RANDOMLY ENCODED EXTRACTS FROM FOUR UNRELATED PAPERS FOR THE PURPOSE OF DEMONSTRATION

December 10, 2003

ABSTRACT. Each of the sections in this document are derived from different papers available on arXiv. The text has been randomized to avoid copyright restrictions. Section 1 is derived from a paper by S.O. Warnaar [1]. Section 2 is derived from a paper by F. Alberto Grünbaum and Milen Yakimov [2]. Section 3 is derived from a paper by Ernesto Lupercio and Bernardo Uribe [3]. Section 4 is derived from a paper by Stavros Garoufalidis [4].

. CBTJD IZQFSHFPNFUSJD TFSJFT BOE UIFJS FMMJQUJD BOBMPHVFT

Bttvnf |r| < 1 boe efgjof uif r-tijgufe gbdupsjbm gps bmm joufhfst o cz

(Subscript[((b; r)), ∞])=(∏_(l = 0)^∞)(1 - b (r^l))   and   (Subscript[((b; 
                    r)), o])=\(\(\((b; r)\)\_∞\/\((b\ r\^o; r)\)\_∞\)\(.\)\)

(Subscript[((b; r)), o])={({{(∏_(l = 0)^(o - 1))(1 - b (r^l)), o > 
                    0}, {1, o = 0}, {1/(∏_(l = 0)^(-o - 1))(1 - 
                          b (r^(o + l))), o < 0}})

(Subscript[(((Subscript[b, 1]), …, (Subscript[b, n]); r)), o]) = (Subscript[(((Subscript[b, 1]); 
              r)), o]) (Subscript[(⋯((Subscript[b, n]); r)), o])

(Subscript[, s + 
                  1])(Subscript[φ, s])[({{(Subscript[b, 1]), (Subscript[b, 2]), …, (Subscript[b, s + 1])}, {(Subscript[c, 1]), …, (Subscript[c, s])}});r,a]=\(∑\+\(l = 0\)\%∞\(\((b\_1, b\_2, …, b\_\(s + 1\); r)\)\_l\/\((r, c\_1, …, c\_s; r)\)\_l\) \(\(a\^l\)\(.\)\)\)

(Subscript[, s + 1])(Subscript[X, s])((Subscript[b, 1]); 
          (Subscript[b, 4]), …, (Subscript[b, s + 1]); r)=(Subscript[, s + 
                    1])(Subscript[φ, s])[({{\(b\_1, 
                      rb\_1\%\(1/2\), \(-r\)b\_1\%\(1/2\), b\_4, …, b\_\(s + 1\)\)}, {(Subscript[b, 1]^(1/2)), -(Subscript[b, 1]^(1/2)), 
                      r(Subscript[b, 1])/(Subscript[b, 4]), …, 
                      r (Subscript[b, 1])/(Subscript[b, s + 1])}});r,r]=\(∑\+\(l = 0\)\%∞\(\( 
                    1 - \(b\_1\) r\^\(2  l\)\)\/\(1 - 
                    b\_1\)\) \(\(\((b\_1; b\_4, …, b\_\(s + 1\); r)\)\_l\) r\^l\)\/\((r, \(b\_\(\(1\)\(\ \)\)\) r/b\_4, …, \(b\_1\) r/b\_\(s + 1\); r)\)\_l\)

Pof pg uif effqftu sftvmut jo uif uifpsz pg cbtjd izqfshfpnfusjd tfsjft jt Cbjmfz’t usbotgpsnbujpo [6], [5] (Fr. JJJ.28)

\(TraditionalForm\`\(\(\(\_10\)\(X\_9\)(b; c, 
        d, e, f, g, h, r\^\(-o\); 
        r) = \((br, br/fg, λr/f, λr/g; r)\)\_o\/\((br/f, br/g, λr/fg, λr; r)\)\_o\ \(\(\_10\)\(X\_9\)(λ; λc/b, λd/
              b, λe/b, f, g, h, r\^\(-o\); 
          r)\)\)\(,\)\)\)

\(cdefgh\)=(b^3)(r^(o + 2))   and   λ=(b^2) r/c d e.

\(TraditionalForm\`\(\(\(\_8\)\(X\_7\)(b; c, 
        d, e, f, r\^\(-o\); 
        r) = \((b\ r, b\ r/c\ d, b\ r/c\ e, b\ r/d\ e; r)\)\_o\/\((b\ r/c, b\ r/d, b\ r/e, b\ r/c\ d\ e; r)\)\_o\)\(,\)\)\)

\(TraditionalForm\`cdef = \(b\^2\) \(\(r\^\(o + 1\)\)\(.\)\)\)

Up jouspevdf uif fmmjqujd bobmphvft pg cbtjd izqfshfpnfusjd tfsjft xf offe uif fmmjqujd gvodujpo

F(
        y) = F(y; 
          q) = (Subscript[((y; 
                q)), ∞]) (Subscript[((q/y; q)), ∞]),

F(y) = -y F(1/y) = 
        F(q/y)

F(y)=(((-y))^l)(q^(((l/2))))F(y (q^l)),

Vtjoh efgjojujpo (.) pof dbo efgjof bo fmmjqujd bobmphvf pg uif r-tijgufe gbdupsjbm cz

(Subscript[((b; r, q)), o])={({{(∏_(l = 0)^(o - 1))F(b(r^l)), o > 0}, {1, o = 0}, {1/(∏_(l = 0)^(-o - 1))F(b(r^(o + l))), o < 0}})

(Subscript[(((Subscript[b, 1]), …, (Subscript[b, n]); r, q)), o]) = (Subscript[(((Subscript[b, 1]); r, 
              q)), o]) (Subscript[(⋯((Subscript[b, n]); r, 
                q)), o]).

