INTERSECTION-NUMBER OPERATORS

FOR CURVES ON DISCS II

Stephen P. Humphries

Abstract. Let the braid group B_{n} act as (isotopy classes of) diffeomorphisms of an n-punctured disc D_{n}. Then there is an action of B_{n} on a polynomial algebra R = ℂ[a_{1}, …, a_{N}] and a way of representing simple closed curves on D_{n} as elements of R. Fix k ∈ 2ℕ. Using this approach we show that the image in Aut(R) of each Dehn twist τ about a simple closed γ in D_{n} satisfies a kind of characteristic equation when its action is restricted to the image in R of the set of curves γ having geometric intersection number k with γ. AMS Classification: 57M50. Key words: Intersection number, polynomial ring, braid group, simple closed

curve.

§1. Introduction.

Let D_{n} be the disc with n punctures π_{1}, …, π_{n} and let C_{m} denote the set of isotopy classes of oriented simple closed curves on D_{n} which surround m ≥ 2 of the punctures. Let C = ∪_{m=2}^{n}C_{m}. Then the braid group B_{n} acts as (isotopy classes of) diffeomorphisms of D_{n} [Bi, Ch.1]. In fact B_{n} acts transitively on C_{m} for all 2 ≤ m ≤ n. The group B_{n} has standard generators σ_{1}, …, σ_{n−1} and presentation

σ_{i}σ_{i+1}σ_{i} = σ_{i+1}σ_{i}σ_{i+1}, for i = 1, …, n − 2
σ_{i}σ_{j} = σ_{j}σ_{i}, for |i − j| > 1.

Here the generator σ_{i} acts as a half-twist [Bi] on D_{n} interchanging π_{i} and π_{i+1} and has a representative diffeomorphism which is supported in a tubular neighbourhood of the arc a_{i} (see Figure 1). For 1 ≤ i < j ≤ n let γ_{ij} be a simple closed curve isotopic to the boundary of a tubular neighbourhood of a_{i} ∪a_{i+1} ∪⋯∪a_{j−1}. Given any c, d ∈ C we let ι(c, d) denote the geometric intersection-number of c and d. This is the minimum number of points of c′ ∩ d′, where c′ and d′ are any simple closed curves isotopic to c and d. Note that ι(c, d) is always even since D_{n} is planar.

Let R be a commutative ring with identity. In a previous paper [H2, §3] we have shown that by representing the free group π_{1}(D_{n}) using transvections (see below) and looking at certain traces we obtain an injective map ϕ : C → R_{n}, where

R_{n} = R[a_{12}, a_{13}, …, a_{1n}, a_{21}, a_{23}, …, a_{2n}, …, a_{n1}, …, a_{n n−1}]

Typeset by AMS-TEX

is a polynomial ring in commuting indeterminates a_{ij}, 1 ≤ i ≠ j ≤ n. It will be convenient for us to put a_{ii} = 0 for i = 1, …, n. In this situation we obtain an action of B_{n} on the ring R_{n} i.e. we have a homomorphism

ψ_{n} : B_{n} → Aut(R_{n});

the kernel of ψ_{n} is the centre of B_{n} [H1]. We note that to each curve γ ∈ C_{m} there is a 1/m twist τ_{γ} whose mth power is the Dehn twist [Bi] about the curve γ. In particular for 1 ≤ k < m ≤ n we have

τ_{γ_{km}} = σ_{m−1} … σ_{k+1}σ_{k}.

In general if γ ∈ C_{m}, then there is α ∈ B_{n} such that α(γ_{1m}) = γ; in this situation we have τ_{γ} = ατ_{γ_{1m}}α^{−1}. In this paper we prove:

Theorem 1.1. Let n, m, r ∈ ℤ, n ≥ m ≥ 2, r ≥ 0. Then there exist polynomials 𝔅_{mr}(x) ∈ R_{n}[x] such that for all γ ∈ C with ι(γ_{1m}, γ) = 2r we have

𝔅_{mr}(ψ_{n}(τ_{γ_{1m}}^{m}))ϕ(γ) = 0.

Further, for γ ∈ C there is an integer 2r such that 𝔅_{mr}(ψ_{n}(τ_{γ_{1m}}^{m}))ϕ(γ) = 0 and the minimal such 2r is equal to ι(γ_{1m}, γ).

In part I of this paper [H4] we proved a slightly different version of this result, but only in the case m = 2. The polynomials obtained were different from those that we obtain in Theorem 1.1 for m = 2. The difference is best described by saying that the polynomials obtained in I for m = 2 are like the minimal polynomials for the action of τ_{γ_{12}}^{2}, while those obtained in Theorem 1.1 are like the characteristic equation.

The polynomials 𝔅_{mr} can be determined algorithmically (see Examples 5.5). Since B_{n} acts transitively on C_{m}, m ≤ n and γ_{1m} ∈ C_{m} Theorem 1.1 gives and algorithm for calculating the intersection numbers ι(γ, γ′) for any simple closed curves γ, γ′. Existing algorithms for calculating intersection number include those of Reinhart [R], Zieschang [Z1, Z2], Chillingworth [C1, C2], Birman and Series [BS], Cohen and Lustig [CL] and Tan [T].

We now give some more information about these polynomials 𝔅_{mr}. For fixed m ≥ 2 we let Π_{m} = T_{1}T_{2} … T_{m} where the T_{i} are certain m × m matrices (transvections):

T_{i} = ( \begin{matrix}1 & 0 & … & 0 & … & 0 & 0 \\ 0 & 1 & … & 0 & … & 0 & 0 \\ ⋮ & ⋮ & ... & ⋮ & … & ⋮ & ⋮ \\ a_{i1} & a_{i2} & … & 1 & … & a_{im−1} & a_{im} \\ ⋮ & ⋮ & … & ⋮ & ... & ⋮ & ⋮ \\ 0 & 0 & … & 0 & … & 1 & 0 \\ 0 & 0 & … & 0 & … & 0 & 1 \end{matrix} )

where the non-zero off-diagonal entries occur in the ith row. Here a transvection [A] is a matrix M = I_{m} + A where I_{m} is the m × m identity matrix, det(M) = 1 and A^{2} = 0. In particular, conjugates of transvections are transvections.

Now in [H2, Theorem 2.8] we have shown that for all 2 ≤ m ≤ n the characteristic polynomial

χ_{m}(x) = ∑_{i=0}^{m} c_{mi}x^{i}

of the matrix Π_{m} has coefficients c_{mi} which are invariant under the action of B_{m}. We note that the c_{mi} are polynomials of degree m for 1 ≤ i < m [H2; Theorem 2.8]. If m < 6, then they generate the ring of invariants for the action of B_{m} on a certain subring of R_{m} [H3]. Note that we have c_{mm} = 1 and c_{m0} = m.

