結び目理論における圏論とコンピュータ計算

Transcript

1. Khovanov homology ݁ͼ໨ཧ࿦ͷݍ࿦ͱίϯϐϡʔλܭࢉ ౦ژେֶ ਺ཧՊֶݚڀՊ ത࢜՝ఔҰ೥ ࠤ໺ַਓ
2. Jones (1984) ݁ͼ໨ K Jones ଟ߲ࣜ
 VK (t) Khovanov homology Kh(K) Khovanov (2000) ∑ i,j (−1)iqj dimℚ Khi,j( − ) +POFT
   ଟ߲ࣜͷ
   lݍ࿦Խz
3. ଟ໘ମ X ΦΠϥʔ਺
 χ(X) (Simplicial) Homology H* (X) Poincaré (1895ʙ) Euler (1785ʙ) V − E + F lݍ࿦Խz
   ͷϓϩτλΠϓ ∑ i (−1)idimℚ Hi ( − )
4. $POUFOUT 1. τϙϩδʔͱݍ࿦ͷ໷໌͚ 2. ݁ͼ໨ཧ࿦ʹ͓͚Δݍ࿦ ʙ Jones ଟ߲ࣜͷݍ࿦Խ
5. 
   τϙϩδʔͱݍ࿦ͷ໷໌͚
6. έʔχώεϕϧΫͷࣣͭͷڮ໰୊
   ΦΠϥʔɼ೥ • ਤܗͷ େ͖͞΍ܗ Λ๨Εͯ ͭͳ͕Γํ ʹͷΈ஫໨ͯ͠໰୊Λղ͍ͨɽ → τϙϩδʔɾάϥϑཧ࿦ͷىݯ
7. ΦΠϥʔͷଟ໘ମެࣜ
   ೥ʙ • (݀ͷۭ͍͍ͯͳ͍)ଟ໘ମ͸ɼ ͕ඞͣ੒Γཱͭɽ • ΑΓҰൠʹ ݸͷ͕ۭ͍݀ͨଟ໘ମ͸ ͱͳΔɽ V − E + F = 2 g V − E + F = 2 − 2g g = 0 g = 1 g = 3
8. Henri Poincaré (1854–1912) ʮ਺ֶͱ͸ɺҟͳΔ΋ͷΛಉ͡΋ͷͱΈͳٕ͢ज़Ͱ͋Δʯ “la mathématique est l'art de donner le même nom à des choses différentes.” Science et Méthode, 1908.
9. lॊΒ͔͍زԿֶz
   τϙϩδʔ τϙϩδʔͰ͸
   ಉ૬ࣸ૾
   ͰҠΓ߹͏ਤܗ͸
   lಉ͡z
   ͱΈͳ͢ɽ
10. زԿֶ ม׵܈ ෆมྔ ϢʔΫϦουزԿֶ ߹ಉม׵ ௕͞ɺ໘ੵɺମੵ ૬ࣅม׵ ௕͞ɾ໘ੵɾମੵͷൺɺ
    ֯౓ɺฏߦ τϙϩδʔʢҐ૬زԿֶʣ ಉ૬ࣸ૾
    ϗϞτϐʔಉ஋ࣸ૾ ʁ
11. ʮҐ૬ෆมྔʯͷྫ ࿈݁ੑʗ࿈݁੒෼ͷݸ਺ ≈ ≈   
12. l݀z
    ͷݸ਺ ≈ ≈    ʮҐ૬ෆมྔʯͷྫ
13. ޲͖෇͚Մೳੑ ≈ ޲͖෇͚Մೳ ޲͖෇͚ෆՄೳ ԁப ϝϏ΢εͷଳ ʮҐ૬ෆมྔʯͷྫ
