The Alternating Knot 10₁₀₉

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Acknowledgement

KnotPlot

PD Presentation: X₆₂₇₁ X_10,4,11,3 X_18,11,19,12 X_16,7,17,8 X_8,17,9,18 X_20,15,1,16 X_12,19,13,20 X_14,6,15,5 X_2,10,3,9 X_4,14,5,13

Gauss Code: {1, -9, 2, -10, 8, -1, 4, -5, 9, -2, 3, -7, 10, -8, 6, -4, 5, -3, 7, -6}

DT (Dowker-Thistlethwaite) Code: 6 10 14 16 2 18 4 20 8 12

Minimum Braid Representative:

Length is 10, width is 3
Braid index is 3

A Morse Link Presentation:

3D Invariants:

Symmetry Type Unknotting Number 3-Genus Bridge/Super Bridge Index Nakanishi Index

NegativeAmphicheiral 2 4 3 / NotAvailable 1

Alexander Polynomial: t^-4 - 4t^-3 + 10t^-2 - 17t^-1 + 21 - 17t + 10t² - 4t³ + t⁴

Conway Polynomial: 1 + 3z² + 6z⁴ + 4z⁶ + z⁸

Other knots with the same Alexander/Conway Polynomial: {...}

Determinant and Signature: {85, 0}

Jones Polynomial: - q^-5 + 3q^-4 - 7q^-3 + 11q^-2 - 13q^-1 + 15 - 13q + 11q² - 7q³ + 3q⁴ - q⁵

Other knots (up to mirrors) with the same Jones Polynomial: {10₈₁, ...}

A2 (sl(3)) Invariant: - q^-14 + q^-12 - 3q^-10 + q^-8 - q^-4 + 5q^-2 - 1 + 5q² - q⁴ + q⁸ - 3q¹⁰ + q¹² - q¹⁴

HOMFLY-PT Polynomial: - 3a^-2 - 6a^-2z² - 4a^-2z⁴ - a^-2z⁶ + 7 + 15z² + 14z⁴ + 6z⁶ + z⁸ - 3a² - 6a²z² - 4a²z⁴ - a²z⁶

Kauffman Polynomial: a^-5z - 2a^-5z³ + a^-5z⁵ + 2a^-4z² - 5a^-4z⁴ + 3a^-4z⁶ - a^-3z + 4a^-3z³ - 8a^-3z⁵ + 5a^-3z⁷ + 3a^-2 - 7a^-2z² + 6a^-2z⁴ - 7a^-2z⁶ + 5a^-2z⁸ - 5a^-1z + 13a^-1z³ - 16a^-1z⁵ + 6a^-1z⁷ + 2a^-1z⁹ + 7 - 18z² + 22z⁴ - 20z⁶ + 10z⁸ - 5az + 13az³ - 16az⁵ + 6az⁷ + 2az⁹ + 3a² - 7a²z² + 6a²z⁴ - 7a²z⁶ + 5a²z⁸ - a³z + 4a³z³ - 8a³z⁵ + 5a³z⁷ + 2a⁴z² - 5a⁴z⁴ + 3a⁴z⁶ + a⁵z - 2a⁵z³ + a⁵z⁵

V₂ and V₃, the type 2 and 3 Vassiliev invariants: {3, 0}

Khovanov Homology:
(The squares with yellow highlighting are those on the "critical diagonals", where j-2r=s+1 or j-2r=s+1, where s=0 is the signature of 10₁₀₉. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

t^rq^j r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5

j = 11 1

j = 9 2

j = 7 5 1

j = 5 6 2

j = 3 7 5

j = 1 8 6

j = -1 6 8

j = -3 5 7

j = -5 2 6

j = -7 1 5

j = -9 2

j = -11 1

n Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q^-15 - 3q^-14 + 2q^-13 + 8q^-12 - 20q^-11 + 6q^-10 + 40q^-9 - 60q^-8 - 7q^-7 + 105q^-6 - 98q^-5 - 45q^-4 + 169q^-3 - 108q^-2 - 87q^-1 + 195 - 87q - 108q² + 169q³ - 45q⁴ - 98q⁵ + 105q⁶ - 7q⁷ - 60q⁸ + 40q⁹ + 6q¹⁰ - 20q¹¹ + 8q¹² + 2q¹³ - 3q¹⁴ + q¹⁵

3 - q^-30 + 3q^-29 - 2q^-28 - 3q^-27 + q^-26 + 13q^-25 - 7q^-24 - 30q^-23 + 11q^-22 + 70q^-21 - 10q^-20 - 133q^-19 - 27q^-18 + 235q^-17 + 97q^-16 - 333q^-15 - 237q^-14 + 421q^-13 + 429q^-12 - 474q^-11 - 643q^-10 + 462q^-9 + 870q^-8 - 412q^-7 - 1055q^-6 + 306q^-5 + 1216q^-4 - 203q^-3 - 1289q^-2 + 57q^-1 + 1337 + 57q - 1289q² - 203q³ + 1216q⁴ + 306q⁵ - 1055q⁶ - 412q⁷ + 870q⁸ + 462q⁹ - 643q¹⁰ - 474q¹¹ + 429q¹² + 421q¹³ - 237q¹⁴ - 333q¹⁵ + 97q¹⁶ + 235q¹⁷ - 27q¹⁸ - 133q¹⁹ - 10q²⁰ + 70q²¹ + 11q²² - 30q²³ - 7q²⁴ + 13q²⁵ + q²⁶ - 3q²⁷ - 2q²⁸ + 3q²⁹ - q³⁰

4 q^-50 - 3q^-49 + 2q^-48 + 3q^-47 - 6q^-46 + 6q^-45 - 12q^-44 + 13q^-43 + 19q^-42 - 37q^-41 + 3q^-40 - 45q^-39 + 75q^-38 + 125q^-37 - 109q^-36 - 108q^-35 - 259q^-34 + 211q^-33 + 581q^-32 + 37q^-31 - 351q^-30 - 1131q^-29 - 41q^-28 + 1474q^-27 + 1097q^-26 - 23q^-25 - 2765q^-24 - 1618q^-23 + 1862q^-22 + 3157q^-21 + 1998q^-20 - 4013q^-19 - 4524q^-18 + 507q^-17 + 4939q^-16 + 5545q^-15 - 3595q^-14 - 7303q^-13 - 2387q^-12 + 5220q^-11 + 9051q^-10 - 1716q^-9 - 8692q^-8 - 5391q^-7 + 4146q^-6 + 11245q^-5 + 514q^-4 - 8632q^-3 - 7516q^-2 + 2477q^-1 + 11939 + 2477q - 7516q² - 8632q³ + 514q⁴ + 11245q⁵ + 4146q⁶ - 5391q⁷ - 8692q⁸ - 1716q⁹ + 9051q¹⁰ + 5220q¹¹ - 2387q¹² - 7303q¹³ - 3595q¹⁴ + 5545q¹⁵ + 4939q¹⁶ + 507q¹⁷ - 4524q¹⁸ - 4013q¹⁹ + 1998q²⁰ + 3157q²¹ + 1862q²² - 1618q²³ - 2765q²⁴ - 23q²⁵ + 1097q²⁶ + 1474q²⁷ - 41q²⁸ - 1131q²⁹ - 351q³⁰ + 37q³¹ + 581q³² + 211q³³ - 259q³⁴ - 108q³⁵ - 109q³⁶ + 125q³⁷ + 75q³⁸ - 45q³⁹ + 3q⁴⁰ - 37q⁴¹ + 19q⁴² + 13q⁴³ - 12q⁴⁴ + 6q⁴⁵ - 6q⁴⁶ + 3q⁴⁷ + 2q⁴⁸ - 3q⁴⁹ + q⁵⁰