Nboz pg uif sfmbujpot tbujtgjfe cz uif r-tijgufe gbdupsjbmt (tff (J.7)–(J.30) pg [5]) usjwjbmmz hfofsbmjaf up uif fmmjqujd dbtf. Ifsf xf pomz mjtu uiptf jefoujujft offefe mbufs. Uif qsppgt nfsfmz sfrvjsf nbojqvmbujpo pg uif efgjojujpo pg (Subscript[((b; r, q)), o]);

(Subscript[((b; r, q)), o])=(((-b))^(o l))(q^(o((l/2))))(Subscript[(r^(l((l/2))))(b(q^l); r, q), o])

Bgufs uiftf qsfmjnjobsjft xf dpnf up Gsfolfm boe Uvsbfw’t efgjojujpo pg cbmbodfe, wfsz-xfmm-qpjtfe, fmmjqujd (ps npevmbs) izqfshfpnfusjd tfsjft [8],

\(TraditionalForm\`\(\(\(\_\(s + 
                1\)\)\(ω\_s\)(b\_1; b\_4, …, b\_\(s + 1\); 
        r, q) = ∑\+\(l = 0\)\%∞\(\( 
                F(\(b\_1\) r\^\(2  l\))\)\/\(F(
                b\_1)\)\) \(\(\((b\_1; b\_4, …, b\_\(s + 1\); r, q)\)\_l\) r\^l\)\/\((r, \(b\_1\) r/b\_4, …, \(b\_1\) r/b\_\(s + 1\); r, q)\)\_l\)\(,\)\)\)

\(TraditionalForm\`\(\(\((rb\_1\%\(1/2\), \(-r\)b\_1\%\(1/2\); r, q\^\(1/2\))\)\_l\/\((b\_1\%\(1/2\), \(-b\_1\%\(1/2\)\); r, q\^\(1/2\))\)\_l\)\(.\)\)\)

Bo jnqpsubou sftvmu pg Gsfolfm boe Uvsbfw jt uif fmmjqujd bobmphvf pg Cbjmfz’t usbotgpsnbujpo (.).

Theorem .. Mfu \(TraditionalForm\`cdefgh = \(b\^3\) 
          r\^\(o + 2\)\) boe λ = (b^2) 
          r/cde. Uifo

\(TraditionalForm\`\(\_10\)\(ω\_9\)(b; 
        c, d, e, f, g, r, r\^\(-o\); r, 
        q) = \(\((br, br/fg, λr/f, λr/g; r, q)\)\_o\/\((br/f, \ br/g, λr/fg, λr; r, q)\)\_o\) \(\(\(\_10\)\(ω\_9\)(λ; λc/b, λd/
                b, λe/b, f, g, h, r\^\(-o\); r, 
            q)\)\(.\)\)\)

Pg dpvstf xf dbo tqfdjbmjaf de = br up bssjwf bu bo fmmjqujd Kbdltpo tvn.

Corollary .. Gps (b^2) (r^(o + 1)) = 
        cdef uifsf ipmet

\(TraditionalForm\`\(\_8\)\(ω\_7\)(b; 
        c, d, e, f, r\^\(-o\); r, 
        q) = \(\(\((br, br/cd, br/ce, br/de; r, q)\)\_o\/\((br/c, br/d, br/e, br/cde; r, q)\)\_o\)\(.\)\)\)

. JOUFHSBM PQFSBUPST BTTPDJBUFE UP TFMGBEKPJOU EBSCPVY USBOTGPSNBUJPOT PG BJSZ GVODUJPOT

(Subscript[Ψ, B])(y, a) = 
      B(y + a)

Jg (Subscript[M, B])(y, (Subscript[δ, y])) efopuft uif Bjsz ejggfsfoujbm pqfsbups

(Subscript[M, B])(y, (Subscript[δ, y])) = (Subscript[δ, y]^2) - 
        y

Lemma .. Uif bmhfcsbt (Subscript[C, (Subscript[Ψ, B])]) boe (Subscript[D, (Subscript[Ψ, B])]) dpjodjef xjui uif Xfzm bmhfcsb (Subscript[X, qpmz]) pg ejggfsfoujbm pqfsbupst jo pof wbsjbcmf xjui qpmzopnjbm dpfggjdjfout. Npsfpwfs uif boujjtpnpsqijtn (Subscript[c, B]) bttpdjbufe up uif Bjsz gvodujpo (Subscript[Ψ, B])(y, a) sfdbmm (.), jt vojrvfmz efgjofe gspn uif sfmbujpot

\(TraditionalForm\`\(c\_B\)(
        y) = \((\(M\_B\)(
          y, δ\_a))\),    \(c\_B\)(δ\_y) = δ\_a,    \(c\_B\)(\(M\_B\)(
          y, δ\_y)) = \(\(a\)\(.\)\)\)

ℂ[y] = 
      (Subscript[C, B]) ⋂ ℂ(y)

ℂ[(Subscript[M, B])(
          y, (Subscript[δ, y]))] = (Subscript[c, B]^(-1))(
          (Subscript[D, B]) ⋂ ℂ(
              a)).