Theorem 1.2. Let n ≥ m ≥ 2 and r ≥ 0. Then the coefficients of the polynomials 𝔅_{mr}(x) are polynomials in ℚ[c_{m1}, c_{m2}, …, c_{m−1m}].

These polynomials have the property that 𝔅_{mr} divides 𝔅_{ms} whenever r ≤ s. Also, if α is a root of 𝔅_{mr} and β is a root of 𝔅_{ms}, then αβ is a root of 𝔅_{m(r+s)}.

The representation of B_{n} in Aut(R_{n}) will be described in detail later, but should be thought of in the following way. Let F_{n} =< x_{1}, …, x_{n} > denote the free group of rank n, which we identify with the fundamental group of D_{n}. The Magnus expansion M of F_{n} [Ma, MKS] is defined as follows: Let P_{n} be the algebra of formal power series in non-commutative variables X_{1}, …, X_{n} over ℂ. Then M is the homomorphism M : F_{n} → P_{n} given on generators by

M(x_{i}) = 1 + X_{i}, M(x_{i}^{−1}) = 1 − X_{i} + X_{i}^{2} − X_{i}^{3} + ….

Then M is injective, has connections with Fox’s free differential calculus and is used to define interesting representations of the braid groups [Bi]. We obtain our representation of B_{n} in Aut(R_{n}) by looking at the situation where in P_{n} we have the extra relations X_{i}^{2} = 0 and X_{i}X_{j} = X_{j}X_{i} for all i, j = 1, …, n. This is accomplished concretely by representing the free group F_{n} using transvections. The standard action of B_{n} on F_{n} [Bi] then gives rise to an action of B_{n} on R_{n} which is a homomorphic image of P_{n}. This is all explained in more detail in §§2, 4. A quotient of this algebra was used by Milnor [Mi1, Mi2] to study links.

For i, j, k, …, r, s ∈ {1, 2, …, n} let c_{ijk...rs} denote the cycle a_{ij}a_{jk} … a_{rs}a_{si} ∈ R_{n}. Then the cycles generate a subalgebra of R_{n} denoted Y_{n}. A cycle c_{ijk...rs} will be called simple if i, j, k, ..., r, s are all distinct. The ring Y_{n} is generated by the (finite number of) simple cycles. We also show

Theorem 1.3. For m ≥ 2 and all α ∈ Y_{n} there is r ∈ ℤ such that

𝔅_{mr}(ψ_{n}(τ_{γ_{1m}}^{m}))(α) = 0.

In what follows we will usually write 𝔅_{mr}(τ_{γ_{1m}}^{m}) instead of 𝔅_{mr}(ψ_{n}(τ_{γ_{1m}}^{m})).

§2 Preliminary results on the action of B_{n} on R_{n}

In this section we construct the representation ψ_{n} : B_{n} → Aut(R_{n}). In §4 we will give a more explicit description of the action.

Note that the action of B_{n} on D_{n} fixes the boundary of D_{n} and so there is an induced action of B_{n} on the fundamental group of D_{n} (where we choose a base point p on the boundary of D_{n}). This fundamental group is the free group F_{n} of rank n. We choose a standard set of generators x_{1}, …, x_{n} for F_{n}, where x_{i} is a simple closed curve enclosing the ith puncture π_{i} and such that x_{1}x_{2} … x_{n} is parallel to the boundary of D_{n}. See Figure 1.

In Figure 1 we have shown arcs a_{i} and the generators x_{i}. The action of B_{n} on the generators x_{i} is as follows: let 1 ≤ j < n; then

σ_{j}(x_{i}) = x_{i} if i ≠ j, j + 1, σ_{j}(x_{j}) = x_{j}x_{j+1}x_{j}^{−1}, σ_{j}(x_{j+1}) = x_{j}.

We now find that a particularly convenient representation of the free group F_{n} is given by the transvections T_{i}. That < T_{1}, …, T_{n} > is a free group of rank n is shown in [H2]. This allows us to identify x_{i} and T_{i} for i = 1, …, n and so to identify F_{n} and < T_{1}, …, T_{n} > .

Now, from the above, B_{n} acts by automorphisms on F_{n} in such a way that for α ∈ B_{n} the matrix α(T_{i}) is a conjugate of some T_{j}, 1 ≤ j ≤ n i.e. α(T_{i}) is also a transvection. Further, if c ∈ C, then c represents a conjugacy class in F_{n} and so its trace is well-defined (the trace of the corresponding product of transvections in F_{n} =< T_{1}, …, T_{n} >). In fact one easily sees that trace(c) ∈ Y_{n} [H1]. Then a map ϕ = ϕ_{n} : C → R_{n} is defined uniquely by

ϕ_{n}(c) = trace(c) − n.  (2.1)

Thus ϕ_{n} can be thought of as being defined on certain conjugacy classes of elements of F_{n} (namely those representing simple closed curves). The map ϕ can be extended to all of F_{n}, by the requirement that for s ∈ F_{n} we have ϕ(s) = trace(s) − n. From [H1] (see also §4) we have

Lemma 2.1. If w = T_{i_{1}}^{k_{1}}T_{i_{2}}^{k_{2}} … T_{i_{r}}^{k_{r}} ∈< T_{1}, …, T_{n} > where r > 1, k_{u} ≠ 0, i_{u} ≠ i_{u+1} for u = 1, …, r − 1 and i_{r} ≠ i_{1}, then ϕ(w) is a polynomial in Y_{n} of degree r with a unique monomial of highest degree.

Now for m ≥ n and s ∈ F_{n} we may also consider s as an element of F_{m} under the natural inclusion of F_{n} in F_{m}. In this case we note that ϕ(s) has the same value whether we consider s as an element of F_{n} or F_{m}.

Now for all i, j we have trace(T_{i}T_{j}) = a_{ij}a_{ji} + n and in general if A, B ∈ F_{n}, then

trace(AT_{i}A^{−1}BT_{j}B^{−1}) = b_{ij}b_{ji} + n

where b_{ij} ∈ R_{n} (see [H1] and §4). It is also easy to see that there is a natural choice so that

b_{ij} = a_{ij} + terms of higher degree.

