Alexander Invariant of Tangles

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1 The.nb (source) file is at based on the original program at drorbn.net/academicpensieve/ /polypoly/. 1. Γ-Calculus Alexander Invariant of Tangles The following code expresses an element of Γ calculus in a nice format. In[1]:= ΓCollect[Γ[ω_, λ_]] := Γ[Simplify[ω], Collect[λ, x _, Collect[#,y _, Factor]&]]; Format[Γ[ω_,λ_]] := Module[{S, M}, S = Union@Cases[Γ[ω,λ], (x y) a_ a, ]; M = Outer[Factor[ x#1y#2λ]&, S, S]; M = Prepend[M, y # & /@ S] // Transpose; M = Prepend[M, Prepend[x # & /@ S, ω]]; M // MatrixForm]; For instance, we can display an element of Γ calculus as follows. In[3]:= Γ ω,{y a,y b }. g 11 g 12 g 21 g 22.{x a,x b } Out[3]= ω x a x b y a g 11 g 12 y b g 21 g 22 Next we program the stitching operation In[4]:= Γ /: Γ[ω1_, λ1_]γ[ω2_, λ2_] := Γ[ω1*ω2, λ1+λ2]; m a_,e_ c_ [Γ[ω_, λ_]] := Module {α,β,γ,δ,θ,ϵ,ϕ,ψ,ξ,}, α β θ ya,xaλ ya,xeλ ya λ γ δ ϵ = ye,xaλ ye,xeλ ye λ /.(y x) a e 0; ϕ ψ Ξ xa λ xe λ λ Γ (=1-γ)ω, {y β+α δ/ θ+α ϵ/ c,1}. ψ+δ ϕ/ Ξ+ϕ ϵ/.{xc,1} /. {t a t c, t e t c } // ΓCollect ; R+ a_,e_ := Γ 1, {y a,y e }. 1 1-t a.{x 0 t a,x e } ; a R- a_,e_ := R+ a,e /. t a 1/t a ; Checking meta-associativity

2 2 GammaCalculusDemoFull.nb In[8]:= ζ=γ ω, {y 1,y 2,y 3,y S }. α 11 α 12 α 13 θ 1 α 21 α 22 α 23 θ 2 α 31 α 32 α 33 θ 3 ϕ 1 ϕ 2 ϕ 3 Ξ.{x 1,x 2,x 3,x S } (ζ//m 1,2 1 //m 1,3 1 ) (ζ//m 2,3 2 //m 1,2 1 ) Out[8]= Out[9]= ω x 1 x 2 x 3 x S y 1 α 11 α 12 α 13 θ 1 y 2 α 21 α 22 α 23 θ 2 y 3 α 31 α 32 α 33 θ 3 y S ϕ 1 ϕ 2 ϕ 3 Ξ Checking the RIII relation In[10]:= R+ 1,4 R+ 2,5 R- 6,3 //m 1,6 1 //m 2,4 2 //m 3,5 3 Out[10]= 1 x 1 x 2 x 3 (-1+t1) t2 y t 1 t1 y 2 0 t t 2 y t2 t1 In[11]:= R- 1,4 R+ 5,2 R+ 6,3 //m 1,5 1 //m 2,6 2 //m 3,4 3 Out[11]= 1 x 1 x 2 x 3 (-1+t1) t2 y t 1 t1 y 2 0 t t 2 y t2 t1 Checking the RII relations In[12]:= R+ i,j R- k,l //m i,k i //m j,l j Out[12]= 1 x i x j y i 1 0 y j 0 1 Checking the Overcrossings Commute (OC) relation In[13]:= R+ 4,2 R+ 1,3 //m 1,4 1 Out[13]= 1 x 1 x 2 x 3 y t t 1 y 2 0 t 1 0 y t 1