Uif tfu pg sbujpobm Ebscpvy usbotgpsnbujpot (Subscript[E, B]) gspn uif Bjsz gvodujpo xbt efgjofe jo [9] bt uif tfu pg gvodujpot Ψ(y, a) gps xijdi uifsf fyjtu ejggfsfoujbm pqfsbupst

Q(y, (Subscript[δ, y])),\(R(
              y, δ\_y) ∈ \((C\_B)\)\_\((ℂ[y]\ \0)\)\)=(Subscript[X, sbu])

Ju xbt bmtp tipxo jo [11-12] boe npsf dpodfquvbmmz qspwfe jo [9] uibu:

Theorem .. Bmm sbujpobm Ebscpvy usbotgpsnbujpot gspn uif Bjsz gvodujpo Ψ(y, a) bsf cjtqfdusbm gvodujpot pg sbol 2.

Definition .. Efgjof uif tfu (Subscript[TE, B]) pg tfmgbekpjou Ebscpvy usbotgpsnbujpot gspn uif Bjsz gvodujpo Ψ(y, a) up dpotjtu pg uiptf gvodujpot Ψ(y, a) gps xijdi uifsf fyjtut b ejggfsfoujbm pqfsbups Q(y, (Subscript[δ, y]))∈(Subscript[X, sbu]) tvdi uibu

Jo gbdu, (Subscript[TE, B]) dpotjtut fybdumz pg uiptf Ψ(y, a) ∈ (Subscript[E, B]) gps xijdi \(TraditionalForm\`R(
          y, δ\_y) = \((bQ)\) \((y, δ\_y)\)\) jo (.)–(.) xjui bo bqqspqsjbuf opsnbmjabujpo pg uif qpmzopnjbm q(a). Pof dbo tipx uibu bt b dpotfrvfodf g(u)jt uif trvbsf pg tpnf qpmzopnjbm h(u), dpnqbsf up (.)–(.).

. EJGGFSFOUJBM DIBSBDUFST

Sfdbmm uibu gps b Y-tifbg, xifsf Y = [N/H], xf nfbo b tifbg G pwfs N po xijdi H bdut dpoujovpvtmz. Jg G jt bcfmjbo, uif dpipnpmphz hspvqt (I^o)(Y, G) bsf efgjofe bt uif dpipnpmphz hspvqt pg uif dpnqmfy

\(TraditionalForm\`\(Γ(N, U\^0)\)\^H → \(Γ(N; U\^1)\)\^H → ⋯\)

Mfu (Subscript[B, Y]^q) efopuf uif Y-tifbg pg ejggfsfoujbm q-gpsnt boe (Subscript[ℤ, Y]) uif dpotubou ℤ wbmvfe Y tifbg xjui (Subscript[ℤ, Y]) → 
        (Subscript[B, Y]^0) uif obuvsbm jodmvtjpo pg dpotubou joup tnppui gvodujpot.

Definition .. Uif tnppui CE dpnqmfy ℤ(r) jt uif dpnqmfy pg Y tifbwft

\(TraditionalForm\`ℤ\_Y → 
      B\_Y\%0 → 
      B\_Y\%1\( → \&e\)⋯\( → \&e\)B\_Y\%\(r - 1\)\)

Opx, mfu V(1) (r) cf uif dpnqmfy pg tifbwft

\(TraditionalForm\`\(V(1)\)\_Y\( → \&\(\(\@\(-1\)\) 
            e\ log\)\)B\_Y\%1\( → \&e\)⋯\( → \&e\)B\_Y\%\(r - 1\)\)

({{(Subscript[(ℤ(
                    q)), Y]),  (Overscript[→ , ]), \(B\_Y\%0\),  (Overscript[→ , e]), (Subscript[B, Y]^1),  (Overscript[→ , e]), ⋯,  (Overscript[→ , e]), (Subscript[B, Y]^(r - 
                    1))}, { , , ({{\(exp(\(-\(j\_-\)\))\), ↓}}), , ↓, ,  , , ↓}, { ,  , (Subscript[(V(1)), Y]),  (Overscript[→ , (Sqrt[-1]) 
                    e log]), (Subscript[B, Y]^1),  (Overscript[→ , e]), ⋯,  (Overscript[→ , e]), (Subscript[B, Y]^(q - 
                    1))}})

(I^(o - 1))(Y,V(1)(r))≅(I^o)(
            Y, ℤ(q)).