Now for α ∈ B_{n} the image α(T_{i}) is a conjugate AT_{j}A^{−1} for some A ∈ F_{n} and 1 ≤ j ≤ n. Here the action of α on the a_{ij} is defined by

trace(α(T_{i})α(T_{j})) = α(a_{ij})α(a_{ji}) + n,

(see §4 for more details) so that it has the following naturality property (with respect to the action of B_{n} on F_{n}): for all w ∈< T_{1}, …, T_{n} > we have

ϕ(α(w)) = α(ϕ(w)).

For example the action of σ_{i} is given by

σ_{i}(a_{i i+1}) = a_{i+1 i}, σ_{i}(a_{i+1 i}) = a_{i i+1},
σ_{i}(a_{h i}) = a_{h i+1} + a_{hi}a_{i i+1}, σ_{i}(a_{h i+1}) = a_{h i},  (2.2)
σ_{i}(a_{i h}) = a_{i+1 h} − a_{i+1 i}a_{ih}, σ_{i}(a_{i+1 h}) = a_{i h},

where 1 ≤ h ≤ n and h ≠ i, i + 1.

It follows from [H1,Theorem 2.5 and Theorem 6.2] that the kernel of the action of B_{n} on R_{n} is the centre of B_{n} and that if B_{n} and R_{n} are thought of as sub-objects of B_{n+1} and R_{n+1} (respectively), then the action of B_{n} on R_{n+1} is faithful. The proof is essentially and application of Lemma 2.1. We note as in [H2] that there is a natural ring involution ∗ on R_{n} which commutes with the action of B_{n}, so that for α ∈ B_{n} we have

α(w)^{∗} = α(w^{∗})  (2.3)

for all w ∈ R_{n}. This involution is determined by its action on the generators a_{ij} which is as follows:

a_{ij}^{∗} = −a_{ji}.

This involution has the following property:

trace(A^{−1}) = trace(A)^{∗},

for all A ∈ F_{n}. Thus for c ∈ C we have ϕ(c^{−1}) = ϕ(c)^{∗}, where c^{−1} is the curve c with its orientation reversed. We also have b_{ji} = −b_{ij}^{∗}, for b_{ij}, b_{ji} as in (2.1).

§3 Preliminaries on symmetric functions

We will need the following results on symmetric functions; these results can all be found in [M].

Let a_{1}, …, a_{n} be algebraically independent indeterminates. For (α_{1}, …, α_{n}) ∈ ℕ^{n} we let a^{α} denote the monomial

a_{1}^{α_{1}}a_{2}^{α_{2}} … a_{n}^{α_{n}}.

Recall that a partition is a sequence λ = (λ_{1}, λ_{2}, …, λ_{r}) of non-negative integers such that λ_{1} ≥ λ_{2} ≥ . . .. The number r is called the length of the partition and the weight of the partition λ is the sum |λ| = λ_{1} + λ_{2} + ⋯ + λ_{r}. The polynomial

m_{λ}(a_{1}, a_{2}, …, a_{r}) = ∑ a^{α},

where the sum is taken over all distinct permutations α of λ = (λ_{1}, …, λ_{r}). The m_{λ} form a basis for the ring of symmetric polynomials in the variables a_{1}, …, a_{r}.

The elementary symmetric functions e_{i} are defined as follows: for r ≥ 0 let

e_{r} = ∑_{i_{1}<i_{2}<···<i_{r}} a_{i_{1}}a_{i_{2}} … a_{i_{r}} = m_{(1^{r})}.

For a partition λ we let

e_{λ} = e_{λ_{1}}e_{λ_{2}} … e_{λ_{r}}.

Note that the degree of m_{λ} or e_{λ} is |λ|. The elementary symmetric functions satisfy:

∏_{i=1}^{r}(1 − ta_{i}) = ∑_{i=0}^{r} e_{i}(−t)^{i}.

From [M, p.35] we get

Lemma 3.1. For all r ≥ 0 we have:

∏_{i,j=1}^{r}(1 + a_{i}b_{j}) = ∑_{λ} e_{λ}(a_{1}, …, a_{r})m_{λ}(b_{1}, …, b_{r})

= ∑_{λ} m_{λ}(a_{1}, …, a_{r})e_{λ}(b_{1}, …, b_{r}).

Here the sum in the right hand side is over all partitions of weight at most r. □

We note that since the μ_{λ} and the e_{λ} are both algebraically independent generators for the ring of symmetric polynomials, that it is possible to express the μ_{λ} as linear combinations of the e_{λ} with rational coefficients, and vice versa.

We have the following easy result:

Lemma 3.2. Suppose that p is a polynomial in ℂ[a_{1}, a_{2}, …, a_{n}, b_{1}, b_{2}, …, b_{r}] of degree m in a_{1}, a_{2}, …, a_{n}. Suppose that p is invariant under the natural action of S_{n} on a_{1}, a_{2}, …, a_{n}. Then p is a polynomial in ℂ[e_{λ}(a_{1}, a_{2}, …, a_{n}), b_{1}, b_{2}, …, b_{r}] where the degree of each of the e_{λ}(a_{1}, a_{2}, …, a_{n}) that we need is at most m. □

§4 Action of B_{n} on R_{n} continued

In this section we describe in greater detail the action of B_{n} on R_{n} as outlined in §2 so as to be able to give explicit formulae for the action of τ_{γ_{1m}}^{m}. In general [A] a transvection in SL_{n}(Q) (for a commutative ring Q with identity) can be defined as a pair T = (ϕ, d) where d ∈ Q^{n} and ϕ is an element of the dual space of Q^{n} satisfying ϕ(d) = 0. The action is given by

T (x) = x + ϕ(x)d for all x ∈ Q^{n}.

Then we have [H1, Lemma 2.1]

Lemma 4.1. Let T = (ϕ, d) and U = (ψ, e) be two transvections. Then for all λ ∈ ℤ we have

U^{λ}T U^{−λ} = (ϕ − λϕ(e)ψ, U^{λ}(d)). □

Let {T_{1} = (ϕ_{1}, d_{1}), …, T_{n} = (ϕ_{n}, d_{n})} be a fixed set of transvections in SL_{n}(R_{n}) where ϕ_{i}(d_{j}) = a_{ij} for all 1 ≤ i ≠ j ≤ n. For any set of transvections T = {T_{1} = (ϕ_{1}, e_{1}), …, T_{n} = (ϕ_{n}, e_{n})} we let M(T) denote the n × n matrix (ϕ_{i}(e_{j})) and we call M(T) the M-matrix of the set of transvections T.