14. ಉ૬
    㱺
    Ґ૬ෆมྔ͸౳͍͠ Ґ૬ෆมྔ͕ҟͳΔ
    㱺
    ಉ૬Ͱͳ͍ 㱻 ରۮ ʮҐ૬ෆมྔʯͷ࢖͍ํ
15. 2
    ਤܗ͔ΒҐ૬ෆมྔΛͲ͏΍ͬͯऔΓग़͢ʁ
16. ୯ମత
    ϗϞϩδʔ ਤܗΛࡾ֯ܗʹ෼ׂͯ͠ɺ֤࣍ݩ͝ͱʹ
    ʮಠཱͳ݀ʯ͕͍ͭ͋͘Δ͔ௐ΂Δɻ
17. X = C* (X) H* (X) ࡾ֯ܗ෼ׂ͞Εͨਤܗ νΣΠϯෳମ ϗϞϩδʔ܈
18. ߟ͑ํ • ݀ ͸ த਎ͷ٧·ͬͯͳ͍αΠΫϧ ʹΑͬͯܭΒΕΔɽ • 1࣍αΠΫϧ ͸ลΛ݁ΜͰ࡞ΒΕΔด࿏ͷ͜ͱɽ • 2࣍αΠΫϧ ͸໘ͷ૊Έ߹ΘͤͰڥք͕ଧͪফ͋͠͏΋ͷͷ͜ͱɽ • ศٓతʹ௖఺͸શͯ 0࣍αΠΫϧ Ͱ͋Δͱ͢Δɽ ݀ αΠΫϧ αΠΫϧͰͳ͍
19. ߟ͑ํ • ຊ࣭తͳαΠΫϧ ͸Ͳ͏΍ͬͨΒऔΓग़ͤΔʁ • த਎͕٧·ͬͨαΠΫϧ͸ແࢹ͍ͨ͠ɽ • ೋͭͷαΠΫϧͷ͕ؒຒΊΒΕΔͱ͖ɼͦΕΒ͸ಉ͡΋ͷͱݟͳ͍ͨ͠ɽ • ແࢹ͢ΔʗಉҰࢹ͢Δ Λ਺ֶతʹѻ͏ํ๏͕ ঎ (quotient) ͷ֓೦ɽ ∼ 0 ∼
20. ਤܗΛࡾ֯ܗʹ෼ׂͯ͠ɼ֤࣍ݩ͝ͱʹϕΫτϧۭؒΛߏ੒ɽ
    ڥքࣸ૾
    㲅
    ʹΑΔ߱ԼྻΛߏ੒͠ɼ
    ΛऔΓͩ͢ɽ ker ∂/im ∂ 㲅 $ $ $ 㲅 ୯ମతෳମ͔Β୅਺తͳෳମΛߏ੒
21. ٿ໘
    㲔
    ࢛໘ମ
    ͷ৔߹
23. ͜ͷ··Ͱ͸Α͘Θ͔Βͳ͍ʜ
24. ߦྻͷجຊมܗΛߦ͍جఈΛऔΓସ͑Δͱ
    㲅
    ͸͜Μͳ؆୯ͳରԠʹͳΔɽ
25. ࣍ݩͷαΠΫϧ͸Ұ͚ͭͩ
    ʢҰ൪্ͳͷͰ͞Βʹ্͔Β͸དྷͳ͍ʣ ̌
26. ࣍ݩʹ͸ඇࣗ໌ͳαΠΫϧ͸ͳ͍
    ʢͲΕ΋Ŗ୯ମͷڥքͱͳΔʣ ̌
27. ࣍ݩͰ͸೚ҙͷ௖఺͕
    lಉ͡z
    αΠΫϧ
    ʢ͜Ε͸ٿ໘͕࿈݁Ͱ͋Δ͜ͱʹରԠʣ ̌
28. τʔϥε
    5
    ͷ৔߹
29. షΓ߹ΘͤΔ షΓ߹ΘͤΔ
32. ϗϞϩδʔ܈ͷҐ૬ෆมੑ w ͍·ϗϞϩδʔ܈Λܭࢉ͢ΔͨΊʹਤܗΛࡾ֯ܗʹ෼ׂ͕ͨ͠ɺ࣮͸ϗϞϩδʔ ܈ͷ܈ߏ଄͸ࡾ֯ܗ෼ׂͷऔΓํʹΑΒͳ͍ʂ
    w ΑΓߴ࣍ͷਤܗʹରͯ͠΋ɺ͞ΒʹҰൠͷҐ૬ۭؒʹରͯ͠΋ϗϞϩδʔ܈͸ఆ ٛͰ͖ͯɺͦͷ܈ߏ଄͸Ґ૬ෆมͰ͋Δ͜ͱ͕ূ໌Ͱ͖Δɽ
    w ϗϞϩδʔ܈͸๛෋ͳҐ૬త৘ใΛ࣋ͭ୅਺తର৅Ͱ͋Δɿ
    w
    