5 - q^-75 + 3q^-74 - 2q^-73 - 3q^-72 + 6q^-71 - q^-70 - 7q^-69 + 6q^-68 - 2q^-67 - 9q^-66 + 24q^-65 + 16q^-64 - 31q^-63 - 29q^-62 - 34q^-61 - 3q^-60 + 117q^-59 + 164q^-58 + 7q^-57 - 240q^-56 - 397q^-55 - 243q^-54 + 366q^-53 + 945q^-52 + 822q^-51 - 266q^-50 - 1712q^-49 - 2127q^-48 - 494q^-47 + 2443q^-46 + 4250q^-45 + 2631q^-44 - 2417q^-43 - 7114q^-42 - 6512q^-41 + 594q^-40 + 9625q^-39 + 12430q^-38 + 4136q^-37 - 10696q^-36 - 19448q^-35 - 12146q^-34 + 8453q^-33 + 26148q^-32 + 23290q^-31 - 2075q^-30 - 30685q^-29 - 35998q^-28 - 8638q^-27 + 31341q^-26 + 48438q^-25 + 22835q^-24 - 27631q^-23 - 58682q^-22 - 38496q^-21 + 19719q^-20 + 65298q^-19 + 53987q^-18 - 9004q^-17 - 68162q^-16 - 67224q^-15 - 3092q^-14 + 67469q^-13 + 77778q^-12 + 14812q^-11 - 64396q^-10 - 84931q^-9 - 25496q^-8 + 59690q^-7 + 89713q^-6 + 34344q^-5 - 54379q^-4 - 91894q^-3 - 42017q^-2 + 48392q^-1 + 92885 + 48392q - 42017q² - 91894q³ - 54379q⁴ + 34344q⁵ + 89713q⁶ + 59690q⁷ - 25496q⁸ - 84931q⁹ - 64396q¹⁰ + 14812q¹¹ + 77778q¹² + 67469q¹³ - 3092q¹⁴ - 67224q¹⁵ - 68162q¹⁶ - 9004q¹⁷ + 53987q¹⁸ + 65298q¹⁹ + 19719q²⁰ - 38496q²¹ - 58682q²² - 27631q²³ + 22835q²⁴ + 48438q²⁵ + 31341q²⁶ - 8638q²⁷ - 35998q²⁸ - 30685q²⁹ - 2075q³⁰ + 23290q³¹ + 26148q³² + 8453q³³ - 12146q³⁴ - 19448q³⁵ - 10696q³⁶ + 4136q³⁷ + 12430q³⁸ + 9625q³⁹ + 594q⁴⁰ - 6512q⁴¹ - 7114q⁴² - 2417q⁴³ + 2631q⁴⁴ + 4250q⁴⁵ + 2443q⁴⁶ - 494q⁴⁷ - 2127q⁴⁸ - 1712q⁴⁹ - 266q⁵⁰ + 822q⁵¹ + 945q⁵² + 366q⁵³ - 243q⁵⁴ - 397q⁵⁵ - 240q⁵⁶ + 7q⁵⁷ + 164q⁵⁸ + 117q⁵⁹ - 3q⁶⁰ - 34q⁶¹ - 29q⁶² - 31q⁶³ + 16q⁶⁴ + 24q⁶⁵ - 9q⁶⁶ - 2q⁶⁷ + 6q⁶⁸ - 7q⁶⁹ - q⁷⁰ + 6q⁷¹ - 3q⁷² - 2q⁷³ + 3q⁷⁴ - q⁷⁵