Any monomial that can be written in the form a_{j_{1}j_{2}}a_{j_{2}j_{3}} … a_{j_{r−1}j_{r}} will be called a j_{1}j_{r}-word. Note that by (2.2) if α ∈ B_{n} and 1 ≤ i ≠ j ≤ n, then α(a_{ij}) is a sum of rs-words, where α(T_{i}) is a conjugate of T_{r} and α(T_{j}) is a conjugate of T_{s}. Let α ∈ B_{n} where α(T_{i}) = w_{i}T_{j}w_{i}^{−1} in freely reduced form for i = 1, ..., n and where w_{i} = w_{i}(T_{1}, …, T_{n}). Then for i = 1, …, n we have w_{i}T_{i}w_{i}^{−1} = (ψ_{i}, f_{i}) for some ψ_{i}, f_{i} determined by Lemma 4.1, which shows that

ψ_{i} = q_{1}ϕ_{1} + ⋯ + q_{n}ϕ_{n} and f_{i} = p_{1}d_{1} + ⋯ + p_{n}d_{n},

where p_{1}, …, p_{n}, q_{1}, …, q_{n} ∈ R_{n}. We define the action of B_{n} on R_{n} by

α(a_{ij}) = ψ_{i}(f_{j}).

This agrees with the previous definition. Thus the M-matrix M(α(T_{1}), …, α(T_{n})) is α(M(T)).

From Lemma 2.3 of [H1] we have:

Lemma 4.2. Let 1 ≤ i, j ≤ n, α ∈ B_{n} where α(T_{i}) = C_{1}T_{k}C_{1}^{−1}, α(T_{j}) = C_{2}T_{p}C_{2}^{−1}, with C_{1}, C_{2} ∈< T_{1}, …, T_{n} > and let C = C_{1}^{−1}C_{2} = T_{j_{1}}^{q_{1}} … T_{j_{r}}^{q_{r}} be freely reduced with j_{r} ≠ p, j_{1} ≠ k, q_{s} ≠ 0 for s = 1, …, r and j_{s} ≠ j_{s+1}, for s = 1, …, r − 1. Then

α(a_{ij}) = ∑_{h=1}^{n} A_{h}a_{hp}

where A_{h} is equal to the sum of all the products of the form

q_{r_{1}}q_{r_{2}} … q_{r_{m}}a_{kj_{r_{1}}} a_{j_{r_{1}} j_{r_{2}}} … a_{j_{r_{m−1}} j_{rm}}

where 1 ≤ r_{1} < r_{2} < ⋯ < r_{m} ≤ r and j_{r_{m}} = h. If p ≠ j_{r}, then the summand of α(a_{ij}) of highest degree is unique and is equal to

±q_{1}q_{2} … q_{r}a_{kj_{1}}a_{j_{1}j_{2}} … a_{j_{r−1}j_{r}}a_{j_{r}p}. □

For example if α(T_{1}) = T_{3}T_{2}^{−1}T_{1}T_{2}T_{3}^{−1} and α(T_{2}) = T_{2}^{−1}T_{3}T_{2}, then we would have C = T_{2}T_{3}^{−1}T_{2}^{−1} and

α(a_{12}) = a_{13} + a_{13}a_{32}a_{23} + a_{12}a_{23}a_{32}a_{23}.

For each m ≥ 2 we let τ_{m} = τ_{γ_{1m}} = σ_{m−1} … σ_{2}σ_{1}, the 1/m twist about the curve γ_{1,m}. Then we have:

Lemma 4.3. (i) For all 1 ≤ i ≤ m we have

τ_{m}^{m}(x_{i}) = (x_{1}x_{2} … x_{m})x_{i}(x_{1}x_{2} … x_{m})^{−1};

(ii) For all 1 ≤ i ≠ j ≤ m we have τ_{m}^{m}(a_{ij}) = a_{ij}; (iii) For 1 ≤ i ≤ m < r ≤ n we have

τ_{m}^{m}(a_{ri}) = a_{ri} + ∑_{s=1}^{m} ∑_{1≤j_{1}<j_{2}<···<j_{s}≤m} a_{rj_{1}}a_{j_{1}j_{2}}a_{j_{2}j_{3}} … a_{j_{s−1}j_{s}}a_{j_{s}i}.

(iv) For 1 ≤ i ≤ m < r ≤ n we have

τ_{m}^{m}(a_{ir}) = a_{ir} + ∑_{s=1}^{m} ∑_{1≤j_{1}<j_{2}<···<j_{s}≤m}(−1)^{s}a_{ij_{s}}a_{j_{s}j_{s−1}} … a_{j_{3}j_{2}}a_{j_{2}j_{1}}a_{j_{1}r}.

(v) For all m < u ≠ v ≤ n we have τ_{m}(a_{uv}) = a_{uv}.

Proof. (i) follows from the action of B_{n} given in §2. The rest all follow from (i) and Lemma 4.2. □

For y > m let v_{y} be the vector (a_{y1}, a_{y2}, …, a_{ym}). Let w_{y} = (a_{1y}, a_{2y}, …, a_{my}) = −v_{y}^{∗}. Then we have :

Lemma 4.4. For all 2 ≤ m ≤ n there is a m × m matrix M_{m} ∈ SL(m, R_{m}) such that

τ_{m}^{hm}(v_{y}) = v_{y}M_{m}^{h},

for all h ∈ ℤ and all n ≥ y > m. In fact M_{m} = T_{1}T_{2} … T_{m}.

We also have τ_{m}^{hm}(w_{y}) = w_{y}(M_{m}^{∗})^{h} for all h ∈ ℤ.

Proof. Fix such a y. Consider the matrix Π_{m} = T_{1}T_{2} … T_{m}. Its (i, k) entry is

∑_{s=0}^{m} ∑_{i≤j_{1}<j_{2}<···<j_{s}≤m} a_{ij_{1}}a_{j_{1}j_{2}} … a_{j_{s−1}j_{s}}a_{j_{s}k}.

But by Lemma 4.3 this is exactly equal to the coefficient of a_{yi} in τ_{m}^{m}(a_{yk}), as required for h = 1. But the entries of Π_{m} are invariant under the action of τ_{m}^{m} since these entries are in R_{m} (use Lemma 4.3 again). So we see that τ_{m}^{hm}(v_{y}) = v_{y}M_{m}^{h}, for all h ∈ ℤ.

Now v_{y}^{∗} = −w_{y} and if we apply the automorphism ^{∗} to the equation τ_{m}^{hm}(v_{y}) = v_{y}M_{m}^{h}, and use (2.3), then we obtain w_{y}(M_{m}^{∗})^{h} = τ_{m}^{hm}(w_{y}) as required. □

Lemma 4.5. The matrix Π_{n} = T_{1}T_{2} … T_{n} has n distinct roots.