3 GammaCalculusDemoFull.nb 3 In[14]:= R+ 4,3 R+ 1,2 //m 1,4 1 Out[14]= 1 x 1 x 2 x 3 y t t 1 y 2 0 t 1 0 y t 1 Compute the invariant of the figure-eight knot In[15]:= R+ 1,6 R+ 5,2 R- 3,8 R- 7,4 //m 1,2 1 //m 1,3 1 //m 1,4 1 //m 1,5 1 //m 1,6 1 //m 1,7 1 //m 1,8 1 Out[15]= 3-1 t1 - t 1 x 1 y 1 1 As another example, let us compute the invariant of the following tangle In[16]:= R 7,2 R 10,6 R 5,11 R 3,12 R 4,8 R 9,1 // m 1,4 1 // m 2,5 2 // m 2,6 2 // m 2,7 2 // m 3,8 3 // m 3,9 3 // m 3,10 3 // m 3,11 3 // m 3,12 3 Out[16]= t2 t3 (-t 1 (-1 + t 3 ) + t 3 ) x 1 x 2 x 3 y 1 - y 2 0 y 3 t3 -t1-t3+t1 t3 t1 (-1+t3) -t1-t3+t1 t3 - (-1+t1) (-1+t3) t3 (-1+t2+t3) (-t1-t3+t1 t3) t2-1+t2+t3 t1 (-1+t3) (-1+t2+t3) (-t1-t3+t1 t3) - (-1+t1) (1-t2-2 t3+t2 t3) (-1+t2+t3) (-t1-t3+t1 t3) -1+t2-1+t2+t3-1+t1+t2-t1 t2+2 t3-3 t1 t3-t2 t3+t1 t2 t (-1+t2+t3) (-t1-t3+t1 t3) Now let us compute the invariant of the knot 7 7 in the Knot Atlas.

4 4 GammaCalculusDemoFull.nb First we compute the invariant of the tangle inside the circle In[18]:= τ 1 = R 1,2 R 14,4 R 5,13 R 3,6 R 12,9 R 7,11 R 10,8 // m 1,4 1 // m 2,5 2 // m 2,6 2 // m 2,7 2 // m 2,8 2 // m 3,9 3 // m 3,10 3 // m 3,11 3 // m 3,12 3 // m 3,13 3 // m 3, t t3 x 1 Out[18]= y 1-1+t1+2 t2-2 t1 t2-t 2 2+t1 t t3-2 t1 t3-4 t2 t3+4 t1 t2 t3+t 2 2 t3-2 t1 t 2 2 t3-t 3 2+t1 t 3 2+t2 t t1 t2 t 3 2+t1 t 2 2 t 3 2 t2 (-t2-t3+t2 t3) y 2 - t1 (-1+t2)2 (-1+t3) 2 y 3 - t2 (-t2-t3+t2 t3) (-1+t3) (1-2 t2-t3+t2 t3) t2 (-t2-t3+t2 t3) The to obtain the invariant of the knot we perform two extra stitchings - In[19]:= τ 1 // m 2,1 2 // m 3,2 1 Out[19]= t t1-5 t 1 + t 12 x 1 y Extended Γ-Calculus First we format extended Γ-calculus In[20]:= eγcollect[eγ[ω_, λ_, σ_]] := eγ[simplify[ω], Collect[λ, x _, Collect[#, y _, Factor] &], σ]; Format[eΓ[ω_, λ_, σ_]] := Module[{S, M}, S = Union@Cases[eΓ[ω, λ, σ], (x y) a_ a, ]; M = Outer[Factor[ x#1 y #2 λ] &, S, S]; M = Prepend[M, y # & /@ S] // Transpose; M = Prepend[M, Prepend[x # & /@ S, ω]]; {M // MatrixForm, σ}]; eγ[ω1_, λ1_, σ1_] eγ[ω2_, λ2_, σ2_] := Simplify[PowerExpand[ω1 ω2 λ1 λ2 σ1 σ2 ]]; For example

