renumerate Dehn Twists
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@ -13,7 +13,7 @@ function Base.getindex(rl::ΡΛ, i::Integer, j::Integer)
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rl.id == :ϱ && return Word([rl.A[ϱ(i, j)]])
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end
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function Te_diagonal(λ::ΡΛ, ϱ::ΡΛ, i::Integer)
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function Te_diagonal(λ::Groups.ΡΛ, ϱ::Groups.ΡΛ, i::Integer)
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@assert λ.N == ϱ.N
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@assert λ.id == :λ && ϱ.id == :ϱ
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@ -25,15 +25,20 @@ function Te_diagonal(λ::ΡΛ, ϱ::ΡΛ, i::Integer)
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A = λ.A
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# comments are for i,j = 1,2
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NI = (2n - 2i) + 1
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NJ = (2n - 2j) + 1
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I = (2n - 2i) + 2
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J = (2n - 2j) + 2
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g = one(Word(Int[]))
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g *= λ[n+j, n+i] # β ↦ α*β
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g *= λ[n+i, i] * inv(A, ϱ[n+i, j]) # α ↦ a*α*b^-1
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g *= inv(A, λ[n+j, n+i]) # β ↦ b*α^-1*a^-1*α*β
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g *= λ[j, n+i] * inv(A, λ[j, i]) # b ↦ α
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g *= inv(A, λ[j, n+i]) # b ↦ b*α^-1*a^-1*α
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g *= inv(A, ϱ[j, n+i]) * ϱ[j, i] # b ↦ b*α^-1*a^-1*α*b*α^-1
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g *= ϱ[j, n+i] # b ↦ b*α^-1*a^-1*α*b*α^-1*a*α*b^-1
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g *= λ[NJ, NI] # β ↦ α*β
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g *= λ[NI, I] * inv(A, ϱ[NI, J]) # α ↦ a*α*b^-1
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g *= inv(A, λ[NJ, NI]) # β ↦ b*α^-1*a^-1*α*β
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g *= λ[J, NI] * inv(A, λ[J, I]) # b ↦ α
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g *= inv(A, λ[J, NI]) # b ↦ b*α^-1*a^-1*α
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g *= inv(A, ϱ[J, NI]) * ϱ[J, I] # b ↦ b*α^-1*a^-1*α*b*α^-1
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g *= ϱ[J, NI] # b ↦ b*α^-1*a^-1*α*b*α^-1*a*α*b^-1
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return g
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end
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@ -45,10 +50,10 @@ function Te_lantern(A::Alphabet, b₀::T, a₁::T, a₂::T, a₃::T, a₄::T, a
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return inv(A, Y) * b₁ * Y # b₂
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end
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Ta(λ::ΡΛ, i::Integer) = (@assert λ.id == :λ;
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λ[λ.N÷2+i, i])
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Tα(λ::ΡΛ, i::Integer) = (@assert λ.id == :λ;
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inv(λ.A, λ[i, λ.N÷2+i]))
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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 Te(λ::ΡΛ, ϱ::ΡΛ, i, j)
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@assert i ≠ j
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@ -7,6 +7,8 @@ using Groups.KnuthBendix
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G = SpecialAutomorphismGroup(FreeGroup(2genus))
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T = Groups.mcg_twists(autF)
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# symplectic pairing in the free Group goes like this:
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# f1 ↔ f5
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# f2 ↔ f6
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@ -43,8 +45,8 @@ using Groups.KnuthBendix
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@testset "preserving relator" begin
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F = Groups.object(G)
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R = prod(commutator(gens(F, i), gens(F, i+genus)) for i in 1:genus)
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R = prod(commutator(gens(F,2i+1), gens(F,2i+2)) for i in 0:genus-1)
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for g in T
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@test g(R) == R
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end
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@ -88,19 +90,17 @@ using Groups.KnuthBendix
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# some additional tests, checking what explicitly happens to the generators of the π₁ under b2
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d = Groups.domain(b2)
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im = evaluate(b2)
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z = im[7] * d[7]^-1
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@test im[1] == d[1]
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@test im[2] == z * d[2] * z^-1
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@test im[3] == z * d[3] * z^-1
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@test im[4] == d[4]
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@test im[5] == d[5] * z^-1
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@test im[6] == z * d[6] * z^-1
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@test im[7] == z * d[7]
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@test im[8] == d[8]
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img = evaluate(b2)
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z = img[3] * d[3]^-1
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@test img[1] == d[1]
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@test img[2] == d[2]