    㱺
    
    
    ͸ހঢ়࿈݁
    w
    
    ࣍
    CFUUJ
    ਺
    
    ͸
    
    ͷ
    l
    ࣍ݩͷ݀ͷ਺z
    Λ༩͑Δ
    w 
    ޲͖෇͚Մೳͳ
    O
    ࣍ݩดଟ༷ମ
    
    㱺
    
    
    
    ͦͷੜ੒ݩΛ
    
    ͷجຊྨͱ͍͏ H0 (X) = ℝ X bk (X) = dimℝ Hk (X) k X k X Hn (X) ≅ ℝ X
33. ϗϞϩδʔ܈ͱΦΠϥʔ਺ w ΦΠϥʔ਺
    
    ͸ɺ֤νΣΠϯ܈ͷ࣍ݩͷަ୅࿨ɻ
    w ͜ͷྔ͸
    ϗϞϩδʔ܈
    ͷ࣍ݩͷަ୅࿨Ͱ΋͋Δ͜ͱ͕Θ͔Δɻ
    w ϗϞϩδʔ܈͸Ґ૬ෆมͳͷͰɺΦΠϥʔ਺΋Ґ૬ෆมͩͱ෼͔ͬͨʂ χ = V − E + F
34. lݍ࿦Խz
    ͱ͍͏ࢦ਑ • ੔਺஋ʗଟ߲ࣜ஋ͷෆมྔ͸Ұݟ෼͔Γ΍͍͕͢ɺزԿతͳ৘ใ ͕ࣦΘΕͯ͠·͏͜ͱ͕ଟ͍ɽ • ෆมྔΛݍ࿦ͷ࿮૊ΈͷதͰଊ͑Δ͜ͱ͕Ͱ͖Ε͹ɺؔ܎ੑ΋ؚ Ίͯ୅਺తʹऔΓѻ͏͜ͱ͕Ͱ͖ΔΑ͏ʹͳΔɽ • ಛʹෆมྔ ͕ɺ͋ΔϗϞϩδʔཧ࿦ ͷ “ΦΠϥʔඪ਺” ͱ͠ ͯऔΓग़͢͜ͱ͕Ͱ͖Δͱ͖ɺ ͸ ͷݍ࿦ԽͰ͋Δͱߟ͑Δɽ x H H x
35. Ґ૬ۭؒ X χ(X) H* (X) ࠶ܝ
    lݍ࿦Խz
    ͷϓϩτλΠϓ ∑ i (−1)idimℚ Hi ( − ) H* χ ݍ࿦Խ
36. 
    ݁ͼ໨ཧ࿦ʹ͓͚Δݍ࿦
    ʙ
    +POFT
    ଟ߲ࣜͷݍ࿦Խ
37. ݁ͼ໨ͱ͸ w ۭؒ಺ʹු͔Ϳབྷ·ͬͨྠ͔ͬɻ
    w ௨Γൈ͚Λڐ͞ͳ͍࿈ଓมܗʢΠιτϐʔʣͰҠΓ͋͏΋ͷ͸ಉ͡ͱݟͳ͢ɻ
38. جຊతͳ໰͍ 
    ݁ͼ໨͕༩͑ΒΕͨͱ͖ɺͦΕ͕ղ͚Δ͔Ͳ͏͔൑ఆͤΑɽ
    