5 GammaCalculusDemoFull.nb 5 In[23]:= eγ 1, {y a, y e } ta 0 t a.{x a, x e }, {s a, s e }.{v a, v e } Out[23]= 1 x a x e y a t a, s a v a + s e v e y e 0 t a Extended stitching In[24]:= eγ /: eγ[ω1_, λ1_, σ1_] eγ[ω2_, λ2_, σ2_] := eγ[ω1 * ω2, λ1 + λ2, σ1 + σ2]; em a_,e_ c_ [eγ[ω_, λ_, σ_]] := Module {α, β, γ, δ, θ, ϵ, ϕ, ψ, Ξ, }, α β θ γ δ ϵ = ya,x a λ ya,x e λ ya λ ye,x a λ ye,x e λ ye λ /. (y x) a e 0; ϕ ψ Ξ xa λ xe λ λ eγ = 1 - γ ω, {y c, 1}. β + α δ / θ + α ϵ / ψ + δ ϕ / Ξ + ϕ ϵ /.{x c, 1}, σ /. v a e 0 + v c ( va σ) ( ve σ) /. {t a t c, t e t c, b a b c, b e b c } // eγcollect ; + er a_,e_ := eγ 1, {y a, y e } ta.{x a, x e }, v a + t a v e ; 0 t a - + er a_,e_ := er a,e /. t a t 1 ā ; Cheking meta-associativity In[28]:= eζ = eγ ω, {y 1, y 2, y 3, y S }. α 11 α 12 α 13 θ 1 α 21 α 22 α 23 θ 2 α 31 α 32 α 33 θ 3 ϕ 1 ϕ 2 ϕ 3 Ξ.{x 1, x 2, x 3, x S }, s 1 v 1 + s 2 v 2 + s 3 v 3 + s S v S (eζ // em 1,2 1 // em 1,3 1 ) (eζ // em 2,3 2 // em 1,2 1 ) Out[28]= ω x 1 x 2 x 3 x S y 1 α 11 α 12 α 13 θ 1 y 2 α 21 α 22 α 23 θ 2 y 3 α 31 α 32 α 33 θ 3 y S ϕ 1 ϕ 2 ϕ 3 Ξ, s 1 v 1 + s 2 v 2 + s 3 v 3 + s S v S Out[29]= Strand reversal In[30]:= dh[a_][eγ[ω_, λ_, σ_]] := Module {α, θ, ϕ, Ξ, sa}, α θ ϕ Ξ = y a,x a λ ya λ xa λ λ /. (y x) a 0; sa = va σ; eγ α ω / sa, {y a, 1}. 1 α θ / α -ϕ / α (α Ξ - ϕ θ) / α.{x a, 1}, v a sa + σ /. {va 0} /. t a 1 t a, b a -b a // eγcollect ; Verifying the crossings

6 6 GammaCalculusDemoFull.nb + - In[31]:= er 1,2 // dh[1] (er 1,2 ) + - er 1,2 // dh[2] (er 1,2 ) - + er 1,2 // dh[1] (er 1,2 ) - + er 1,2 // dh[2] (er 1,2 ) Out[31]= Out[32]= Out[33]= Out[34]= Verifying the stitching relations In[35]:= ζ = eγ ω, {y b, y c, y S }. α β θ γ δ ϵ ϕ ψ Ξ.{x b, x c, x S }, s b v b + s c v c + s S v S (ζ // em b,c a // dh[a]) (ζ // dh[b] // dh[c] // em c,b a ) Out[35]= ω x b x c x S y b α β θ y c γ δ ϵ y S ϕ ψ Ξ, s b v b + s c v c + s S v S Out[36]= Strand doubling In[37]:= qδ[i_, j_, k_][eγ[ω_, λ_, σ_]] := Module ; {α, θ, ϕ, Ξ, si, M, ti,, ν}, α θ ϕ Ξ = y i,x i λ yi λ xi λ λ /. (y x) i 0 /. t i ti; si = vi σ /. t i ti; = -1 + ti; ν = α - si; M = -α+ti si+t j ν t j (-1+t k ) ν (-1+t j ) ν -si+ti α-t j ν (-1+t j ) θ t j (-1+t k ) θ ϕ ϕ Ξ eγ ω /. {t i t j t k }, {y j, y k, 1}.M.{x j, x k, 1} /. {ti t j t k }, σ /. {v i 0} + (v j + v k ) si /. t i ; ti t j t k // eγcollect Verifying the crossings

7 GammaCalculusDemoFull.nb In[38]:= qδ[1, 1, 2][eR 1,3 ] (er 2,3 er 1,4 // em 3,4 3 ) qδ[1, 1, 2][eR 1,3 ] (er 1,3 er 2,4 // em 3,4 3 ) qδ[2, 2, 3][eR 1,2 ] (er 1,2 er 4,3 // em 1,4 1 ) qδ[2, 2, 3][eR 1,2 ] (er 1,3 er 4,2 // em 1,4 1 ) Out[38]= Out[39]= Out[40]= Out[41]= Verifying the stitching relations In[42]:= ζ = eγ ω, {y i, y j, y S }. α β θ γ δ ϵ ϕ ψ Ξ.{x i, x j, x S }, s i v i + s j v j + s S v S (ζ // qδ[i, i 1, i 2 ] // qδ[j, j 1, j 2 ] // em i1,j 1 k 1 // em i2,j 2 k 2 ) (ζ // em i,j k // qδ[k, k 1, k 2 ]) Out[42]= ω x i x j x S y i α β θ y j γ δ ϵ y S ϕ ψ Ξ, s i v i + s j v j + s S v S Out[43]= 3. The Lie Algebra G 0 Again first we format G 0 In[44]:= CF[expr_] := expr // Simplify; /: CF[ [ω_, λ_]] := [CF[ω], CF[λ]]; /: [ω1_, λ1_] [ω2_, λ2_] := CF@ [ω1 ω2, λ1 + λ2]; [ω1_, λ1_] [ω2_, λ2_] := CF[ω1 ω2 λ1 λ2]; So for instance In[48]:= [ω, Sum[l x,y b x c y + q x,y u x w y, {x, {i, j}}, {y, {i, j}}]] Out[48]= [ω, b i c i l i,i + b i c j l i,j + b j c i l j,i + b j c j l j,j + u i w i q i,i + u i w j q i,j + u j w i q j,i + u j w j q j,j ] The switching operators

