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add rotation_element and correct Te, Ta, Tα
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@ -5,8 +5,8 @@ struct ΡΛ
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end
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function Base.getindex(rl::ΡΛ, i::Integer, j::Integer)
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@assert 1 ≤ i ≤ rl.N
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@assert 1 ≤ j ≤ rl.N
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@assert 1 ≤ i ≤ rl.N "Got $i > $(rl.N)"
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@assert 1 ≤ j ≤ rl.N "Got $j > $(rl.N)"
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@assert i ≠ j
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@assert rl.id ∈ (:λ, :ϱ)
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rl.id == :λ && return Word([rl.A[λ(i, j)]])
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@ -19,11 +19,16 @@ function Te_diagonal(λ::Groups.ΡΛ, ϱ::Groups.ΡΛ, i::Integer)
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N = λ.N
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@assert iseven(N)
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A = λ.A
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n = N ÷ 2
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j = i + 1
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@assert 1 <= i < n
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A = λ.A
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if i == n
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τ = rotation_element(λ, ϱ)
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return inv(A, τ) * Te_diagonal(λ, ϱ, 1) * τ
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end
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@assert 1 <= i < n
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NI = (2n - 2i) + 1
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NJ = (2n - 2j) + 1
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@ -44,43 +49,104 @@ end
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function Te_lantern(A::Alphabet, b₀::T, a₁::T, a₂::T, a₃::T, a₄::T, a₅::T) where {T}
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a₀ = (a₁ * a₂ * a₃)^4 * inv(A, b₀)
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X = a₄ * a₅ * a₃ * a₄
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b₁ = inv(A, X) * a₀ * X
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X = a₄ * a₅ * a₃ * a₄ # from Primer
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b₁ = inv(A, X) * a₀ * X # from Primer
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Y = a₂ * a₃ * a₁ * a₂
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return inv(A, Y) * b₁ * Y # b₂
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return inv(A, Y) * b₁ * Y # b₂ from Primer
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end
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Ta(λ::Groups.ΡΛ, i::Integer) = (@assert λ.id == :λ; λ[λ.N-2i+1, λ.N-2i+2])
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Tα(λ::Groups.ΡΛ, i::Integer) = (@assert λ.id == :λ; inv(λ.A, λ[λ.N-2i+2, λ.N-2i+1]))
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function Ta(λ::Groups.ΡΛ, i::Integer)
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@assert λ.id == :λ;
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return λ[mod1(λ.N-2i+1, λ.N), mod1(λ.N-2i+2, λ.N)]
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end
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function Tα(λ::Groups.ΡΛ, i::Integer)
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@assert λ.id == :λ;
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return inv(λ.A, λ[mod1(λ.N-2i+2, λ.N), mod1(λ.N-2i+1, λ.N)])
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end
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function Te(λ::ΡΛ, ϱ::ΡΛ, i, j)
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@assert i ≠ j
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i, j = i < j ? (i, j) : (j, i)
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@assert λ.N == ϱ.N
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@assert λ.A == ϱ.A
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@assert λ.id == :λ && ϱ.id == :ϱ
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@assert iseven(λ.N)
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genus = λ.N÷2
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i = mod1(i, genus)
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j = mod1(j, genus)
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@assert 1 ≤ i ≤ λ.N
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@assert 1 ≤ j ≤ λ.N
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if j == i + 1
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A = λ.A
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if mod(j - (i + 1), genus) == 0
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return Te_diagonal(λ, ϱ, i)
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else
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return Te_lantern(
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λ.A,
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Ta(λ, i + 1),
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Ta(λ, i),
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Tα(λ, i),
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Te(λ, ϱ, i, i + 1),
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Tα(λ, i + 1),
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Te(λ, ϱ, i + 1, j),
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)
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return inv(A, Te_lantern(
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A,
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# Our notation: # Primer notation:
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inv(A, Ta(λ, i + 1)), # b₀
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inv(A, Ta(λ, i)), # a₁
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inv(A, Tα(λ, i)), # a₂
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inv(A, Te_diagonal(λ, ϱ, i)), # a₃
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inv(A, Tα(λ, i + 1)), # a₄
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inv(A, Te(λ, ϱ, i + 1, j)), # a₅
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))
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end
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end
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"""
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rotation_element(G::AutomorphismGroup{<:FreeGroup})
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Return the element of `G` which corresponds to shifting generators of the free group.
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In the corresponding mapping class group this element acts by rotation of the surface anti-clockwise.
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"""
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function rotation_element(G::AutomorphismGroup{<:FreeGroup})
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A = alphabet(G)
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@assert iseven(ngens(object(G)))
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genus = ngens(object(G)) ÷ 2
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λ = ΡΛ(:λ, A, 2genus)
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ϱ = ΡΛ(:ϱ, A, 2genus)
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return G(rotation_element(λ, ϱ))
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end
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function rotation_element(λ::ΡΛ, ϱ::ΡΛ)
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@assert iseven(λ.N)
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genus = λ.N÷2
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A = λ.A
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halftwists = map(1:genus-1) do i
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j = i + 1
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x = Ta(λ, j) * inv(A, Ta(λ, i)) * Tα(λ, j) * Te_diagonal(λ, ϱ, i)
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δ = x * Tα(λ, i) * inv(A, x)
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c =
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inv(A, Ta(λ, j)) *
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Te(λ, ϱ, i, j) *
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Tα(λ, i)^2 *
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inv(A, δ) *
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inv(A, Ta(λ, j)) *
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Ta(λ, i) *
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δ
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z =
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Te_diagonal(λ, ϱ, i) *
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inv(A, Ta(λ, i)) *
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Tα(λ, i) *
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Ta(λ, i) *
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inv(A, Te_diagonal(λ, ϱ, i))
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Ta(λ, i) * inv(A, Ta(λ, j) * Tα(λ, j))^6 * (Ta(λ, j) * Tα(λ, j) * z)^4 * c
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end
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τ = (Ta(λ, 1) * Tα(λ, 1))^6 * prod(halftwists)
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return τ
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end
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function mcg_twists(G::AutomorphismGroup{<:FreeGroup})
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@assert iseven(ngens(object(G)))
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