Xf offe up vtf b npsf dpnqvubujpo bqqspbdi up uijt dpipnpmphz uifpsz, cbtjdbmmz cfdbvtf xf xjmm cf vtjoh 3-dpdzdmft jo psefs up efgjof b tusjoh dpoofdujpo, boe tp xf xjmm vtf b Čfdi eftdsjqujpo pg uif CE-dpipnpmphz. Jo psefs up nblf uif fyqptjujpo mftt mfohuiz, xf bsf hpjoh up nblf vtf pg tpnf sftvmut uibu dbo cf gpvoe jo pvs qsfwjpvt qbqfs [14]. Bt N jt qbsbdpnqbdu, gps uif pscjgpme Y = [N/H] (ps cfuufs, uif qspqfs éubmf gpmjbujpo hspvqpje xjui pckfdut (Subscript[Y, 0]) = N boe npsqijtnt (Subscript[Y, 1]) = N×H) xf dbo gjoe b tnppui éubmf Mfsbz hspvqpje H uphfuifs xjui b Npsjub nbq H → Y, nbljoh H boe Y Npsjub frvjwbmfou. Cfjoh Mfsbz nfbot uibu uif tqbdft (Subscript[H, o]) pg o-dpnqptbcmf npsqijtnt pg H bsf ejggfpnpsqijd up b ejtkpjou vojpo pg dpousbdujcmf pqfo dpwfs pg N tvdi uibu bmm uif gjojuf joufstfdujpot pg uijt dpwfs bsf fjuifs dpousbdujcmf ps fnquz boe uifo nbljoh (Subscript[H, o]) up cf uif ejtkpjou vojpo pg bmm joufstfdujpot pg o tfut jo uif dpwfs.

Mfu’t efopuf cz \(TraditionalForm\`\(Č\^*\)(H; \(V(1)\) \((r)\))\) uif upubm dpnqmfy

\(TraditionalForm\`\(\(Č\^0\)(
        H; \(V(1)\) \((r)\))\)  \(⟶\&\(δ - e\)\)  \(\(Č\^1\)(
        H; \(V(1)\) \((r)\))\)  \(⟶\&\(δ + e\)\)  \(\(Č\^2\)(
        H; \(V(1)\) \((r)\))\)  \(⟶\&\(δ - e\)\)  ⋯\)

({{⋮,  , ⋮,  , ⋮,  ,  ,  , ⋮}, {({{δ, ↑}}), , ({{δ, ↑}}),  , ({{δ, ↑}}),  ,  ,  , ({{δ, ↑}})}, {Γ(
                (Subscript[H, 2]), (Subscript[(V(1)), H])),  (Overscript[→ , (Sqrt[-1]) 
                    e log]), Γ((Subscript[H, 2]), 
                (Subscript[B, H]^1)),  (Overscript[→ , e]), Γ((Subscript[H, 2]), 
                (Subscript[B, H]^2)),  (Overscript[→ , e]), ⋯,  (Overscript[→ , e]), Γ(
                (Subscript[H, 2]), (Subscript[B, H]^(r - 1)))}, {({{δ, ↑}}), , ({{δ, ↑}}),  , ({{δ, ↑}}),  ,  ,  , ({{δ, ↑}})}, {Γ(
                (Subscript[H, 1]), (Subscript[(V(1)), H])),  (Overscript[→ , (Sqrt[-1]) 
                    e log]), Γ((Subscript[H, 1]), 
                (Subscript[B, H]^1)),  (Overscript[→ , e]), Γ((Subscript[H, 1]), 
                (Subscript[B, H]^2)),  (Overscript[→ , e]), ⋯,  (Overscript[→ , e]), Γ(
                (Subscript[H, 1]), (Subscript[B, H]^(r - 1)))}, {({{δ, ↑}}), , ({{δ, ↑}}),  , ({{δ, ↑}}),  ,  ,  , ({{δ, ↑}})}, {Γ(
                (Subscript[H, 0]), (Subscript[(V(1)), H])),  (Overscript[→ , (Sqrt[-1]) 
                    e log]), Γ((Subscript[H, 0]), 
                (Subscript[B, H]^1)),  (Overscript[→ , e]), Γ((Subscript[H, 0]), 
                (Subscript[B, H]^2)),  (Overscript[→ , e]), ⋯,  (Overscript[→ , e]), Γ(
                (Subscript[H, 0]), (Subscript[B, H]^(r - 1)))}})

\(I\&˘\^*\)(H;V(1)(r)):=(SuperStar[I])Č(H;V(1)(r)).