8 8 GammaCalculusDemoFull.nb In[49]:= N ui_ c j_ k_[ [ω_, λ_]] := CF ω, e -γ β u k + γ c k + λ /. c j u i 0 /. {γ cj λ, β ui λ} ; N wi_ c j_ k_[ [ω_, λ_]] := CF ω, e γ α w k + γ c k + λ /. c j w i 0 /. {γ cj λ, α wi λ} ; N wi_ u j_ k_[ [ω_, λ_]] := CF ν ω, -b k ν α β + ν β u k + ν δ u k w k + ν α w k + λ /. w i u j 0 /. ν 1 + b k δ -1 /. {α wi λ /. u j 0, β uj λ /. w i 0, δ wi,u j λ} ; The stitching operation In[52]:= gm i_,j_ k_ [ [ω_, λ_]] := CF[Module[{x}, ( [ω, λ] // N wi c j x // N ui c x x // N wx u j x) /. {c i c k, w j w k, y_ x y k, b i j b k }]] + gr i_,j_ = 1, b i c j + b 1 ī e b i u i w j ; gr i_,j_ = 1, -b i c j + b 1 ī e -b i - 1 u i w j ; Next we program a map from G 0 to extended Γ-calculus. In[54]:= G0toΓ[e_] := Module {A, λ, L, ω, Q, n, II, DD, T, M, σ, i, j}, ω = e 1 /. e x_ e Simplify[x /. b i_ Log[t i ]] // Simplify; λ = e 2 ; A = Union@Cases[λ, (b c) a_ a, ]; L = Outer[Factor[ b#1 c#2 λ] &, A, A]; Q = Outer[Factor[ u#1 w #2 λ] &, A, A]; n = Length[A]; II = IdentityMatrix[n]; DD = DiagonalMatrix[Table[b i, {i, A}]]; L i,j T = DiagonalMatrix Table Product t A i, {i, 1, n}, {j, 1, n} ; L i,j σ = Sum Product t A i, {i, 1, n} v A j, {j, 1, n} ; M = T.(II - DD.Q) /. e x_ e Simplify[x /. b i_ Log[t i ]] // Simplify; eγ ω -1, Table[y i, {i, A}].M.Table[x i, {i, A}], σ ; Verifying the crossings In[55]:= + + er i,j gr i,j // G0toΓ - - er i,j gr i,j // G0toΓ Out[55]= Out[56]= Verifying the stitching relations

9 GammaCalculusDemoFull.nb 9 In[57]:= ζ = [ω, Sum[l x,y b x c y + q x,y u x w y, {x, {i, j, S}}, {y, {i, j, S}}]] ζ // G0toΓ // em i,j k ζ // gm i,j k // G0toΓ Out[57]= [ω, b i c i l i,i + b i c j l i,j + b i c S l i,s + b j c i l j,i + b j c j l j,j + b j c S l j,s + b S c i l S,i + b S c j l S,j + b S c S l S,S + u i w i q i,i + u i w j q i,j + u i w S q i,s + u j w i q j,i + u j w j q j,j + u j w S q j,s + u S w i q S,i + u S w j q S,j + u S w S q S,S ] Out[58]= Orientation reversal In[59]:= gh[a_][e_ ] := (e /. {c a -c a, w a -w a, b a -b a, u a -u a }) // N ua c a a // N wa c a a // N wa u a a

ALEXANDER INVARIANTS OF TANGLES VIA EXPANSIONS. Huan Vo

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