The Alternating Knot 10₇₉

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Acknowledgement

KnotPlot

PD Presentation: X₆₂₇₁ X₈₄₉₃ X_12,6,13,5 X_18,13,19,14 X_16,9,17,10 X_10,17,11,18 X_20,15,1,16 X_14,19,15,20 X₂₈₃₇ X_4,12,5,11

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

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

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--3 4 3 / NotAvailable 1

Alexander Polynomial: t^-4 - 3t^-3 + 7t^-2 - 12t^-1 + 15 - 12t + 7t² - 3t³ + t⁴

Conway Polynomial: 1 + 5z² + 9z⁴ + 5z⁶ + z⁸

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

Determinant and Signature: {61, 0}

Jones Polynomial: - q^-5 + 2q^-4 - 5q^-3 + 8q^-2 - 9q^-1 + 11 - 9q + 8q² - 5q³ + 2q⁴ - q⁵

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

A2 (sl(3)) Invariant: - q^-14 - 3q^-10 + 5q^-2 + 1 + 5q² - 3q¹⁰ - q¹⁴

HOMFLY-PT Polynomial: - 5a^-2 - 9a^-2z² - 5a^-2z⁴ - a^-2z⁶ + 11 + 23z² + 19z⁴ + 7z⁶ + z⁸ - 5a² - 9a²z² - 5a²z⁴ - a²z⁶

Kauffman Polynomial: 2a^-5z - 3a^-5z³ + a^-5z⁵ + a^-4z² - 4a^-4z⁴ + 2a^-4z⁶ - 2a^-3z + 4a^-3z³ - 6a^-3z⁵ + 3a^-3z⁷ + 5a^-2 - 13a^-2z² + 12a^-2z⁴ - 7a^-2z⁶ + 3a^-2z⁸ - 11a^-1z + 22a^-1z³ - 15a^-1z⁵ + 4a^-1z⁷ + a^-1z⁹ + 11 - 28z² + 32z⁴ - 18z⁶ + 6z⁸ - 11az + 22az³ - 15az⁵ + 4az⁷ + az⁹ + 5a² - 13a²z² + 12a²z⁴ - 7a²z⁶ + 3a²z⁸ - 2a³z + 4a³z³ - 6a³z⁵ + 3a³z⁷ + a⁴z² - 4a⁴z⁴ + 2a⁴z⁶ + 2a⁵z - 3a⁵z³ + a⁵z⁵

V₂ and V₃, the type 2 and 3 Vassiliev invariants: {5, 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 10₇₉. 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 1

j = 7 4 1

j = 5 4 1

j = 3 5 4

j = 1 6 4

j = -1 4 6

j = -3 4 5

j = -5 1 4

j = -7 1 4

j = -9 1

j = -11 1

n Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q^-15 - 2q^-14 + q^-13 + 5q^-12 - 11q^-11 + 2q^-10 + 21q^-9 - 30q^-8 - 5q^-7 + 53q^-6 - 48q^-5 - 24q^-4 + 85q^-3 - 53q^-2 - 44q^-1 + 99 - 44q - 53q² + 85q³ - 24q⁴ - 48q⁵ + 53q⁶ - 5q⁷ - 30q⁸ + 21q⁹ + 2q¹⁰ - 11q¹¹ + 5q¹² + q¹³ - 2q¹⁴ + q¹⁵

3 - q^-30 + 2q^-29 - q^-28 - q^-27 - q^-26 + 7q^-25 - 3q^-24 - 10q^-23 + q^-22 + 27q^-21 - 4q^-20 - 43q^-19 - 13q^-18 + 80q^-17 + 31q^-16 - 107q^-15 - 80q^-14 + 135q^-13 + 143q^-12 - 152q^-11 - 212q^-10 + 143q^-9 + 291q^-8 - 131q^-7 - 348q^-6 + 90q^-5 + 412q^-4 - 67q^-3 - 427q^-2 + 12q^-1 + 455 + 12q - 427q² - 67q³ + 412q⁴ + 90q⁵ - 348q⁶ - 131q⁷ + 291q⁸ + 143q⁹ - 212q¹⁰ - 152q¹¹ + 143q¹² + 135q¹³ - 80q¹⁴ - 107q¹⁵ + 31q¹⁶ + 80q¹⁷ - 13q¹⁸ - 43q¹⁹ - 4q²⁰ + 27q²¹ + q²² - 10q²³ - 3q²⁴ + 7q²⁵ - q²⁶ - q²⁷ - q²⁸ + 2q²⁹ - q³⁰