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@test img[3] == z * d[3]
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@test img[4] == z * d[4] * z^-1
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@test img[5] == z * d[5] * z^-1
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@test img[6] == z * d[6] * z^-1
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@test img[7] == d[7] * z^-1
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@test img[8] == d[8]
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end
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@testset "b2: commutation relations" begin
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@ -128,54 +128,6 @@ using Groups.KnuthBendix
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end
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end
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@testset "Te₁₂ definition" begin
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G = parent(first(T))
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F₈ = Groups.object(G)
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(a, b, c, d, α, β, γ, δ) = Groups.gens(F₈)
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A = alphabet(G)
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λ = [i == j ? one(G) : G([A[Groups.λ(i, j)]]) for i in 1:8, j in 1:8]
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ϱ = [i == j ? one(G) : G([A[Groups.ϱ(i, j)]]) for i in 1:8, j in 1:8]
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g = one(G)
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# β ↦ α*β
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g *= λ[6, 5]
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@test evaluate(g)[6] == α * β
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# α ↦ a*α*b^-1
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g *= λ[5, 1] * inv(ϱ[5, 2])
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@test evaluate(g)[5] == a * α * b^-1
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# β ↦ b*α^-1*a^-1*α*β
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g *= inv(λ[6, 5])
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@test evaluate(g)[6] == b * α^-1 * a^-1 * α * β
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# b ↦ α
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g *= λ[2, 5] * inv(λ[2, 1])
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@test evaluate(g)[2] == α
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# b ↦ b*α^-1*a^-1*α
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g *= inv(λ[2, 5])
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@test evaluate(g)[2] == b * α^-1 * a^-1 * α
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# b ↦ b*α^-1*a^-1*α*b*α^-1
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g *= inv(ϱ[2, 5]) * ϱ[2, 1]
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@test evaluate(g)[2] == b * α^-1 * a^-1 * α * b * α^-1
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# b ↦ b*α^-1*a^-1*α*b*α^-1*a*α*b^-1
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g *= ϱ[2, 5]
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@test evaluate(g)[2] == b * α^-1 * a^-1 * α * b * α^-1 * a * α * b^-1
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x = b * α^-1 * a^-1 * α
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@test evaluate(g) ==
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(a, x * b * x^-1, c, d, α * x^-1, x * β, γ, δ)
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# (a, b, c, d, α, β, γ, δ)
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@test g == T[9]
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end
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Base.conj(t::Groups.Transvection, p::Perm) =
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Groups.Transvection(t.id, t.i^p, t.j^p, t.inv)
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@ -189,13 +141,15 @@ using Groups.KnuthBendix
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Te₁₂, Te₂₃ = T[9], T[12]
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G = parent(Te₁₂)
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F₈ = Groups.object(G)
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(a, b, c, d, α, β, γ, δ) = Groups.gens(F₈)
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(δ, d, γ, c, β, b, α, a) = Groups.gens(F₈)
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Groups.domain(Te₁₂)
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img_Te₂₃ = evaluate(Te₂₃)
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y = c * β^-1 * b^-1 * β
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@test img_Te₂₃ == (a, b, y * c * y^-1, d, α, β * y^-1, y * γ, δ)
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@test img_Te₂₃ == (δ, d, y * γ, y * c * y^-1, β * y^-1, b, α, a)
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σ = perm"(1,2,3)(5,6,7)(8)"
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σ = perm"(7,5,3)(8,6,4)"
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Te₂₃_σ = conj(Te₁₂, σ)
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# @test word(Te₂₃_σ) == word(Te₂₃)
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@ -207,9 +161,9 @@ using Groups.KnuthBendix
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Te₁₂, Te₃₄ = T[9], T[14]
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G = parent(Te₁₂)
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F₈ = Groups.object(G)
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(a, b, c, d, α, β, γ, δ) = Groups.gens(F₈)
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(δ, d, γ, c, β, b, α, a) = Groups.gens(F₈)
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σ = perm"(1,3)(2,4)(5,7)(6,8)"
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σ = perm"(7,3)(8,4)(5,1)(6,2)"
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Te₃₄_σ = conj(Te₁₂, σ)
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@test Te₃₄ == Te₃₄_σ
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end
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