    ೋͭͷ݁ͼ໨͕༩͑ΒΕͨͱ͖ɺͦΕ͕ಉ͔͡൱͔൑ఆͤΑɽ
    㱺
    ͱͯ΋جຊతͳ໰͍͕ͩɺ౴͑Δͷ͕ۃΊͯ೉͍͠ɻ ˢ࣮͸ղ͚Δ
39. ࣹӨਤͷަ఺਺ʹΑΔ෼ྨ
40. ,
    ͱ
    ,`
    ͸ಉ͡
    㱺
    ෆมྔ͸౳͍͠ ෆมྔ͕ҟͳΔ
    㱺
    ,
    ͱ
    ,`
    ͸ҟͳΔ 㱻 ରۮ ݁ͼ໨ͷෆมྔ
41. ݁ͼ໨ͷෆมྔɿ+POFT
    ଟ߲ࣜ
     w ଟ߲ࣜͷܗͷෆมྔͰɺࣹӨਤΛ༻͍ͯ૊Έ߹ΘͤతʹܭࢉͰ͖Δɻ
    w ڧྗͳෆมྔͰ͋Δ͕ɺزԿతͳҙຯ͸Α͘෼͔Βͳ͍ɻ
    w
    ͷͱ͖
    
    Ͱ͋Δ͔ʁʢະղܾʣ VK = 1 K = ◯ 1 −q−8 + q−6 + q−2 q−4 − q−2 + 1 − q2 + q4
42. ,IPWBOPW
    IPNPMPHZ
     w ೥ɼ,IPWBOPW
    ͸
    +POFT
    ଟ߲ࣜΛ
    lݍ࿦Խz
    ͨ͠ʂ
43. ४උ
44. ४උ
    ࣍ݩίϘϧσΟζϜͷݍ • ର৅ɿෳ਺ͷԁप (1࣍ݩดଟ༷ମ) • ɹࣹɿԁपͨͪͷؒΛுΔۂ໘ (2࣍ݩͷίϘϧσΟζϜ) X Y S ʢͨͩ͠ଟ༷ମ͸޲͖͚ͮΒΕͨ΋ͷͱ͠ɼίϘϧσΟζϜ͸ڥքΛݻఆ͢Δඍ෼ಉ૬ྨΛߟ͑Δʣ
45. ४උ
    ࣍ݩίϘϧσΟζϜͷݍ X Y S • ߹੒ ʹ ۂ໘Λͭͳ͛Δ Y Z S′ ∘
46. ४උ
    ࣍ݩίϘϧσΟζϜͷݍ X • ߹੒ ʹ ۂ໘Λͭͳ͛Δ Z S′∘ S
47. ४උ
    ࣍ݩίϘϧσΟζϜͷݍ • ߃౳ࣹ ʹ ? X X idX ?
48. ४උ
    ࣍ݩίϘϧσΟζϜͷݍ • ߃౳ࣹ ʹ ౵ X X idX = X × I
49. ४උ
    'SPCFOJVT
    ୅਺ • ମ ্ͷϕΫτϧۭؒ ͕ɼ - ੵ ͱ ୯Ґ Λ࣋ͭ -୅਺Ͱ͋Γɼ - ༨ੵ ͱ ༨୯Ґ Λ࣋ͭ -༨୅਺Ͱ͋Γɼ - Frobenius ؔ܎ࣜɿ Λຬͨ͢ͱ͖ɼ Λ ্ͷ Frobenius ୅਺ ͱ͍͏ɽ k A m : A ⊗ A → A ι : k → A k Δ : A → A ⊗ A ϵ : A → k k Δ ∘ m = (id ⊗ m) ∘ (Δ ⊗ id) = (m ⊗ id) ∘ (id ⊗ Δ) A k
50. ४උ
    Մ׵
    'SPCFOJVT
    ୅਺ͱ
    52'5 • ্ͷՄ׵ͳ Frobenius ୅਺ ͕༩͑ΒΕΔͱɼ 2࣍ݩίϘϧσΟζϜͷݍ͔Β -ϕΫτϧۭؒͷݍ΁ͷؔख (TQFT) ͕ఆ·Δɽ k A k ℱ : Cob2 ⟶ Vectk
51. ४උ
    Մ׵
    'SPCFOJVT
    ୅਺ͱ
    52'5 • ର৅ͷରԠɿ • ࣹͷରԠɿ ◯ ⊔ ⋯ ⊔ ◯ r ⟶ A ⊗ ⋯ ⊗ A r
52. ४උ
    Մ׵
    'SPCFOJVT
    ୅਺ͱ
    52'5 J. Kock, Frobenius algebras and 2D topological quantum field theories
53. ,IPWBOPW
    IPNPMPHZ
    ͷߏ੒ 2. Ezak¥iF7¥F7a&AHtzBEz# 23 . ) n =3 100 110 000 010 10 I 11 1 •P@@.; D = ݁ͼ໨ͷࣹӨਤ CKh(D) νΣΠϯෳମ ϗϞϩδʔ܈ Kh(D)
54. ,IPWBOPW
    IPNPMPHZ
    ͷߏ੒ 1 . ¥t¥a's # a III. EFFE :* Ed 0 1 ← as to w ݁ͼ໨ͷਤࣜ
    