Proof. Let T′_{i} be the n × n matrix T_{i} with all the indeterminates a_{ij} replaced by an indeterminate b_{i}. We prove by induction that there are positive real numbers b_{i} such that for all m ≥ 2 the matrix T′_{1}T′_{2} … T′_{m} has n − m eigenvalues equal to 1, the rest being distinct and not equal to 1. The case m = 2 is easy (take b_{1} = b_{2} = 1). Now assume that T′_{1}T′_{2} … T′_{m}, 2 ≤ m < n satisfies the inductive hypothesis for some positive real choice of b_{1}, …, b_{m}. Consider P = T′_{1}T′_{2} … T′_{m}T′_{m+1}. Since b_{1}, …, b_{m} > 0, the matrix T′_{1}T′_{2} … T′_{m} has all strictly positive entries in its first m rows. Thus the trace of P is a non-constant function of b_{m+1}. Thus the characteristic polynomial of P is a function which depends continuously and non-constantly on b_{m+1}. When b_{m+1} = 0, this polynomial has m distinct roots not equal to 1, and so we can choose b_{m+1} small enough so that P has m + 1 distinct eigenvalues not equal to 1. (In fact if one takes b_{1} = 1 and b_{i} = 1/(i − 1) for n ≥ i ≥ 2, then the eigenvalues are \frac{n−2}{n−1}, \frac{n−3}{n−2}, …, \frac{2}{1} together with the two positive real and distinct roots of the quadratic z^{2} − (2n − 1)z + n − 1). □

Lemma 4.6. Suppose that the matrix T_{1}T_{2} … T_{n} has characteristic polynomial χ_{n}(x). Then the matrix (T_{1}T_{2} … T_{n})^{∗} has characteristic polynomial

χ_{n}(x)^{∗} = (−x)^{n}χ(1/x).

The eigenvalues of (T_{1}T_{2} … T_{n})^{∗} are the inverses of the eigenvalues of T_{1}T_{2} … T_{n}.

Proof. The first claim is an immediate consequence of [H2; Corollary 2.7], which says that the same result is true for all matrices in < T_{1}, …, T_{n} >. The second claim follows from the first. □

Now suppose that μ is a monomial in R_{n}. Then we can write μ = μ_{1}μ_{2}μ_{3}μ_{4} where each μ_{i} is a monomial with

μ_{1} ∈ R_{m}, μ_{2} ∈ R[a_{ik}|1 ≤ i ≤ m < k], μ_{3} ∈ R[a_{ki}|1 ≤ i ≤ m < k]
and μ_{4} ∈ R[a_{ik}|m < i, k ≤ n].  (4.1)

Now τ_{m}^{m} fixes μ_{1} and μ_{4} and its action on the generators a_{ik} occurring in μ_{2} and μ_{3} is given by Lemma 4.3. Further, we note that if μ ∈ Y_{n}, then μ_{2} and μ_{3} have the same degrees.

§5 Kronecker products

We now wish to recall some elementary facts about Kronecker products of matrices. The basic reference is [HJ, §4.2]. Let A = (a_{ij}) be an n × n matrix and B = (b_{ij}) be an m × m matrix. Then the Kronecker (or tensor) product A ⊗ B is the nm × nm block matrix

A ⊗ B = ( \begin{matrix}a_{11}B & a_{12}B & … & a_{1n}B \\ a_{21}B & a_{22}B & … & a_{2n}B \\ ⋮ & ⋮ & ... & ⋮ \\ a_{n1}B & a_{n2}B & … & a_{nn}B \end{matrix} ) .

This can be interpreted as follows: Suppose that A and B are the matrices of linear transformations α : V → V and β : W → W acting on vector spaces V and W with respect to bases u_{i} and v_{i} (respectively). Then the action of α ⊗ β on V ⊗ W with respect to the basis u_{i} ⊗ v_{j} is given by A ⊗ B.

Lemma 5.1. [HJ; p. 245] If the eigenvalues of A are a_{1}, …, a_{n} and the eigenvalues of B are b_{1}, …, b_{m} (both counting multiplicities), then the eigenvalues of A ⊗ B are a_{i}b_{j} for all i = 1, …, n, j = 1, …, m (counting multiplicities). □

We apply this to the situation where y > m is fixed and τ_{m}^{m} is acting on the subring R[a_{iy}, a_{yj}|1 ≤ i, j ≤ m] of R_{n}. Then by Lemma 4.3 τ_{m}^{m} fixes each element of R_{m} and there are matrices M_{m}, M_{m}^{∗} such that τ_{m}^{m}(v_{y}) = v_{y}M_{m} and τ_{m}^{m}(w_{y}) = w_{y}M_{m}^{∗}. But then, by the above remarks, the action of τ_{m}^{m} on products like a_{iy}a_{yj} is completely determined by the Kronecker product N_{m} = M_{m} ⊗ M_{m}^{∗}. In fact, since the matrix M_{m} does not depend on y the Kronecker product N_{m} also determines the action of τ_{m}^{m} on products like a_{iy}a_{zj} for all n ≤ y, z > m. Thus we have

Lemma 5.2. Let χ_{N}(t) denote the characteristic polynomial of N_{m}. If we have a monomial μ = μ_{1}μ_{2}μ_{3}μ_{4} ∈ Y_{n} as in (4.1), where μ_{2} and μ_{3} have degree 1, then

χ_{N}(τ_{m}^{m})(μ) = 0.

Proof. If μ = μ_{1}μ_{2}μ_{3}μ_{4}, then τ_{m}^{m} fixes each of μ_{1} and μ_{4}; thus χ_{N}(τ_{m}^{m})(μ) = μ_{1}μ_{4}χ_{N}(τ_{m}^{m})(μ_{2}μ_{3}), the latter being 0 by the above discussion. □

Now the action of τ_{m}^{m} on an arbitrary monomial in Y_{n} is given by

Proposition 5.3. The action of τ_{m}^{m} on any monomial μ = μ_{1}μ_{2}μ_{3}μ_{4} ∈ Y_{n} where μ_{2} and μ_{3} have degree r is determined by the r-fold Kronecker product N_{m}^{⊗r}. If χ_{N^{⊗r}}(t) denotes the characteristic polynomial of N_{m}^{⊗r}, then

χ_{N^{⊗r}}(τ_{m}^{m})(μ) = 0. □

Corollary 5.4. If α ∈ Y_{n}, then there is r ∈ ℤ such that

χ_{N^{⊗r}}(τ_{m}^{m})(α) = 0. □

This proves Theorem 1.3, since we can let

𝔅_{m0}(x) = x − 1 and 𝔅_{mr}(x) = χ_{N^{⊗r}}(x) for r ≥ 1.