6 q^-105 - 3q^-104 + 2q^-103 + 3q^-102 - 6q^-101 + q^-100 + 2q^-99 + 13q^-98 - 17q^-97 - 8q^-96 + 22q^-95 - 27q^-94 + 21q^-92 + 71q^-91 - 34q^-90 - 75q^-89 + 16q^-88 - 129q^-87 - 36q^-86 + 126q^-85 + 400q^-84 + 145q^-83 - 158q^-82 - 208q^-81 - 865q^-80 - 703q^-79 + 55q^-78 + 1611q^-77 + 1858q^-76 + 1210q^-75 + 101q^-74 - 3327q^-73 - 4931q^-72 - 3798q^-71 + 1805q^-70 + 6774q^-69 + 9727q^-68 + 8450q^-67 - 2638q^-66 - 14630q^-65 - 21452q^-64 - 13030q^-63 + 4180q^-62 + 26230q^-61 + 40159q^-60 + 25383q^-59 - 9352q^-58 - 50139q^-57 - 63640q^-56 - 43374q^-55 + 15871q^-54 + 84539q^-53 + 105774q^-52 + 63311q^-51 - 36248q^-50 - 126670q^-49 - 161655q^-48 - 88027q^-47 + 66592q^-46 + 197978q^-45 + 224393q^-44 + 97674q^-43 - 106500q^-42 - 289269q^-41 - 295599q^-40 - 95461q^-39 + 189377q^-38 + 390494q^-37 + 345386q^-36 + 74611q^-35 - 302530q^-34 - 502861q^-33 - 374848q^-32 + 13010q^-31 + 434079q^-30 + 586654q^-29 + 368325q^-28 - 149880q^-27 - 585134q^-26 - 640685q^-25 - 265026q^-24 + 320369q^-23 + 705269q^-22 + 640606q^-21 + 91283q^-20 - 522554q^-19 - 789043q^-18 - 516704q^-17 + 130277q^-16 + 693197q^-15 + 803318q^-14 + 306604q^-13 - 392185q^-12 - 820783q^-11 - 673048q^-10 - 40674q^-9 + 619164q^-8 + 865087q^-7 + 447497q^-6 - 268553q^-5 - 795142q^-4 - 750697q^-3 - 163412q^-2 + 538093q^-1 + 877141 + 538093q - 163412q² - 750697q³ - 795142q⁴ - 268553q⁵ + 447497q⁶ + 865087q⁷ + 619164q⁸ - 40674q⁹ - 673048q¹⁰ - 820783q¹¹ - 392185q¹² + 306604q¹³ + 803318q¹⁴ + 693197q¹⁵ + 130277q¹⁶ - 516704q¹⁷ - 789043q¹⁸ - 522554q¹⁹ + 91283q²⁰ + 640606q²¹ + 705269q²² + 320369q²³ - 265026q²⁴ - 640685q²⁵ - 585134q²⁶ - 149880q²⁷ + 368325q²⁸ + 586654q²⁹ + 434079q³⁰ + 13010q³¹ - 374848q³² - 502861q³³ - 302530q³⁴ + 74611q³⁵ + 345386q³⁶ + 390494q³⁷ + 189377q³⁸ - 95461q³⁹ - 295599q⁴⁰ - 289269q⁴¹ - 106500q⁴² + 97674q⁴³ + 224393q⁴⁴ + 197978q⁴⁵ + 66592q⁴⁶ - 88027q⁴⁷ - 161655q⁴⁸ - 126670q⁴⁹ - 36248q⁵⁰ + 63311q⁵¹ + 105774q⁵² + 84539q⁵³ + 15871q⁵⁴ - 43374q⁵⁵ - 63640q⁵⁶ - 50139q⁵⁷ - 9352q⁵⁸ + 25383q⁵⁹ + 40159q⁶⁰ + 26230q⁶¹ + 4180q⁶² - 13030q⁶³ - 21452q⁶⁴ - 14630q⁶⁵ - 2638q⁶⁶ + 8450q⁶⁷ + 9727q⁶⁸ + 6774q⁶⁹ + 1805q⁷⁰ - 3798q⁷¹ - 4931q⁷² - 3327q⁷³ + 101q⁷⁴ + 1210q⁷⁵ + 1858q⁷⁶ + 1611q⁷⁷ + 55q⁷⁸ - 703q⁷⁹ - 865q⁸⁰ - 208q⁸¹ - 158q⁸² + 145q⁸³ + 400q⁸⁴ + 126q⁸⁵ - 36q⁸⁶ - 129q⁸⁷ + 16q⁸⁸ - 75q⁸⁹ - 34q⁹⁰ + 71q⁹¹ + 21q⁹² - 27q⁹⁴ + 22q⁹⁵ - 8q⁹⁶ - 17q⁹⁷ + 13q⁹⁸ + 2q⁹⁹ + q¹⁰⁰ - 6q¹⁰¹ + 3q¹⁰² + 2q¹⁰³ - 3q¹⁰⁴ + q¹⁰⁵

7 - q^-140 + 3q^-139 - 2q^-138 - 3q^-137 + 6q^-136 - q^-135 - 2q^-134 - 8q^-133 - 2q^-132 + 27q^-131 - 5q^-130 - 19q^-129 + 11q^-128 - 6q^-127 - 3q^-126 - 39q^-125 - 21q^-124 + 123q^-123 + 54q^-122 - 28q^-121 - 19q^-120 - 113q^-119 - 83q^-118 - 202q^-117 - 127q^-116 + 423q^-115 + 523q^-114 + 433q^-113 + 187q^-112 - 528q^-111 - 932q^-110 - 1542q^-109 - 1445q^-108 + 481q^-107 + 2262q^-106 + 3828q^-105 + 3942q^-104 + 1189q^-103 - 2557q^-102 - 7683q^-101 - 10796q^-100 - 7503q^-99 + 356q^-98 + 12216q^-97 + 22258q^-96 + 22340q^-95 + 11840q^-94 - 11241q^-93 - 37619q^-92 - 50791q^-91 - 43098q^-90 - 6817q^-89 + 46720q^-88 + 90727q^-87 + 103530q^-86 + 62047q^-85 - 27592q^-84 - 127434q^-83 - 194314q^-82 - 174586q^-81 - 54007q^-80 + 125828q^-79 + 294953q^-78 + 350927q^-77 + 235030q^-76 - 29753q^-75 - 353701q^-74 - 569223q^-73 - 535658q^-72 - 219825q^-71 + 291598q^-70 + 761077q^-69 + 932844q^-68 + 663707q^-67 - 16844q^-66 - 821391q^-65 - 1351760q^-64 - 1288831q^-63 - 532987q^-62 + 629230q^-61 + 1660551q^-60 + 2013990q^-59 + 1366120q^-58 - 89043q^-57 - 1713017q^-56 - 2697937q^-55 - 2404293q^-54 - 825147q^-53 + 1383588q^-52 + 3167221q^-51 + 3501745q^-50 + 2054700q^-49 - 619529q^-48 - 3275526q^-47 - 4475059q^-46 - 3455196q^-45 - 538986q^-44 + 2940592q^-43 + 5154249q^-42 + 4842520q^-41 + 1968432q^-40 - 2175340q^-39 - 5435305q^-38 - 6036651q^-37 - 3489453q^-36 + 1077612q^-35 + 5294987q^-34 + 6908832q^-33 + 4926397q^-32 + 200479q^-31 - 4798379q^-30 - 7408002q^-29 - 6137036q^-28 - 1494262q^-27 + 4059766q^-26 + 7557368q^-25 + 7049027q^-24 + 2668753q^-23 - 3217285q^-22 - 7435250q^-21 - 7652071q^-20 - 3640243q^-19 + 2387724q^-18 + 7144243q^-17 + 7993632q^-16 + 4380913q^-15 - 1655058q^-14 - 6781116q^-13 - 8143362q^-12 - 4913199q^-11 + 1050675q^-10 + 6420493q^-9 + 8184351q^-8 + 5287929q^-7 - 573356q^-6 - 6100215q^-5 - 8177528q^-4 - 5571466q^-3 + 179641q^-2 + 5825888q^-1 + 8172629 + 5825888q + 179641q² - 5571466q³ - 8177528q⁴ - 6100215q⁵ - 573356q⁶ + 5287929q⁷ + 8184351q⁸ + 6420493q⁹ + 1050675q¹⁰ - 4913199q¹¹ - 8143362q¹² - 6781116q¹³ - 1655058q¹⁴ + 4380913q¹⁵ + 7993632q¹⁶ + 7144243q¹⁷ + 2387724q¹⁸ - 3640243q¹⁹ - 7652071q²⁰ - 7435250q²¹ - 3217285q²² + 2668753q²³ + 7049027q²⁴ + 7557368q²⁵ + 4059766q²⁶ - 1494262q²⁷ - 6137036q²⁸ - 7408002q²⁹ - 4798379q³⁰ + 200479q³¹ + 4926397q³² + 6908832q³³ + 5294987q³⁴ + 1077612q³⁵ - 3489453q³⁶ - 6036651q³⁷ - 5435305q³⁸ - 2175340q³⁹ + 1968432q⁴⁰ + 4842520q⁴¹ + 5154249q⁴² + 2940592q⁴³ - 538986q⁴⁴ - 3455196q⁴⁵ - 4475059q⁴⁶ - 3275526q⁴⁷ - 619529q⁴⁸ + 2054700q⁴⁹ + 3501745q⁵⁰ + 3167221q⁵¹ + 1383588q⁵² - 825147q⁵³ - 2404293q⁵⁴ - 2697937q⁵⁵ - 1713017q⁵⁶ - 89043q⁵⁷ + 1366120q⁵⁸ + 2013990q⁵⁹ + 1660551q⁶⁰ + 629230q⁶¹ - 532987q⁶² - 1288831q⁶³ - 1351760q⁶⁴ - 821391q⁶⁵ - 16844q⁶⁶ + 663707q⁶⁷ + 932844q⁶⁸ + 761077q⁶⁹ + 291598q⁷⁰ - 219825q⁷¹ - 535658q⁷² - 569223q⁷³ - 353701q⁷⁴ - 29753q⁷⁵ + 235030q⁷⁶ + 350927q⁷⁷ + 294953q⁷⁸ + 125828q⁷⁹ - 54007q⁸⁰ - 174586q⁸¹ - 194314q⁸² - 127434q⁸³ - 27592q⁸⁴ + 62047q⁸⁵ + 103530q⁸⁶ + 90727q⁸⁷ + 46720q⁸⁸ - 6817q⁸⁹ - 43098q⁹⁰ - 50791q⁹¹ - 37619q⁹² - 11241q⁹³ + 11840q⁹⁴ + 22340q⁹⁵ + 22258q⁹⁶ + 12216q⁹⁷ + 356q⁹⁸ - 7503q⁹⁹ - 10796q¹⁰⁰ - 7683q¹⁰¹ - 2557q¹⁰² + 1189q¹⁰³ + 3942q¹⁰⁴ + 3828q¹⁰⁵ + 2262q¹⁰⁶ + 481q¹⁰⁷ - 1445q¹⁰⁸ - 1542q¹⁰⁹ - 932q¹¹⁰ - 528q¹¹¹ + 187q¹¹² + 433q¹¹³ + 523q¹¹⁴ + 423q¹¹⁵ - 127q¹¹⁶ - 202q¹¹⁷ - 83q¹¹⁸ - 113q¹¹⁹ - 19q¹²⁰ - 28q¹²¹ + 54q¹²² + 123q¹²³ - 21q¹²⁴ - 39q¹²⁵ - 3q¹²⁶ - 6q¹²⁷ + 11q¹²⁸ - 19q¹²⁹ - 5q¹³⁰ + 27q¹³¹ - 2q¹³² - 8q¹³³ - 2q¹³⁴ - q¹³⁵ + 6q¹³⁶ - 3q¹³⁷ - 2q¹³⁸ + 3q¹³⁹ - q¹⁴⁰

Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):

In[1]:=

<< KnotTheory`

Loading KnotTheory` (version of August 30, 2005, 10:15:35)...

In[2]:=
PD[Knot[10, 109]]

Out[2]=
PD[X[6, 2, 7, 1], X[10, 4, 11, 3], X[18, 11, 19, 12], X[16, 7, 17, 8], > X[8, 17, 9, 18], X[20, 15, 1, 16], X[12, 19, 13, 20], X[14, 6, 15, 5], > X[2, 10, 3, 9], X[4, 14, 5, 13]]

In[3]:=
GaussCode[Knot[10, 109]]

Out[3]=
GaussCode[1, -9, 2, -10, 8, -1, 4, -5, 9, -2, 3, -7, 10, -8, 6, -4, 5, -3, 7, > -6]

In[4]:=
DTCode[Knot[10, 109]]

Out[4]=
DTCode[6, 10, 14, 16, 2, 18, 4, 20, 8, 12]

In[5]:=
br = BR[Knot[10, 109]]

Out[5]=
BR[3, {-1, -1, 2, -1, 2, 2, -1, -1, 2, 2}]

In[6]:=
{First[br], Crossings[br]}

Out[6]=
{3, 10}

In[7]:=
BraidIndex[Knot[10, 109]]

Out[7]=
3

In[8]:=
Show[DrawMorseLink[Knot[10, 109]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[10, 109]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

Out[9]=
{NegativeAmphicheiral, 2, 4, 3, NotAvailable, 1}

In[10]:=
alex = Alexander[Knot[10, 109]][t]

Out[10]=
-4 4 10 17 2 3 4 21 + t - -- + -- - -- - 17 t + 10 t - 4 t + t 3 2 t t t

In[11]:=
Conway[Knot[10, 109]][z]

Out[11]=
2 4 6 8 1 + 3 z + 6 z + 4 z + z

In[12]:=
Select[AllKnots[], (alex === Alexander[#][t])&]

Out[12]=
{Knot[10, 109]}

In[13]:=
{KnotDet[Knot[10, 109]], KnotSignature[Knot[10, 109]]}

Out[13]=
{85, 0}

In[14]:=
Jones[Knot[10, 109]][q]

Out[14]=
-5 3 7 11 13 2 3 4 5 15 - q + -- - -- + -- - -- - 13 q + 11 q - 7 q + 3 q - q 4 3 2 q q q q

In[15]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]

Out[15]=
{Knot[10, 81], Knot[10, 109]}

In[16]:=
A2Invariant[Knot[10, 109]][q]

Out[16]=
-14 -12 3 -8 -4 5 2 4 8 10 12 14 -1 - q + q - --- + q - q + -- + 5 q - q + q - 3 q + q - q 10 2 q q

In[17]:=
HOMFLYPT[Knot[10, 109]][a, z]

Out[17]=
2 4 6 3 2 2 6 z 2 2 4 4 z 2 4 6 z 7 - -- - 3 a + 15 z - ---- - 6 a z + 14 z - ---- - 4 a z + 6 z - -- - 2 2 2 2 a a a a 2 6 8 > a z + z

In[18]:=
Kauffman[Knot[10, 109]][a, z]