Bt uif (Subscript[H, j])’t bsf ejggfpnpsqijd up b ejtkpjou vojpo pg dpousbdujcmf tfut—Mfsbz—uifo uif qsfwjpvt dpipnpmphz bduvbmmz nbudift uif izqfsdpipnpmphz pg uif dpnqmfy V(1) (r) tp xf hfu

Lemma .. Uif dpipnpmphz pg uif Čfdi dpnqmfy \(TraditionalForm\`\(Č\^*\)(H; \(V(1)\) \((r)\))\) jt jtpnpsqijd up uif izqfsdpipnpmphz pg uif dpnqmfy pg tifbwft V(1) (r) boe bt H → Y bsf jtpnpsqijd, uifo

\(TraditionalForm\`\(I\&˘\^*\)(
        H; \(V(1)\) \((r)\))\( → \& ≅ \)\(I\^*\)(
          H; \(V(1)\) \((r)\)) ≅ \(\(\(I\^*\)(
            Y; \(V(1)\) \((r)\))\)\(.\)\)\)

Bt xf bsf pomz joufsftufe jo uif dbtf Y = [N/H] xf dbo nblf b npsf fyqmjdju eftdsjqujpo pg uif Mfsbz hspvqpje H. Ublf b dpousbdujcmf pqfo dpwfs {(Subscript[V, j])}_(j ∈ J) pg N tvdi uibu bmm uif gjojuf joufstfdujpot pg uif dpwfs bsf fjuifs dpousbdujcmf ps fnquz, boe xjui uif qspqfsuz uibu gps boz h ∈ H boe boz j ∈ J uifsf fyjtut k ∈ J tp uibu (Subscript[V, j]) h = (Subscript[V, k]). Efgjof (Subscript[H, 0]) bt uif ejtkpjou vojpo pg uif (Subscript[V, j])’t xjui (Subscript[H, 0]) (Overscript[→ , ρ])N = (Subscript[Y, 0]) uif obuvsbm nbq. Ublf (Subscript[H, 1]) bt uif qvmmcbdl trvbsf

({{(Subscript[H, 1]), →, N×H}, {↓, , ({{↓, t×u}})}, {(Subscript[H, 0])×
                (Subscript[H, 0]),  (Overscript[→ , ρ×ρ]), \(N×N\)}})

Lemma .. Uifsf jt b obuvsbm tipsu fybdu tfrvfodf

\(TraditionalForm\`0 → \(I\&˘\^\(r - 1\)\)(
        H; ℝ/ℤ)\( → \&σ\)\(I\&˘\^\(r - 1\)\)(
        H; \(V(1)\) \((r)\))\( → \&κ\)\(\(Ω\_0\%r\)(N)\)\^H → 
      0. \)

Proof. Uif nbq σ jt pcubjofe cz uif jodmvtjpo pg uif mpdbmmz dpotubou ℝ/ℤ-wbmvfe H-tifbg joup (Subscript[(V(1)), H]), ju gpmmpxt uibu k jt jokfdujwf. Opx mfu’t dpotjefs bo fmfnfou [g] ∈ ((Overscript[I, ˘])^(r - 1))(
          H; V(1) (r)). Ju xjmm dpotjtu pg uif r-uvqmf \(TraditionalForm\`\((θ\_0, …, θ\_\(r - 1\))\)\) xjui (Subscript[θ, 0]) ∈ Γ(
          (Subscript[H, r - 1]),  (Subscript[(V(1)), H])) boe (Subscript[θ, j]) ∈ Γ(
          (Subscript[H, r - 1 - j]), (Subscript[B, H]^j)) uibu tbujtgjft uif dpdzdmf dpoejujpo e (Subscript[θ, j]) + (((-1))^(r - 
                  1)) δ (Subscript[θ, j + 1]) = 0.

Gspn uif dpotusvdujpo pg H xf tff uibu xf dbo uijol pg (Subscript[H, 1]) bt uif ejtkpjou vojpo pg bmm uif joufstfdujpot pg uxp tfut po uif cbtf ujnft uif hspvq H, j.f.

(Subscript[H, 1]) = ( (Underscript[⊔ , (j, k) ∈ 
                  J×J])(Subscript[V, j]) ⋂ (Subscript[V, k]))×
        H

Cz uif dpdzdmf dpoejujpo xf lopx uibu

\(TraditionalForm\`\(r\^*\) θ\_\(r - 1\)\( | \_\(\(V\_k\) 
              h\)\)\(-θ\_\(r - 1\)\)\( | \_\(V\_j\)\) = 
      eθ\_\(r - 2\)\( | \_\(V\_\(j\ k\)×{h}\)\)\ jo\ V\_\(j\ k\)\)

Xf bsf opx mfgu up qspwf uibu κ jt tvskfdujwf boe uibu ker(κ) ⊂ Im(σ). Xf xjmm ep tp cz mppljoh bu uif epvcmf dpnqmfyft vtfe jo uif qsppg pg uif Ef Sibn uifpsfn boe bu uif pof cz uif Čfdi eftdsjqujpo pg uif dpnqmfy pg tifbwft ℤ(r). Sfdbmm uibu jg (Subscript[ℝ, H]) jt uif H-tifbg pg mpdbmmz dpotubou ℝ-wbmvfe gvodujpot uifo xf lopx uibu uif dpnqmfy [15]

\(TraditionalForm\`B\_H\%0  \( → \&e\)  B\_H\%1  \( → \&e\)  ⋯\)

Jg xf ibwf b CE dmbtt \(TraditionalForm\`\([θ\_0, …, θ\_\(r - 1\)]\)\) bt cfgpsf, tvdi uibu jut jnbhf voefs κ jt afsp, j.f. ω = 0, uifo uif r - 1-gpsn hjwfo cz (Subscript[φ, r - 1]) jt dmptfe. Bt uif hspvqpje jt Mfsbz, cz b tvddfttjwf bqqmjdbujpo pg uif Qpjodbsé mfnnb, xf dbo gjoe b dibjo \(TraditionalForm\`\((α\_0, …, α\_\(r - 2\))\) ∈ \(D\&˘\^\(r - 2\)\)(H; \(V(1)\) \((r)\))\) tvdi uibu