We note that ι(γ_{1m}, γ) = 0 if and only if τ_{m}^{m}(γ) = γ if and only if 𝔅_{m0}(τ_{m}^{m})(ϕ(γ)) = 0.

Now by Lemma 4.6 the eigenvalues of Π_{m} and Π_{m}^{∗} are inverses of each other. Thus by Lemma 5.1 the matrix N_{m} = M_{m} ⊗ M_{m}^{∗} has 1 as an eigen-value with multiplicity at least m ≥ 2. Thus also from Lemma 5.1 we see that any root of χ_{N^{⊗r}} is also a root of χ_{N^{⊗s}} whenever 0 ≤ r < s. It also easily follows that if α is a root of χ_{N^{⊗r}} and β is a root of χ_{N^{⊗s}}, then αβ is a root of χ_{N^{⊗r+s}}. This is the proof of the second part of Theorem 1.2.

Examples 5.5. (i) If m = 2, then we let I = −2 − a_{12}a_{21} so that x^{2} + Ix + 1 is the characteristic polynomial of M_{2}. It has roots a, 1/a say, since det(M_{2}) = 1. Then by Lemma 4.6 M_{2}^{∗} also has roots a, 1/a and so χ_{N}(x) has roots 1, 1, a^{2}, 1/a^{2} (by Lemma 5.1). The roots of χ_{N^{⊗r}}(x) are thus all of the 4^{r} products a_{1}a_{2} … a_{r} for all possible a_{i} = 1, 1, a^{2}, 1/a^{2}. To find χ_{N^{⊗r}}(x) one multiplies out

∏_{a_{1},...,a_{r}=1,1,a^{2},1/a^{2}}(x − a_{1})(x − a_{2}) … (x − a_{r})

and finds that this is a polynomial in x and I.

To be more specific we let s_{0}(x) = x − 1 and for i ≥ 1 we let

s_{i} = (x − a^{2i})(x − 1/a^{2i}).

Thus for example, we have

s_{1}(x) = x^{2} + (2 − I^{2})x + 1;
s_{2}(x) = x^{2} + (−2 + 4I^{2} − I^{4})x + 1;
s_{3}(x) = x^{2} + (2 − 9I^{2} + 6I^{4} − I^{6})x + 1;
s_{4}(x) = x^{2} + (−2 + 16I^{2} − 20I^{4} + 8I^{6} − I^{8})x + 1.

The s_{i} are quadratic polynomials in x related to the polynomials 𝔄_{i} defined in [H4]. We have

𝔅_{2 0}(x) = s_{0}(x), 𝔅_{2 1}(x) = s_{0}(x)^{2}s_{1}(x).

By the above we see that 𝔅_{2 m+1}(x) = χ_{N^{⊗m+1}}(x) will factor into a product of the form

𝔅_{2m}(x)^{2}C_{m1}(x)C_{m2}(x)

where C_{m1}(x) (respectively C_{2m}(x)) is obtained from 𝔅_{2m}(x) by replacing each root α of 𝔅_{2m}(x) by αa^{2} (respectively α/a^{2}). Thus each 𝔅_{2m}(x) is a product of powers of the s_{i}(x). This has the effect of replacing s_{0}(x) by s_{1}(x), s_{1}(x) by s_{0}(x)^{2}s_{2}(x), s_{2}(x) by s_{1}(x)s_{3}(x), s_{3}(x) by s_{2}(x)s_{4}(x) etc. We obtain

𝔅_{21}(x) = s_{0}(x)^{2}s_{1}(x);
𝔅_{22}(x) = s_{0}(x)^{6}s_{1}(x)^{4}s_{2}(x);
𝔅_{23}(x) = s_{0}(x)^{20}s_{1}(x)^{15}s_{2}(x)^{6}s_{3}(x); etc.

Using the fact that 𝔅_{2 m+1}(x) = 𝔅_{2m}(x)^{2}C_{m1}(x)C_{m2}(x) for C_{1m}, C_{2m} as described above one easily sees that the general result is

𝔅_{2n}(x) = ∏_{i=0}^{n} s\binom{2n}{n−i}_{i}.

(ii) For the case m = 3 we see that if M_{3} has roots a, b, c (which are distinct by Lemma 4.5), then abc = 1 since det(M_{3}) = 1. Further, M_{3}^{∗} has roots 1/a, 1/b, 1/c and so N_{3} = M_{3} ⊗ M_{3}^{∗} has roots 1, 1, 1, \frac{a}{b}, \frac{a}{c}, \frac{b}{a}, \frac{b}{c}, \frac{c}{a}, \frac{c}{b} (by Lemma 5.1). This allows one to find that

𝔅_{31}(x) = (x − 1)^{3}[x^{6} + (3 − p_{1}p_{2})(x + x^{5}) + (6 − 5p_{1}p_{2} + p_{1}^{3} + p_{2}^{3})(x^{2} + x^{4})

+ (7 + 2p_{1}^{3} − 6p_{1}p_{2} − p_{1}^{2}p_{2}^{2} + 2p_{2}^{3})x^{3} + 1];

𝔅_{32}(x) = (x − 1)^{15}[x^{3} + (3p_{1}p_{2} − p_{2}^{3} − 3)x^{2} + (p_{1}^{3} − 3p_{1}p_{2} + 3)x − 1]^{2}×

[x^{3} + (−p_{1}^{3} + 3p_{1}p_{2} − 3)x^{2} + (−3p_{1}p_{2} + p_{2}^{3} + 3)x − 1]^{2}×

[x^{6} + (3 − p_{1}p_{2})(x + x^{5}) + (6 − 5p_{1}p_{2} + p_{1}^{3} + p_{2}^{3})(x^{2} + x^{4})

+ (7 + 2p_{1}^{3} − 6p_{1}p_{2} − p_{1}^{2}p_{2}^{2} + 2p_{2}^{3})x^{3} + 1]^{8}×

[x^{6} + (2p_{1}^{3} − p_{1}^{2}p_{2}^{2} − 4p_{1}p_{2} + 2p_{2}^{3} + 3)(x + x^{5})

+ (p_{1}^{6} − 6p_{1}^{4}p_{2} + 2p_{1}^{3} + 19p_{1}^{2}p_{2}^{2} − 6p_{1}p_{2}^{4} − 20p_{1}p_{2} + p_{2}^{6} + 2p_{2}^{3} + 6)(x^{2} + x^{4})

+ (−2p_{1}^{6} + 4p_{1}^{5}p_{2}^{2} − p_{1}^{4}p_{2}^{4} + 4p_{1}^{4}p_{2} − 16p_{1}^{3}p_{2}^{3} − 4p_{1}^{3} + 4p_{1}^{2}p_{2}^{5} + 26p_{1}^{2}p_{2}^{2} + 4p_{1}p_{2}^{4}

− 24p_{1}p_{2} − 2p_{2}^{6} − 4p_{2}^{3} + 7)x^{3} + 1].

where p_{1} = a_{12}a_{21} + a_{13}a_{31} + a_{23}a_{32} − a_{13}a_{32}a_{21} and p_{2} = −p_{1}^{∗}. As in the case m = 2 one can find a (more complicated) recursion for the polynomials 𝔅_{3n}(x).