    ͕༩͑ΒΕͨͱ͢Δɽ
    w ֤ަ఺ʹରͯ͠ೋ௨Γͷղফ͕ߟ͑ΒΕΔɽ D D =
55. ,IPWBOPW
    IPNPMPHZ
    ͷߏ੒ 2. Ezak¥iF7¥F7a&AHtzBEz# 23 . Ho ) Is IB n =3 100 110 add IB Ind KB 000 010 10 I 11 1 •P@@.; iff ( 23=8 # 's ) 001 Of 1 w શͯͷަ఺Λղফ͢Δશͯͷ૊Έ߹ΘͤΛߟ͑Δɽ
56. ,IPWBOPW
    IPNPMPHZ
    ͷߏ੒ w 
    ˠ
    
    ͷ޲͖ͰลΛ݁ΜͰ
    O࣍ݩ
    ཱํମΛ࡞Δɽ 3 . At # Esa III 's a # ¥72213 Is D8 100 1 10 add IB Ind 8.8 000 010 10 I 11 1 IDE did 001 Of 1
57. ,IPWBOPW
    IPNPMPHZ
    ͷߏ੒ 3 . At # Esa III 's a # ¥72213 Is D8 100 1 10 add IB Ind 8.8 000 010 10 I 11 1 IDE did 001 Of 1 3 . At # Esa III 's a # ¥72213 Is D8 100 1 10 add IB Ind 8.8 000 010 10 I 11 1 IDE did 001 Of 1 Oo §??tE ¥¥¥¥o←s NFSHF
58. ,IPWBOPW
    IPNPMPHZ
    ͷߏ੒ 3 . At # Esa III 's a # ¥72213 Is D8 100 1 10 add IB Ind 8.8 000 010 10 I 11 1 IDE did 001 Of 1 3 . At # Esa III 's a # ¥72213 Is D8 100 1 10 add IB Ind 8.8 000 010 10 I 11 1 IDE did 001 Of 1 Oo §??tE ¥¥¥o←s TQMJU
59. ,IPWBOPW
    IPNPMPHZ
    ͷߏ੒ 3 . At # Esa III 's a A ¥72213 #*.€as ***e**¥EE¥a*⇐e¥ Edi did w ࣍ݩίϘϧσΟζϜͷݍʹ͓͚ΔՄ׵ਤࣜͱݟΔɽ
60. ,IPWBOPW
    IPNPMPHZ
    ͷߏ੒ 3 . At # Esa III 's a A ¥72213 #*.€as ***e**¥EE¥a*⇐e¥ Edi did w 'SPCFOJVT
    ୅਺
    
    ʹΑΓఆ·Δ
    52'5
    
    Ͱ
    ϕΫτϧ ۭؒͷݍʹҠ͢ʂ A = ℚ[X]/(X2) ℱ ⟶ ℱ A⊗3 A⊗2 A⊗2 A⊗2 A A A A⊗2 m m m m m m m m m Δ Δ Δ Cob2 Vectℚ
61. ,IPWBOPW
    IPNPMPHZ
    ͷߏ੒ w ลʹූ߸Λ෇͚ͯ൓Մ׵Խͯ͠ɼॎʹ௚࿨ΛऔΔɽ
    w ֤࣍਺ʹ͓͍ͯ
    