Out[18]=
2 2 3 2 z z 5 z 3 5 2 2 z 7 z 7 + -- + 3 a + -- - -- - --- - 5 a z - a z + a z - 18 z + ---- - ---- - 2 5 3 a 4 2 a a a a a 3 3 3 2 2 4 2 2 z 4 z 13 z 3 3 3 5 3 > 7 a z + 2 a z - ---- + ---- + ----- + 13 a z + 4 a z - 2 a z + 5 3 a a a 4 4 5 5 5 4 5 z 6 z 2 4 4 4 z 8 z 16 z 5 > 22 z - ---- + ---- + 6 a z - 5 a z + -- - ---- - ----- - 16 a z - 4 2 5 3 a a a a a 6 6 7 7 3 5 5 5 6 3 z 7 z 2 6 4 6 5 z 6 z > 8 a z + a z - 20 z + ---- - ---- - 7 a z + 3 a z + ---- + ---- + 4 2 3 a a a a 8 9 7 3 7 8 5 z 2 8 2 z 9 > 6 a z + 5 a z + 10 z + ---- + 5 a z + ---- + 2 a z 2 a a

In[19]:=
{Vassiliev[2][Knot[10, 109]], Vassiliev[3][Knot[10, 109]]}

Out[19]=
{3, 0}

In[20]:=
Kh[Knot[10, 109]][q, t]

Out[20]=
8 1 2 1 5 2 6 5 7 6 - + 8 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ---- + --- + q 11 5 9 4 7 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t q t q t 3 3 2 5 2 5 3 7 3 7 4 9 4 > 6 q t + 7 q t + 5 q t + 6 q t + 2 q t + 5 q t + q t + 2 q t + 11 5 > q t

In[21]:=
ColouredJones[Knot[10, 109], 2][q]

Out[21]=
-15 3 2 8 20 6 40 60 7 105 98 45 169 195 + q - --- + --- + --- - --- + --- + -- - -- - -- + --- - -- - -- + --- - 14 13 12 11 10 9 8 7 6 5 4 3 q q q q q q q q q q q q 108 87 2 3 4 5 6 7 8 > --- - -- - 87 q - 108 q + 169 q - 45 q - 98 q + 105 q - 7 q - 60 q + 2 q q 9 10 11 12 13 14 15 > 40 q + 6 q - 20 q + 8 q + 2 q - 3 q + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 10₁₀₉