(iii) For m = 4 similar reasoning allows one to find that

𝔅_{41}(x) =(x − 1)^{4}[(1 + x^{12}) + (x + x^{11})(−p_{2}p_{3} + 4)

+ (x^{2} + x^{10})(−2p_{1}^{2} + p_{1}p_{2}^{2} + p_{1}p_{3}^{2} − 4p_{2}p_{3} + 10)

+ (x^{3} + x^{9})(−p_{1}^{2}p_{2}p_{3} − 8p_{1}^{2} + 7p_{1}p_{2}^{2} + 7p_{1}p_{3}^{2} − p_{2}^{4} − 13p_{2}p_{3} − p_{3}^{4} + 20)

+ (x^{4} + x^{8})(p_{1}^{4} − 8p_{1}^{2}p_{2}p_{3} − 16p_{1}^{2} + p_{1}p_{2}^{3}p_{3} + 18p_{1}p_{2}^{2}

+ p_{1}p_{2}p_{3}^{3} + 18p_{1}p_{3}^{2} − 3p_{2}^{4} − p_{2}^{2}p_{3}^{2} − 24p_{2}p_{3} − 3p_{3}^{4} + 31)

+ (x^{5} + x^{7})(4p_{1}^{4} − p_{1}^{3}p_{2}^{2} − p_{1}^{3}p_{3}^{2} − 19p_{1}^{2}p_{2}p_{3} − 24p_{1}^{2} + 5p_{1}p_{2}^{3}p_{3} + 29p_{1}p_{2}^{2}

+ 5p_{1}p_{2}p_{3}^{3} + 29p_{1}p_{3}^{2} − 6p_{2}^{4} − p_{2}^{3}p_{3}^{3} − 2p_{2}^{2}p_{3}^{2} − 34p_{2}p_{3} − 6p_{3}^{4} + 40)

+ x^{6}(6p_{1}^{4} − 2p_{1}^{3}p_{2}^{2} − 2p_{1}^{3}p_{3}^{2} + p_{1}^{2}p_{2}^{2}p_{3}^{2} − 24p_{1}^{2}p_{2}p_{3} − 28p_{1}^{2} + 6p_{1}p_{2}^{3}p_{3}

+ 34p_{1}p_{2}^{2} + 6p_{1}p_{2}p_{3}^{3} + 34p_{1}p_{3}^{2} − 7p_{2}^{4} − 2p_{2}^{3}p_{3}^{3} − 40p_{2}p_{3} − 7p_{3}^{4} + 44)].

Here

p_{3} = −a_{21}a_{14}a_{43}a_{32} + a_{13}a_{32}a_{21} + a_{21}a_{14}a_{42} + a_{31}a_{14}a_{43} + a_{32}a_{24}a_{43} − a_{12}a_{21}
− a_{13}a_{31} − a_{23}a_{32} − a_{14}a_{41} − a_{24}a_{42} − a_{34}a_{43} − 4,
p_{1} = a_{23}a_{32}a_{14}a_{41} − a_{13}a_{32}a_{24}a_{41} − a_{23}a_{31}a_{14}a_{42} + a_{13}a_{31}a_{24}a_{42}
− a_{13}a_{34}a_{42}a_{21} − a_{12}a_{24}a_{43}a_{31} + a_{12}a_{21}a_{34}a_{43} + a_{12}a_{23}a_{31} − a_{13}a_{21}a_{32}
+ a_{12}a_{24}a_{41} + a_{13}a_{34}a_{41} − a_{21}a_{14}a_{42} + a_{23}a_{34}a_{42} − a_{31}a_{14}a_{43}

− a_{32}a_{24}a_{43} + 2a_{12}a_{21} + 2a_{13}a_{31} + 2a_{23}a_{32} + 2a_{14}a_{41} + 2a_{24}a_{42} + 2a_{34}a_{43} + 6,

p_{2} = p_{3}^{∗}.

These examples, together with other calculations made in the preparation of this paper, were accomplished using MAGMA [MA].

As we are seeing in these examples, the coefficients of 𝔅_{mr}(x) are polynomials in the coefficients of χ_{M_{m}}(x), as stated in Theorem 1.2. That this is the case in general follows from Lemma 3.2 and Lemma 5.2 applied to N_{m} = M_{m} ⊗ M_{m}^{∗}.

§6 Geometric intersection numbers of curves on a disc

Let γ, γ′ ∈ C. We wish to find the geometric intersection number ι(γ, γ′). Since B_{n} acts transitively on each C_{m} we may without loss assume that γ′ = γ_{1m} for some m ≥ 2. For i = 1, …, n let b_{i} denote a vertical arc in D_{n} meeting the arc a_{i} exactly once, meeting no other a_{j}, j ≠ i, and with its endpoints on the boundary of D_{n}. Then D_{n} \ b_{i} has two components, one containing the arcs a_{1}, …, a_{i−1} and part of a_{i}. This component we will think of as the disc D_{m}, so that γ_{1m} is the boundary curve of D_{m} ⊂ D_{n}.

Let μ denote the monomial of greatest degree in ϕ_{n}(γ) and write μ = μ_{1}μ_{2}μ_{3}μ_{4} as we have done in (4.1). Write μ = μ(γ), μ_{i} = μ_{i}(γ). We note that ϕ(γ) ∈ Y_{n} and so deg(μ_{2}) = deg(μ_{3}). Then we have

Proposition 6.1. For γ ∈ C we have

ι(γ_{1m}, γ) = degree(μ_{2}(γ)μ_{3}(γ)) = 2degree(μ_{2}(γ)).