    ͱͳΓ
    
    ͸ෳମΛͳ͢ɽ
    w ͞Βʹྔࢠ࣍਺Λಋೖͯ͠ɼೋॏ࣍਺෇͖ͷෳମΛಘΔɽ d ∘ d = 0 (C*, d) A⊗3 A⊗2 A⊗2 A⊗2 A A A A⊗2 m m m −m m −m m −m m Δ Δ −Δ ⊕ ⊕ ⊕ ⊕ C0 C1 C2 C3 0 0 d d d
62. ܭࢉྫ q−1 + q −q−9 + q−5 + q−3 + q−1 = (q−1 + q)(−q−8 + q−6 + q−2) q−5 + q5 = (q + q−1)(q−4 − q−2 + 1 − q2 + q4)
63. ,IPWBOPW
    ͷఆཧ
     1. ݁ͼ໨ ͷਤࣜ ͔Βఆ·ΔϗϞϩδʔ܈ ͷʢೋॏ࣍ ਺෇͖Ճ܈ͱͯ͠ͷʣಉܕྨ͸ɼ ͷऔΓํʹΑΒͳ͍ɽ → ಉܕྨΛ ͱॻ͖ ͷ Khovanov homology ͱݺͿɽ 2. ͷ࣍਺෇͖ΦΠϥʔ਺͸ Jones ଟ߲ࣜʹҰக͢Δɿ K D {Khi,j(D)} D {Khi,j(K)} K {Khi,j(K)} ∑ i,j (−1)iqj dimℚ (Khi,j(D)) = (q + q−1)V(L)| t=−q
64. ݁ͼ໨ K Jones ଟ߲ࣜ
 VK (t) Khovanov homology Kh(K) ̂ χ +POFT
    ଟ߲ࣜͷ
    lݍ࿦Խz ݍ࿦Խ Kh V
65. ࣄ࣮ ,IPWBOPW
    IPNPMPHZ
    ͸
    +POFT
    ଟ߲ࣜΑΓ΋ਅʹڧ͍ෆมྔ ྫ) 51 10132 −q−15 + q−7 + q−5 + q−3 −q−15 + q−7 + q−5 + q−3 = ≠
66. ࣄ࣮
    ܎਺
    ,IPWBOPW
    IPNPMPHZ
    ͸݁ͼ໨ͷࣗ໌ੑΛ൑ఆͰ͖Δʂ ℤ Kh(K) = ⇒ K = Kronheimer-Mrowka, “Khovanov homology is an unknot-detector”
67. ,IPWBOPW
    IPNPMPHZ
    ͷߏ੒ʹݱΕΔͷ͸ɼ
    ༗ݶ࣍ݩϕΫτϧۭؒͱͦͷؒͷઢܗࣸ૾ɽ
    ˠ
    ίϯϐϡʔλܭࢉͰ͖Δʂ
68. %&.0 https://github.com/taketo1024/SwiftyMath-knots
69. ࢀߟจݙ • τϙϩδʔʢ୯ମతϗϞϩδʔʣʹ͍ͭͯɿ - ాதɾଜ্ʰτϙϩδʔೖ໳ʱɼSGCϥΠϒϥϦ - ాଜҰ࿠ʰτϙϩδʔʱɼؠ೾શॻ - ࠤ໺ͷϒϩάهࣄʰτϙϩδʔ΁ͷট଴ 1ʙ3ʱ • Khovanov homology ʹ͍ͭͯ - େ௬஌஧ʰ݁ͼ໨ͷෆมྔʱɼڞཱग़൛ - D. Bar-Natan, On Khovanov's categorification of the Jones polynomial • Frobenius ୅਺ͱ TQFT ʹ͍ͭͯ - J. Kock, Frobenius algebras and 2D topological quantum field theories
70. 5IBOL
    ZPV