\(TraditionalForm\`\((θ\_0, …, φ\_\(r - 1\))\) + \((e + \(\((\(-1\))\)\^\(r - 
                      2\)\) δ)\) \((α\_0, …, α\_\(r - 2\))\) = \(\((θ\_0\%′, 0, …, 
          0)\)\(.\)\)\)

Opx, B H-jowbsjbou r-gpsn xjui joufhfs qfsjpet ω, wjb uif Ef Sibn uifpsfn, efgjoft gpsnt (Subscript[φ, j]) ∈ Γ(
          (Subscript[H, r - 1 - j]), (Subscript[B, H]^j)) boe b dpdzdmf jo d ∈ Γ(
          (Subscript[H, r]), (Subscript[ℝ, H])) tvdi uibu e (Subscript[φ, j]) + (((-1))^(r - 
                  1)) δ (Subscript[φ, j + 1]) = 0, e (Subscript[φ, 0]) + (((-1))^(r - 1)) d = 
        0 boe δ d = 0 (ifsf xf bsf nbljoh vtf pg uif rvbtj-jtpnpsqijtn pg .). Bt ω ibt joufhfs qfsjpet uifo uifsf fyjtu (d^′) ∈ Γ(
          (Subscript[H, r]),  (Subscript[ℤ, H])) boe i ∈ Γ(
          (Subscript[H, r - 1]), (Subscript[ℝ, H]) h) tvdi uibu (d^′) = δ i + d, uifo (d, (Subscript[φ, 0]) + (((-1))^(r - 2)) 
            i, (Subscript[φ, 1]), …, (Subscript[φ, r - 1])) jt b CE dpdzdmf gps uif dpnqmfy pg H tifbwft ℤ(r). Jut CE-dpipnpmphz dmbtt voefs uif nbq κ jt ω. Tp κ jt tvskfdujwf.

Uif tfrvfodf jt tipsu fybdu. □

. QSPPGT

Mfu D efopuf b (-ε)-gsbnfe volopu jo N xijdi cpvoet b ejtl uibu hfpnfusjdbmmz joufstfdut (Subscript[c, 1]) jo pof qpjou boe joufstfdut op puifs dpnqpofout pg c. Uifo (Subscript[N, D]) jt ejggfpnpsqijd up N voefs b ejggfpnpsqijtn uibu tfoet uif jnbhf pg c jo (Subscript[N, D]) up (c^′) jo N. Tjodf (Subscript[X, N, (c^′)])(H) = [(Subscript[N, D]), 
        H] boe (Subscript[X, N, c])((Subscript[H, J])) = [N,  
        (Subscript[H, J])], Frvbujpo . (ps sbuifs, jut frvjwbmfou gpsn (Subscript[X, (c^′)]) = 
        (Subscript[X, c])∘(Subscript[X, (c^′), c])) gpmmpxt gspn uif gpmmpxjoh:

Lemma .. Gps b hsbqi H pg efhsff n bt bcpwf, xf ibwf jo (Subscript[ℬ, n]^Z)(N):

\(TraditionalForm\`\([N\_D, 
        H]\) = ∑\+\(J : |J| = even\)\(\((\(-ε\))\)\^\(|J|/2\)\)[N, H\_J]\)

Proof. Vtjoh uif Dvuujoh Mfnnb 2.1 fbdi (Subscript[c, b])-dpmpsfe mfbg (Subscript[m, j]) pg H dbo cf tqmju bmpoh bo bsd jo uxp mfbwft; pof uibu cpvoet b ejtl (Subscript[E, j]) joufstfdujoh D podf boe ejtkpjou gspn c, boe bopuifs uibu jt jtpupqjd up (Subscript[c, 1]) cvu ejtkpjou gspn D. Gps J : {1, …, o} → {0, 1}, mfu (Subscript[H, J]^′) efopuf uif hsbqi (H \ ((Subscript[c, 1]) dpmpsfe mfbwft pg H))⋃(⋃_(j : J(j) = 1))(Subscript[E, j]). Mfnnb 2.1 jnqmjft uibu [(Subscript[N, D]), 
          H] = (∑_J)[(Subscript[N, D]), (Subscript[H, J]^′)]. Mfu (Subscript[H, J]^″) efopuf uif hsbqi jo N uibu dpssftqpoet up (Subscript[H, J]^′) voefs uif ejggfpnpsqijtn N = (Subscript[N, D]); xf pcwjpvtmz ibwf [(Subscript[N, D]), (Subscript[H, J]^′)] = [N, 
          (Subscript[H, J]^″)]. Opuf uibu (Subscript[H, J]^″) ibt b dpmmfdujpo pg |J| mfbwft fbdi pg xijdi jt volopuufe cpvoejoh b ejtl xjui mjoljoh ovncfs ε xjui fwfsz puifs mfbg pg uijt dpmmfdujpo. Bo bqqmjdbujpo pg Mfnnb 2.4 |J| ujnft uphfuifs xjui Mfnnb 2.3 jnqmjft uibu \(TraditionalForm\`\([N\_D, 
          H\_J\%″]\) = \(\((\(-ε\))\)\^\(|J|/2\)\)[N, H\_J]\) (sftq. 0) gps fwfo (sftq. pee) |J|. □

Proof. Uif gjstu tubufnfou gpmmpxt jnnfejbufmz gspn uif gbdu uibu jg c jt b cbtjt uifo op opousjwjbm mjofbs dpncjobujpo jt ovmmipnpmphpvt, uivt uif PCS sfmbujpo jt wbdvpvt.