Proof. We may assume that γ meets γ_{1m} in ι(γ_{1m}, γ) points. Then, since γ_{1m} is the boundary curve of D_{m} ⊂ D_{n}, we see that γ meets b_{m} in exactly ι(γ_{1,m}, γ) points. Now let w ∈< T_{1}, …, T_{n} > be a cyclically reduced word representing γ. Then we can write

w = w_{1}y_{1}w_{2}y_{2} … w_{k}y_{k} where w_{i} ∈< T_{1}, …, T_{m} >, y_{i} ∈< T_{m+1}, …, T_{n} > .

Here we may assume that w_{i} and y_{i} are all non-trivial. Now using Lemma 4.2 we see that ϕ(γ) = trace(w) − n is a sum of monomials of the form

±m_{1}a_{i_{1}j_{1}}m′_{1}a_{j′_{1}i′_{1}}m_{2}a_{i_{2}j_{2}}m′_{2}a_{j′_{2}i′_{2}} … m_{h}a_{i_{h}j_{h}}m′_{h}a_{j′_{h}i′_{h}},

where each m_{i} is a monomial in R_{m}, each monomial m′_{i} is in R[a_{uv}|m < u, v ≤ n] and 1 ≤ i_{r}, i′_{r} ≤ m < j_{r}, j′_{r} ≤ n for all 1 ≤ r ≤ h ≤ k. We note that ι(γ_{1m}, γ) = 2k since for every adjacent pair of words w_{1}y_{1}, y_{1}w_{2}, …, w_{k}y_{k}, y_{k}w_{1} in w the curve γ must cross the arc b_{m} and conversely, each crossing must correspond to one of these pairs of adjacent words in w. This proves the result. □

Thus if ι(γ_{1m}, γ) = 2k, then by the proposition 5.3 we see that 𝔅_{mk}(τ_{m}^{m})(ϕ(γ)) = 0. This proves the first part of Theorem 1.1.

Now the above argument never used the fact that γ was simple closed curve, so what we have proved is

Proposition 6.2. Let γ be any closed curve on the disc D_{n} and let

w = T_{i_{1}}^{±1}T_{i_{2}}^{±1} … T_{i_{k}}^{±1} ∈< T_{1}, …, T_{n} >

be any cyclically reduced word representing γ. Then the geometric intersection number ι(γ_{1m}, γ) is equal to the number of j ≤ k such that i_{j} ≤ m and i_{j+1} > m or i_{j} > m and i_{j+1} ≤ m (indices taken mod k). Further, if 2r = ι(γ_{1m}, γ), then 𝔅_{mr}(τ_{m}^{m})(ϕ(γ)) = 0. □

In order to conclude the proof of Theorem 1.1 it will suffice to prove Proposition 6.3. For all m ≥ 2 and r ≥ 1 we have

𝔅_{mr}(τ_{m}^{m})((a_{1 m+1}a_{m+1 1})^{r+1}) ≠ 0.

Proof. One checks the following facts about degrees using Lemma 4.3:

(i) deg(τ_{m}^{km}(a_{1 m+1})) = km + 1; (ii) deg(τ_{m}^{km}(a_{1 m+1}a_{m+1 1})) = 2(km + 1); (iii) 𝔅_{mr}(x) has degree m^{2r} in the variable x; (iv) deg((τ_{m}^{m})^{m^{2r}}(a_{1 m+1}a_{m+1 1})) = 2(m^{2r}m + 1).

We also have

(v) The coefficient of x^{m^{2r}−k} in 𝔅_{mr}(x) is a polynomial in the a_{ij} of degree no greater than 2rmk.

To see this last fact we note as in §5 that if a_{i} are the eigenvalues of M_{m} = T_{1} … T_{m}, and if a′_{i} are the eigenvalues of M^{∗}, then a_{i}a′_{j} are the eigenvalues of N_{m} = M_{m}⊗M_{m}^{∗} and a_{i_{1}}a′_{j_{1}}a_{i_{2}}a′_{j_{2}} … a_{i_{k}}a′_{j_{k}} are the eigenvalues of N_{m}^{⊗k}. Thus the coefficient of x^{m^{2r}−k} in 𝔅_{mr}(x) is the sum of all terms of the form a_{i_{1}}a′_{j_{1}}a_{i_{2}}a′_{j_{2}} … a_{i_{k}}a′_{j_{k}}. Such a sum is clearly invariant under the action of S_{m} × S_{m} and so is a polynomial in the coefficients of the characteristic polynomial of M and M^{∗} (use Lemma 3.2). This polynomial has degree 2k (use Lemma 3.2 again) as a polynomial in these coefficients and so has degree 2mk as a polynomial in the a_{ij} (since, as noted in §1, the coefficients of the characteristic polynomial of M have degree m in R_{n}). This proves statement (v).

Continuing with the proof of Proposition 6.3 we see that if c_{k} is the coefficient of x^{m^{2r}−k} in 𝔅_{mr}(x), then

deg(c_{k}(τ_{m}^{m})^{m^{2r}−k}((a_{1 m+1}a_{m+1 1})^{r+1})) ≤ 2mrk + (r + 1)(2((m^{2r} − k)m + 1))

for all k ≥ 0 and where we have equality for k = 0. It follows that

deg((τ_{m}^{m})^{m^{2r}}((a_{1 m+1}a_{m+1 1})^{r+1})) − deg(c_{k}(τ_{m}^{m})^{m^{2r}−k}((a_{1 m+1}a_{m+1 1})^{r+1})) ≥ 2mk.

This proves Proposition 6.3 and concludes the proof of Theorem 1.1. □

Remark 6.4. Suppose that γ is a multicurve on D_{n} i.e. γ is the disjoint union of simple closed curves on D_{n}. Then the orbit of γ under the action of B_{n} contains a curve of the form γ_{i_{1}j_{1}} ∪ γ_{i_{2}j_{2}} ∪ ⋯ ∪ γ_{i_{s}j_{s}} where i_{k} < j_{k} for all k and the intervals [i_{k}, j_{k}] satisfy

[i_{u}, j_{u}] ∩ [i_{v}, j_{v}] = ∅ or [i_{u}, j_{u}] ⊂ [i_{v}, j_{v}] or [i_{v}, j_{v}] ⊂ [i_{u}, j_{u}].

Thus we can, without loss, assume that γ is the above multicurve.

Now if γ′ ∈ C, then we let r_{k} = ι(γ_{i_{k}j_{k}}, γ′). From Theorem 1.1 it follows that

[𝔅_{j_{1}−i_{1}+1 r_{1}}(τ_{γ_{i_{1}j_{1}}} ) + ⋯ + B_{j_{s}−i_{s}+1 r_{s}}(τ_{γ_{isjs}})](ϕ(γ)) = 0.

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