10₁₀₈
10₁₁₀

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-15 - 3q^-14 + 2q^-13 + 8q^-12 - 20q^-11 + 6q^-10 + 40q^-9 - 60q^-8 - 7q^-7 + 105q^-6 - 98q^-5 - 45q^-4 + 169q^-3 - 108q^-2 - 87q^-1 + 195 - 87q - 108q² + 169q³ - 45q⁴ - 98q⁵ + 105q⁶ - 7q⁷ - 60q⁸ + 40q⁹ + 6q¹⁰ - 20q¹¹ + 8q¹² + 2q¹³ - 3q¹⁴ + q¹⁵
3	- q^-30 + 3q^-29 - 2q^-28 - 3q^-27 + q^-26 + 13q^-25 - 7q^-24 - 30q^-23 + 11q^-22 + 70q^-21 - 10q^-20 - 133q^-19 - 27q^-18 + 235q^-17 + 97q^-16 - 333q^-15 - 237q^-14 + 421q^-13 + 429q^-12 - 474q^-11 - 643q^-10 + 462q^-9 + 870q^-8 - 412q^-7 - 1055q^-6 + 306q^-5 + 1216q^-4 - 203q^-3 - 1289q^-2 + 57q^-1 + 1337 + 57q - 1289q² - 203q³ + 1216q⁴ + 306q⁵ - 1055q⁶ - 412q⁷ + 870q⁸ + 462q⁹ - 643q¹⁰ - 474q¹¹ + 429q¹² + 421q¹³ - 237q¹⁴ - 333q¹⁵ + 97q¹⁶ + 235q¹⁷ - 27q¹⁸ - 133q¹⁹ - 10q²⁰ + 70q²¹ + 11q²² - 30q²³ - 7q²⁴ + 13q²⁵ + q²⁶ - 3q²⁷ - 2q²⁸ + 3q²⁹ - q³⁰
4	q^-50 - 3q^-49 + 2q^-48 + 3q^-47 - 6q^-46 + 6q^-45 - 12q^-44 + 13q^-43 + 19q^-42 - 37q^-41 + 3q^-40 - 45q^-39 + 75q^-38 + 125q^-37 - 109q^-36 - 108q^-35 - 259q^-34 + 211q^-33 + 581q^-32 + 37q^-31 - 351q^-30 - 1131q^-29 - 41q^-28 + 1474q^-27 + 1097q^-26 - 23q^-25 - 2765q^-24 - 1618q^-23 + 1862q^-22 + 3157q^-21 + 1998q^-20 - 4013q^-19 - 4524q^-18 + 507q^-17 + 4939q^-16 + 5545q^-15 - 3595q^-14 - 7303q^-13 - 2387q^-12 + 5220q^-11 + 9051q^-10 - 1716q^-9 - 8692q^-8 - 5391q^-7 + 4146q^-6 + 11245q^-5 + 514q^-4 - 8632q^-3 - 7516q^-2 + 2477q^-1 + 11939 + 2477q - 7516q² - 8632q³ + 514q⁴ + 11245q⁵ + 4146q⁶ - 5391q⁷ - 8692q⁸ - 1716q⁹ + 9051q¹⁰ + 5220q¹¹ - 2387q¹² - 7303q¹³ - 3595q¹⁴ + 5545q¹⁵ + 4939q¹⁶ + 507q¹⁷ - 4524q¹⁸ - 4013q¹⁹ + 1998q²⁰ + 3157q²¹ + 1862q²² - 1618q²³ - 2765q²⁴ - 23q²⁵ + 1097q²⁶ + 1474q²⁷ - 41q²⁸ - 1131q²⁹ - 351q³⁰ + 37q³¹ + 581q³² + 211q³³ - 259q³⁴ - 108q³⁵ - 109q³⁶ + 125q³⁷ + 75q³⁸ - 45q³⁹ + 3q⁴⁰ - 37q⁴¹ + 19q⁴² + 13q⁴³ - 12q⁴⁴ + 6q⁴⁵ - 6q⁴⁶ + 3q⁴⁷ + 2q⁴⁸ - 3q⁴⁹ + q⁵⁰
5	- q^-75 + 3q^-74 - 2q^-73 - 3q^-72 + 6q^-71 - q^-70 - 7q^-69 + 6q^-68 - 2q^-67 - 9q^-66 + 24q^-65 + 16q^-64 - 31q^-63 - 29q^-62 - 34q^-61 - 3q^-60 + 117q^-59 + 164q^-58 + 7q^-57 - 240q^-56 - 397q^-55 - 243q^-54 + 366q^-53 + 945q^-52 + 822q^-51 - 266q^-50 - 1712q^-49 - 2127q^-48 - 494q^-47 + 2443q^-46 + 4250q^-45 + 2631q^-44 - 2417q^-43 - 7114q^-42 - 6512q^-41 + 594q^-40 + 9625q^-39 + 12430q^-38 + 4136q^-37 - 10696q^-36 - 19448q^-35 - 12146q^-34 + 8453q^-33 + 26148q^-32 + 23290q^-31 - 2075q^-30 - 30685q^-29 - 35998q^-28 - 8638q^-27 + 31341q^-26 + 48438q^-25 + 22835q^-24 - 27631q^-23 - 58682q^-22 - 38496q^-21 + 19719q^-20 + 65298q^-19 + 53987q^-18 - 9004q^-17 - 68162q^-16 - 67224q^-15 - 3092q^-14 + 67469q^-13 + 77778q^-12 + 14812q^-11 - 64396q^-10 - 84931q^-9 - 25496q^-8 + 59690q^-7 + 89713q^-6 + 34344q^-5 - 54379q^-4 - 91894q^-3 - 42017q^-2 + 48392q^-1 + 92885 + 48392q - 42017q² - 91894q³ - 54379q⁴ + 34344q⁵ + 89713q⁶ + 59690q⁷ - 25496q⁸ - 84931q⁹ - 64396q¹⁰ + 14812q¹¹ + 77778q¹² + 67469q¹³ - 3092q¹⁴ - 67224q¹⁵ - 68162q¹⁶ - 9004q¹⁷ + 53987q¹⁸ + 65298q¹⁹ + 19719q²⁰ - 38496q²¹ - 58682q²² - 27631q²³ + 22835q²⁴ + 48438q²⁵ + 31341q²⁶ - 8638q²⁷ - 35998q²⁸ - 30685q²⁹ - 2075q³⁰ + 23290q³¹ + 26148q³² + 8453q³³ - 12146q³⁴ - 19448q³⁵ - 10696q³⁶ + 4136q³⁷ + 12430q³⁸ + 9625q³⁹ + 594q⁴⁰ - 6512q⁴¹ - 7114q⁴² - 2417q⁴³ + 2631q⁴⁴ + 4250q⁴⁵ + 2443q⁴⁶ - 494q⁴⁷ - 2127q⁴⁸ - 1712q⁴⁹ - 266q⁵⁰ + 822q⁵¹ + 945q⁵² + 366q⁵³ - 243q⁵⁴ - 397q⁵⁵ - 240q⁵⁶ + 7q⁵⁷ + 164q⁵⁸ + 117q⁵⁹ - 3q⁶⁰ - 34q⁶¹ - 29q⁶² - 31q⁶³ + 16q⁶⁴ + 24q⁶⁵ - 9q⁶⁶ - 2q⁶⁷ + 6q⁶⁸ - 7q⁶⁹ - q⁷⁰ + 6q⁷¹ - 3q⁷² - 2q⁷³ + 3q⁷⁴ - q⁷⁵
6	q^-105 - 3q^-104 + 2q^-103 + 3q^-102 - 6q^-101 + q^-100 + 2q^-99 + 13q^-98 - 17q^-97 - 8q^-96 + 22q^-95 - 27q^-94 + 21q^-92 + 71q^-91 - 34q^-90 - 75q^-89 + 16q^-88 - 129q^-87 - 36q^-86 + 126q^-85 + 400q^-84 + 145q^-83 - 158q^-82 - 208q^-81 - 865q^-80 - 703q^-79 + 55q^-78 + 1611q^-77 + 1858q^-76 + 1210q^-75 + 101q^-74 - 3327q^-73 - 4931q^-72 - 3798q^-71 + 1805q^-70 + 6774q^-69 + 9727q^-68 + 8450q^-67 - 2638q^-66 - 14630q^-65 - 21452q^-64 - 13030q^-63 + 4180q^-62 + 26230q^-61 + 40159q^-60 + 25383q^-59 - 9352q^-58 - 50139q^-57 - 63640q^-56 - 43374q^-55 + 15871q^-54 + 84539q^-53 + 105774q^-52 + 63311q^-51 - 36248q^-50 - 126670q^-49 - 161655q^-48 - 88027q^-47 + 66592q^-46 + 197978q^-45 + 224393q^-44 + 97674q^-43 - 106500q^-42 - 289269q^-41 - 295599q^-40 - 95461q^-39 + 189377q^-38 + 390494q^-37 + 345386q^-36 + 74611q^-35 - 302530q^-34 - 502861q^-33 - 374848q^-32 + 13010q^-31 + 434079q^-30 + 586654q^-29 + 368325q^-28 - 149880q^-27 - 585134q^-26 - 640685q^-25 - 265026q^-24 + 320369q^-23 + 705269q^-22 + 640606q^-21 + 91283q^-20 - 522554q^-19 - 789043q^-18 - 516704q^-17 + 130277q^-16 + 693197q^-15 + 803318q^-14 + 306604q^-13 - 392185q^-12 - 820783q^-11 - 673048q^-10 - 40674q^-9 + 619164q^-8 + 865087q^-7 + 447497q^-6 - 268553q^-5 - 795142q^-4 - 750697q^-3 - 163412q^-2 + 538093q^-1 + 877141 + 538093q - 163412q² - 750697q³ - 795142q⁴ - 268553q⁵ + 447497q⁶ + 865087q⁷ + 619164q⁸ - 40674q⁹ - 673048q¹⁰ - 820783q¹¹ - 392185q¹² + 306604q¹³ + 803318q¹⁴ + 693197q¹⁵ + 130277q¹⁶ - 516704q¹⁷ - 789043q¹⁸ - 522554q¹⁹ + 91283q²⁰ + 640606q²¹ + 705269q²² + 320369q²³ - 265026q²⁴ - 640685q²⁵ - 585134q²⁶ - 149880q²⁷ + 368325q²⁸ + 586654q²⁹ + 434079q³⁰ + 13010q³¹ - 374848q³² - 502861q³³ - 302530q³⁴ + 74611q³⁵ + 345386q³⁶ + 390494q³⁷ + 189377q³⁸ - 95461q³⁹ - 295599q⁴⁰ - 289269q⁴¹ - 106500q⁴² + 97674q⁴³ + 224393q⁴⁴ + 197978q⁴⁵ + 66592q⁴⁶ - 88027q⁴⁷ - 161655q⁴⁸ - 126670q⁴⁹ - 36248q⁵⁰ + 63311q⁵¹ + 105774q⁵² + 84539q⁵³ + 15871q⁵⁴ - 43374q⁵⁵ - 63640q⁵⁶ - 50139q⁵⁷ - 9352q⁵⁸ + 25383q⁵⁹ + 40159q⁶⁰ + 26230q⁶¹ + 4180q⁶² - 13030q⁶³ - 21452q⁶⁴ - 14630q⁶⁵ - 2638q⁶⁶ + 8450q⁶⁷ + 9727q⁶⁸ + 6774q⁶⁹ + 1805q⁷⁰ - 3798q⁷¹ - 4931q⁷² - 3327q⁷³ + 101q⁷⁴ + 1210q⁷⁵ + 1858q⁷⁶ + 1611q⁷⁷ + 55q⁷⁸ - 703q⁷⁹ - 865q⁸⁰ - 208q⁸¹ - 158q⁸² + 145q⁸³ + 400q⁸⁴ + 126q⁸⁵ - 36q⁸⁶ - 129q⁸⁷ + 16q⁸⁸ - 75q⁸⁹ - 34q⁹⁰ + 71q⁹¹ + 21q⁹² - 27q⁹⁴ + 22q⁹⁵ - 8q⁹⁶ - 17q⁹⁷ + 13q⁹⁸ + 2q⁹⁹ + q¹⁰⁰ - 6q¹⁰¹ + 3q¹⁰² + 2q¹⁰³ - 3q¹⁰⁴ + q¹⁰⁵
7	- q^-140 + 3q^-139 - 2q^-138 - 3q^-137 + 6q^-136 - q^-135 - 2q^-134 - 8q^-133 - 2q^-132 + 27q^-131 - 5q^-130 - 19q^-129 + 11q^-128 - 6q^-127 - 3q^-126 - 39q^-125 - 21q^-124 + 123q^-123 + 54q^-122 - 28q^-121 - 19q^-120 - 113q^-119 - 83q^-118 - 202q^-117 - 127q^-116 + 423q^-115 + 523q^-114 + 433q^-113 + 187q^-112 - 528q^-111 - 932q^-110 - 1542q^-109 - 1445q^-108 + 481q^-107 + 2262q^-106 + 3828q^-105 + 3942q^-104 + 1189q^-103 - 2557q^-102 - 7683q^-101 - 10796q^-100 - 7503q^-99 + 356q^-98 + 12216q^-97 + 22258q^-96 + 22340q^-95 + 11840q^-94 - 11241q^-93 - 37619q^-92 - 50791q^-91 - 43098q^-90 - 6817q^-89 + 46720q^-88 + 90727q^-87 + 103530q^-86 + 62047q^-85 - 27592q^-84 - 127434q^-83 - 194314q^-82 - 174586q^-81 - 54007q^-80 + 125828q^-79 + 294953q^-78 + 350927q^-77 + 235030q^-76 - 29753q^-75 - 353701q^-74 - 569223q^-73 - 535658q^-72 - 219825q^-71 + 291598q^-70 + 761077q^-69 + 932844q^-68 + 663707q^-67 - 16844q^-66 - 821391q^-65 - 1351760q^-64 - 1288831q^-63 - 532987q^-62 + 629230q^-61 + 1660551q^-60 + 2013990q^-59 + 1366120q^-58 - 89043q^-57 - 1713017q^-56 - 2697937q^-55 - 2404293q^-54 - 825147q^-53 + 1383588q^-52 + 3167221q^-51 + 3501745q^-50 + 2054700q^-49 - 619529q^-48 - 3275526q^-47 - 4475059q^-46 - 3455196q^-45 - 538986q^-44 + 2940592q^-43 + 5154249q^-42 + 4842520q^-41 + 1968432q^-40 - 2175340q^-39 - 5435305q^-38 - 6036651q^-37 - 3489453q^-36 + 1077612q^-35 + 5294987q^-34 + 6908832q^-33 + 4926397q^-32 + 200479q^-31 - 4798379q^-30 - 7408002q^-29 - 6137036q^-28 - 1494262q^-27 + 4059766q^-26 + 7557368q^-25 + 7049027q^-24 + 2668753q^-23 - 3217285q^-22 - 7435250q^-21 - 7652071q^-20 - 3640243q^-19 + 2387724q^-18 + 7144243q^-17 + 7993632q^-16 + 4380913q^-15 - 1655058q^-14 - 6781116q^-13 - 8143362q^-12 - 4913199q^-11 + 1050675q^-10 + 6420493q^-9 + 8184351q^-8 + 5287929q^-7 - 573356q^-6 - 6100215q^-5 - 8177528q^-4 - 5571466q^-3 + 179641q^-2 + 5825888q^-1 + 8172629 + 5825888q + 179641q² - 5571466q³ - 8177528q⁴ - 6100215q⁵ - 573356q⁶ + 5287929q⁷ + 8184351q⁸ + 6420493q⁹ + 1050675q¹⁰ - 4913199q¹¹ - 8143362q¹² - 6781116q¹³ - 1655058q¹⁴ + 4380913q¹⁵ + 7993632q¹⁶ + 7144243q¹⁷ + 2387724q¹⁸ - 3640243q¹⁹ - 7652071q²⁰ - 7435250q²¹ - 3217285q²² + 2668753q²³ + 7049027q²⁴ + 7557368q²⁵ + 4059766q²⁶ - 1494262q²⁷ - 6137036q²⁸ - 7408002q²⁹ - 4798379q³⁰ + 200479q³¹ + 4926397q³² + 6908832q³³ + 5294987q³⁴ + 1077612q³⁵ - 3489453q³⁶ - 6036651q³⁷ - 5435305q³⁸ - 2175340q³⁹ + 1968432q⁴⁰ + 4842520q⁴¹ + 5154249q⁴² + 2940592q⁴³ - 538986q⁴⁴ - 3455196q⁴⁵ - 4475059q⁴⁶ - 3275526q⁴⁷ - 619529q⁴⁸ + 2054700q⁴⁹ + 3501745q⁵⁰ + 3167221q⁵¹ + 1383588q⁵² - 825147q⁵³ - 2404293q⁵⁴ - 2697937q⁵⁵ - 1713017q⁵⁶ - 89043q⁵⁷ + 1366120q⁵⁸ + 2013990q⁵⁹ + 1660551q⁶⁰ + 629230q⁶¹ - 532987q⁶² - 1288831q⁶³ - 1351760q⁶⁴ - 821391q⁶⁵ - 16844q⁶⁶ + 663707q⁶⁷ + 932844q⁶⁸ + 761077q⁶⁹ + 291598q⁷⁰ - 219825q⁷¹ - 535658q⁷² - 569223q⁷³ - 353701q⁷⁴ - 29753q⁷⁵ + 235030q⁷⁶ + 350927q⁷⁷ + 294953q⁷⁸ + 125828q⁷⁹ - 54007q⁸⁰ - 174586q⁸¹ - 194314q⁸² - 127434q⁸³ - 27592q⁸⁴ + 62047q⁸⁵ + 103530q⁸⁶ + 90727q⁸⁷ + 46720q⁸⁸ - 6817q⁸⁹ - 43098q⁹⁰ - 50791q⁹¹ - 37619q⁹² - 11241q⁹³ + 11840q⁹⁴ + 22340q⁹⁵ + 22258q⁹⁶ + 12216q⁹⁷ + 356q⁹⁸ - 7503q⁹⁹ - 10796q¹⁰⁰ - 7683q¹⁰¹ - 2557q¹⁰² + 1189q¹⁰³ + 3942q¹⁰⁴ + 3828q¹⁰⁵ + 2262q¹⁰⁶ + 481q¹⁰⁷ - 1445q¹⁰⁸ - 1542q¹⁰⁹ - 932q¹¹⁰ - 528q¹¹¹ + 187q¹¹² + 433q¹¹³ + 523q¹¹⁴ + 423q¹¹⁵ - 127q¹¹⁶ - 202q¹¹⁷ - 83q¹¹⁸ - 113q¹¹⁹ - 19q¹²⁰ - 28q¹²¹ + 54q¹²² + 123q¹²³ - 21q¹²⁴ - 39q¹²⁵ - 3q¹²⁶ - 6q¹²⁷ + 11q¹²⁸ - 19q¹²⁹ - 5q¹³⁰ + 27q¹³¹ - 2q¹³² - 8q¹³³ - 2q¹³⁴ - q¹³⁵ + 6q¹³⁶ - 3q¹³⁷ - 2q¹³⁸ + 3q¹³⁹ - q¹⁴⁰

