νa(1) = ν(1) lying on p⁻¹(zi) ⊂ p⁻¹(ρ). Clearly, νa(1) = g^v(μa(0)), where g:D_•→D_•is the generator of the group of covering transformations chosen above andv >0 is the winding number of the loopμν aroundz_i⁻. The diskB bounded byδ=μaνaρhas to include the area between the arcμaνaandp⁻¹(ρ) (this area is shaded in Figure 3.12). Observing Figure 3.12, one immediately concludes that forv ≥2, this area must meet g(μ_aν_a). This contradicts the assumption B^◦∩p⁻¹(N ∪α) = ∅. It follows that v = 1, so that p⁻¹(ρ) is just a line and B is the area between this line and the arc μ_aν_a. Then B projects injectively toD_•, the loopρbounds a small disk containingz_i⁻, and the union of this disk withp(B) is a digon forN, α. This completes the proof of the equalityui,i=ui,j in Case 1.

⎧⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎩

g a

p⁻¹(ρ)

μ ν

∂D_•

D_•

Fig. 3.12.The casev= 3

Case 2: The arc ν approaches α at z_i from the left (in other words, ν passes throughz_i⁻). Let us slightly push the arc ν near z_i⁻ to Σ− {z⁻_i } so that z⁻_i lies on the left side of the resulting arc. Denote by ν this new arc, also leading fromzj tozi. The loopμν does not pass throughz_i⁻and we can consider its winding number,v, aroundz_i⁻. We claim thatv >0. Observe first that the pointz_i⁻ splitsν into two subarcsν1 andν2, whereν1 leads fromzj

to z_i⁻ and ν2 leads from z⁻_i to zi. We have γ_i,j⁻ =ν2γi,i and γ_i,i⁻ =ν₁⁻¹γi,j; see Figure 3.13. As in Case 1, we haveβj ∼βiμ. Therefore,

ξi,j ={θ⁻, θ}{β_i⁻, βj}{γ_i,j⁻, γi,j} ∼ {θ⁻, θ}{β_i⁻, βi}{ν2, μν1}{γ_i,i⁻, γi,i}. The latter loop is homologous inCto the loop

{θ⁻, θ}{β⁻_i , βi}{γ_i,i⁻, γi,i}{ν2, μν1}=ξi,i{ν2, μν1}.

It is easy to deduce from the definitions and the construction of ν that u({ν2, μν1}) =u({z⁻_i , μν})−1 = 2v−1. Therefore,

2v−1 =u({ν2, μν1}) =u(ξi,j)−u(ξi,i) =ui,j−ui,i.

The assumptionui,i < ui,j implies that v >0. The rest of the proof of the equalityui,i=ui,j goes as in Case 1 with the difference that instead ofν one should everywhere useν.

α μ

zi

z_j

N γi,j

z_i⁻

γi,i

ν₁ ν2

Fig. 3.13.Case 2: the pathsγi,i andγi,j

Analogous arguments prove that uj,j =ui,j for any maximal pair (i, j).

This can also be deduced from the results above using the following symmetry for the loopsξi,j defined by (3.19), wherei, j is an arbitrary (not necessarily maximal) pair of elements of the set{1, . . . , m}. Let us write

ξ_i,j=ξ_i,j(N, α, z, z⁺, θ⁻, θ),

stressing the dependence on the data in the parentheses. We will use simi- lar notation for w_i,j = w(ξ_i,j) and u_i,j = u(ξ_i,j). Consider the noodle −N obtained from N by reversing the orientation. Similarly, consider the span- ning arcs−α,−α⁻ on (D, Q) obtained fromα, α⁻, respectively, by reversing the orientation. It is clear that−α lies on the left of −α⁻, so that we can set (−α⁻)⁻ = −α. The noodle−N crosses−α⁻ and (−α⁻)⁻ = −α in the same points as before and we numerate them in the same way, except thatzi

becomesz_i⁻ and vice versa (for alli). It follows from the definitions that ξi,j(N, α, z⁻, z, θ⁻, θ) =ξj,i(−N,−α⁻, z, z⁻, θ, θ⁻)

for all i, j. This implies similar formulas for wi,j and ui,j. Now, if the pair (i, j) is maximal for (N, α), then the pair (j, i) is maximal for (−N,−α⁻) and by the results above,

ui,j(N, α, z⁻, z, θ⁻, θ) =uj,i(−N,−α⁺, z, z⁻, θ, θ⁻)

=uj,j(−N,−α⁺, z, z⁻, θ, θ⁻)

=uj,j(N, α, z⁻, z, θ⁻, θ).

We conclude that ui,i = ui,j = uj,j for any maximal pair (i, j). This

proves (3.21) and the lemma.