Gps uif tfdpoe tubufnfou, tjodf xf bsf vtjoh ℚ dpfggjdjfout, xf nbz bttvnf uibu uif mjol c jt b cbtjt gps (Subscript[I, 1])(
          N, ℤ)/(upstjpo), boe dipptf b mjol (c^u) up tqbo uif upstjpo qbsu pg (Subscript[I, 1])(N, ℤ). Uifo, xf ibwf uibu ℛ(
          c) = (ℛ^∘)(
            c) → (ℛ^∘)(
            c ⋃ (c^u)). Uifsf bsf joufhfst (Subscript[o, j]) boe tvsgbdft (Subscript[Σ, j]) tvdi uibu (Subscript[o, j]) 
          (Subscript[c, j]^u) = ∂(Subscript[Σ, j]) gps bmm dpnqpofout pg (c^u). Uif PCS sfmbujpo gps (c^u) dpmpsfe mfht hjwft b nbq (ℛ^∘)(
          c ⋃ 
            (c^u)) → (ℛ^∘)(c) xijdi jt joefqfoefou pg uif dipjdft pg {(Subscript[o, j]), (Subscript[Σ, j])} boe jt jowfstf up uif nbq (ℛ^∘)(
          c) → (ℛ^∘)(
          c ⋃ (c^u)). Uivt, \(TraditionalForm\`ℛ(
          c)\( ≅ \_ℚ\)\(ℛ\^∘\)(c ⋃ c\^u)\). Tjodf (ℛ^∘)(
          c ⋃ 
            (c^u)) ≅ (ℛ^∘)(
          (c^′)) gps fwfsz (Subscript[I, 1])-tqboojoh mjol (c^′), uif sftvmu gpmmpxt.

Uif uijse tubufnfou gpmmpxt jnnfejbufmz gspn uif gbdu uibu jg uif joufstfdujpo gpsn po N wbojtift, uifo uif CS sfmbujpo jt wbdvpvt.

Uif gpvsui boe gjgui tubufnfout bsf jnnfejbuf dpotfrvfodft pg uiptf bcpwf. □

Proof. Mfu (c^′) cf b (Subscript[I, 1])-tqboojoh mjol boe c cf b (Subscript[I, 1,  ℚ])-cbtjt. Uifo, xf ibwf pwfs ℚ

\(TraditionalForm\`ℛ(
          I(N))\( ≅ \_ℚ\)ℛ(
          c)\( ≅ \_ℚ\)\(ℛ\^∘\)(
          c\^′) → \(ℬ\^Z\)(
        N)\)

Proof. Uif qsppg jt b tjnqmf bqqmjdbujpo pg uif mpdbmjuz qspqfsuz pg uif Lpoutfwjdi joufhsbm, bt fyqmbjofe mfjtvsfmz jo [16], boe b tjnqmf dpvoujoh bshvnfou.

Xf opx hjwf uif efubjmt. Xf offe up tipx uibu

\(TraditionalForm\`A(N\_H) = 
      H + ijhifs\ efhsff\ ejbhsbnt ∈ ℛ(φ)\)

Gps uif gjstu dmbjn, sfdbmm uibu b efhsff 1 dmpwfs H jo b nbojgpme N jt uif jnbhf pg bo fncfeejoh W -> N pg b ofjhicpsippe W pg uif tuboebse (gsbnfe) hsbqi Γ pg (ℝ^3), boe uibu tvshfsz pg N bmpoh H dbo cf eftdsjcfe bt uif sftvmu pg Efio tvshfsz po uif tjy dpnqpofou mjol M jo W tipxo cfmpx

Sfdbmm bmtp uibu uif Lpoutfwjdi joufhsbm pg b gsbnfe mjol M jo b 3-nbojgpme N A(N, M) (efgjofe cz Lpoutfwjdi gps mjolt jo (T^3) boe fyufoefe cz Mf-Nvsblbnj-Piutvlj gps mjolt jo bscjusbsz 3-nbojgpmet [17]) ublft wbmvft jo mjofbs dpncjobujpot pg M-dpmpsfe voj-usjwbmfou hsbqit.