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	PD[Knot[10, 109]]
Out[2]=	PD[X[6, 2, 7, 1], X[10, 4, 11, 3], X[18, 11, 19, 12], X[16, 7, 17, 8], > X[8, 17, 9, 18], X[20, 15, 1, 16], X[12, 19, 13, 20], X[14, 6, 15, 5], > X[2, 10, 3, 9], X[4, 14, 5, 13]]
In[3]:=	GaussCode[Knot[10, 109]]
Out[3]=	GaussCode[1, -9, 2, -10, 8, -1, 4, -5, 9, -2, 3, -7, 10, -8, 6, -4, 5, -3, 7, > -6]
In[4]:=	DTCode[Knot[10, 109]]
Out[4]=	DTCode[6, 10, 14, 16, 2, 18, 4, 20, 8, 12]
In[5]:=	br = BR[Knot[10, 109]]
Out[5]=	BR[3, {-1, -1, 2, -1, 2, 2, -1, -1, 2, 2}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{3, 10}
In[7]:=	BraidIndex[Knot[10, 109]]
Out[7]=	3
In[8]:=	Show[DrawMorseLink[Knot[10, 109]]]

Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 109]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{NegativeAmphicheiral, 2, 4, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 109]][t]
Out[10]=	-4 4 10 17 2 3 4 21 + t - -- + -- - -- - 17 t + 10 t - 4 t + t 3 2 t t t
In[11]:=	Conway[Knot[10, 109]][z]
Out[11]=	2 4 6 8 1 + 3 z + 6 z + 4 z + z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 109]}
In[13]:=	{KnotDet[Knot[10, 109]], KnotSignature[Knot[10, 109]]}
Out[13]=	{85, 0}
In[14]:=	Jones[Knot[10, 109]][q]
Out[14]=	-5 3 7 11 13 2 3 4 5 15 - q + -- - -- + -- - -- - 13 q + 11 q - 7 q + 3 q - q 4 3 2 q q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 81], Knot[10, 109]}
In[16]:=	A2Invariant[Knot[10, 109]][q]
Out[16]=	-14 -12 3 -8 -4 5 2 4 8 10 12 14 -1 - q + q - --- + q - q + -- + 5 q - q + q - 3 q + q - q 10 2 q q
In[17]:=	HOMFLYPT[Knot[10, 109]][a, z]
Out[17]=	2 4 6 3 2 2 6 z 2 2 4 4 z 2 4 6 z 7 - -- - 3 a + 15 z - ---- - 6 a z + 14 z - ---- - 4 a z + 6 z - -- - 2 2 2 2 a a a a 2 6 8 > a z + z
In[18]:=	Kauffman[Knot[10, 109]][a, z]
Out[18]=	2 2 3 2 z z 5 z 3 5 2 2 z 7 z 7 + -- + 3 a + -- - -- - --- - 5 a z - a z + a z - 18 z + ---- - ---- - 2 5 3 a 4 2 a a a a a 3 3 3 2 2 4 2 2 z 4 z 13 z 3 3 3 5 3 > 7 a z + 2 a z - ---- + ---- + ----- + 13 a z + 4 a z - 2 a z + 5 3 a a a 4 4 5 5 5 4 5 z 6 z 2 4 4 4 z 8 z 16 z 5 > 22 z - ---- + ---- + 6 a z - 5 a z + -- - ---- - ----- - 16 a z - 4 2 5 3 a a a a a 6 6 7 7 3 5 5 5 6 3 z 7 z 2 6 4 6 5 z 6 z > 8 a z + a z - 20 z + ---- - ---- - 7 a z + 3 a z + ---- + ---- + 4 2 3 a a a a 8 9 7 3 7 8 5 z 2 8 2 z 9 > 6 a z + 5 a z + 10 z + ---- + 5 a z + ---- + 2 a z 2 a a
In[19]:=	{Vassiliev[2][Knot[10, 109]], Vassiliev[3][Knot[10, 109]]}
Out[19]=	{3, 0}
In[20]:=	Kh[Knot[10, 109]][q, t]
Out[20]=	8 1 2 1 5 2 6 5 7 6 - + 8 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ---- + --- + q 11 5 9 4 7 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t q t q t 3 3 2 5 2 5 3 7 3 7 4 9 4 > 6 q t + 7 q t + 5 q t + 6 q t + 2 q t + 5 q t + q t + 2 q t + 11 5 > q t
In[21]:=	ColouredJones[Knot[10, 109], 2][q]
Out[21]=	-15 3 2 8 20 6 40 60 7 105 98 45 169 195 + q - --- + --- + --- - --- + --- + -- - -- - -- + --- - -- - -- + --- - 14 13 12 11 10 9 8 7 6 5 4 3 q q q q q q q q q q q q 108 87 2 3 4 5 6 7 8 > --- - -- - 87 q - 108 q + 169 q - 45 q - 98 q + 105 q - 7 q - 60 q + 2 q q 9 10 11 12 13 14 15 > 40 q + 6 q - 20 q + 8 q + 2 q - 3 q + q

The Alternating Knot 10109

The Alternating Knot 10₁₀₉