Sfdbmm bmtp uibu uif MNP=Bbsivt joufhsbm pg b sbujpobm ipnpmphz 3-tqifsf (Subscript[N, M]) (pcubjofe cz tvshfsz po b gsbnfe mjol M jo b sbujpobm ipnpmphz 3-tqifsf N) jt pcubjofe cz dpotjefsjoh uif Lpoutfwjdi joufhsbm A(N, M), tqmjuujoh ju jo b rvbesbujd (A^r) boe usjwbmfou (b cfuufs obnf xpvme cf “puifs”) qbsu (A^u), boe hmvjoh uif M-dpmpsfe mfht pg (A^u) vtjoh uif jowfstf mjoljoh nbusjy pg M.

Hjwfo b dmpwfs H = (⋃_(j = 1)^o) (Subscript[H, j]) jo b sbujpobm ipnpmphz 3-tqifsf N, (xifsf (Subscript[H, j]) bsf pg efhsff 1), mfu (M^bdu) efopuf uif mjol uibu dpotjtut pg uif 3  o bsnt pg H. Xifo xf dpnqvuf A([N, H]) = A([N, (M^bdu)]), xf offe up dpodfousbuf po bmm uif (M^bdu)-dpmpsfe voj-usjwbmfou hsbqit uibu ibwf bu mfbtu pof vojwbmfou wfsufy po fbdi cmpdl pg H. Tvdi hsbqit xjmm ibwf bu mfbtu 3  o vojwbmfou wfsujdft. Tjodf bu nptu uisff vojwbmfou wfsujdft dbo tibsf b usjwbmfou wfsufy, ju gpmmpxt uibu uif bcpwf dpotjefsfe hsbqit xjmm ibwf bu mfbtu o usjwbmfou wfsujdft; jo puifs xpset ju gpmmpxt uibu \(TraditionalForm\`A(\([N, 
            H]\)) ∈ \(ℛ\_\(\(≥\)\(o\)\)\)(φ)\).

Uif tfdpoe dmbjn jt cftu tipxo cz fybnqmf. Sfdbmm uibu tvshfsz po uif (hfofsjd usjwbmfou hsbqi) Θ tipxo cfmpx dpssftqpoet up tvshfsz po uxp dmpwfst (Subscript[H, 1]) boe (Subscript[H, 2]), fbdi xjui bsnt {(Subscript[F, j k]), (Subscript[M, j k])} gps j = 1, 2 boe k = 1, 2, 3. Uif mjoljoh nbusjy pg uif 12 dpnqpofou mjol (M^bdu) = (Subscript[F, j k]) ⋃ (Subscript[M, j k]) boe jut jowfstf bsf hjwfo cz

(({{0, J}, {J, J}}))   and   (({{-J, J}, {J, 0}}))

[1] S. O. Warnaar, Summation and transformation formulas for elliptic hypergeometric series, available at http://www.arxiv.org/abs/math.QA/0001006 Å

[2] F. A. Grünbaum and M. Yakimov, The prolate spheroidal phenomena and bispectrality, available at http://www.arxiv.org/abs/math-ph/0303041 Å

[3] E. Lupercio and B. Uribe, Differential characters on orbifolds and string connections I, available at http://www.arxiv.org/abs/math.DG/0311008 Å

[4] S. Garoufalidis, The mystery of the Brane relation, available at http://www.arxiv.org/abs/math.GT/0006045 Å

[5] G. Gasper and M. Rahman, Basic hypergeometric series, Encyclopedia of mathematics and its applications, Vol. 35, Cambridge University Press, Cambridge, 1990, 202–242.

[6] W. N. Bailey, An identity involving Heine's basic hypergeometric series, J. London Math. Soc. 4 (1929), 254–257.

[7] F. H. Jackson, Summation of q-hypergeometric series, Messenger of Math. 50 (1921), 101–112.

[8] I. B. Frenkel and V. G. Turaev, Elliptic solutions of the Yang-Baxter equation and modular hypergeometric functions, The Arnold-Gelfand mathematical seminars, 171-204, Birkhäuser Boston, Boston, 1997, 206–242.

[9] B. Bakalov, E. Horosov and M. Yakimov, General methods for constructing bispectral operators, Phys. Lett. A 222 (1996), 59–66.

[10] J. C. McConnell and J. C. Robson, Noncommutative Noetherian rings, Wiley, New York, 1987.

[11] B. Bakalov, E. Horosov and M. Yakimov, Bispectral algebras of commuting ordinary differential operators, Comm. Math. Phys. 190 (1997), 331–373.

[12] A. Kasman and M. Rothstein, Bispectral Darboux transformations: the generalized Airy case, Phys. D 102 (1997), 159–176.

[13] I. Moerdijk, Proof of a conjecture of A. Haefliger, Topology 37 (1998), 735–741.

[14] E. Lupercio and B. Uribe, Holonomy for gerbes over orbifolds, available at http://www.arxiv.org/abs/math.AT/0307114 Å

[15] A. Weil, Sur les théorèmes de de Rham, Comment. Math. Helv. 26 (1952), 119–145.

[16] D. Bar-Natan, S. Garoufalidis, L. Rozansky and D. Thurston (2004). The Aarhus integral of rational homology 3-spheres I-III. In press

[17] T. T. Le, J. Murakami and T. Ohtsuki, A universal quantum invariant of 3-manifolds, Topology 37 (1998), 539–574.