4 q^-50 - 2q^-49 + q^-48 + q^-47 - 3q^-46 + 5q^-45 - 7q^-44 + 5q^-43 + 6q^-42 - 16q^-41 + 11q^-40 - 18q^-39 + 23q^-38 + 33q^-37 - 47q^-36 - 5q^-35 - 70q^-34 + 66q^-33 + 141q^-32 - 41q^-31 - 50q^-30 - 274q^-29 + 32q^-28 + 353q^-27 + 159q^-26 + 37q^-25 - 655q^-24 - 296q^-23 + 451q^-22 + 578q^-21 + 521q^-20 - 952q^-19 - 932q^-18 + 150q^-17 + 937q^-16 + 1362q^-15 - 873q^-14 - 1548q^-13 - 507q^-12 + 973q^-11 + 2200q^-10 - 474q^-9 - 1867q^-8 - 1182q^-7 + 733q^-6 + 2731q^-5 - 11q^-4 - 1872q^-3 - 1645q^-2 + 386q^-1 + 2903 + 386q - 1645q² - 1872q³ - 11q⁴ + 2731q⁵ + 733q⁶ - 1182q⁷ - 1867q⁸ - 474q⁹ + 2200q¹⁰ + 973q¹¹ - 507q¹² - 1548q¹³ - 873q¹⁴ + 1362q¹⁵ + 937q¹⁶ + 150q¹⁷ - 932q¹⁸ - 952q¹⁹ + 521q²⁰ + 578q²¹ + 451q²² - 296q²³ - 655q²⁴ + 37q²⁵ + 159q²⁶ + 353q²⁷ + 32q²⁸ - 274q²⁹ - 50q³⁰ - 41q³¹ + 141q³² + 66q³³ - 70q³⁴ - 5q³⁵ - 47q³⁶ + 33q³⁷ + 23q³⁸ - 18q³⁹ + 11q⁴⁰ - 16q⁴¹ + 6q⁴² + 5q⁴³ - 7q⁴⁴ + 5q⁴⁵ - 3q⁴⁶ + q⁴⁷ + q⁴⁸ - 2q⁴⁹ + q⁵⁰

5 - q^-75 + 2q^-74 - q^-73 - q^-72 + 3q^-71 - q^-70 - 5q^-69 + 5q^-68 - 4q^-66 + 10q^-65 + 3q^-64 - 19q^-63 - 2q^-62 - q^-61 + 36q^-59 + 33q^-58 - 30q^-57 - 58q^-56 - 65q^-55 - 26q^-54 + 115q^-53 + 184q^-52 + 82q^-51 - 116q^-50 - 321q^-49 - 320q^-48 + 55q^-47 + 496q^-46 + 623q^-45 + 264q^-44 - 531q^-43 - 1137q^-42 - 802q^-41 + 331q^-40 + 1510q^-39 + 1710q^-38 + 371q^-37 - 1752q^-36 - 2759q^-35 - 1527q^-34 + 1389q^-33 + 3772q^-32 + 3240q^-31 - 423q^-30 - 4456q^-29 - 5166q^-28 - 1245q^-27 + 4500q^-26 + 7083q^-25 + 3498q^-24 - 3903q^-23 - 8681q^-22 - 5933q^-21 + 2616q^-20 + 9688q^-19 + 8440q^-18 - 956q^-17 - 10187q^-16 - 10475q^-15 - 962q^-14 + 10093q^-13 + 12223q^-12 + 2709q^-11 - 9719q^-10 - 13272q^-9 - 4355q^-8 + 9034q^-7 + 14136q^-6 + 5574q^-5 - 8372q^-4 - 14366q^-3 - 6711q^-2 + 7511q^-1 + 14645 + 7511q - 6711q² - 14366q³ - 8372q⁴ + 5574q⁵ + 14136q⁶ + 9034q⁷ - 4355q⁸ - 13272q⁹ - 9719q¹⁰ + 2709q¹¹ + 12223q¹² + 10093q¹³ - 962q¹⁴ - 10475q¹⁵ - 10187q¹⁶ - 956q¹⁷ + 8440q¹⁸ + 9688q¹⁹ + 2616q²⁰ - 5933q²¹ - 8681q²² - 3903q²³ + 3498q²⁴ + 7083q²⁵ + 4500q²⁶ - 1245q²⁷ - 5166q²⁸ - 4456q²⁹ - 423q³⁰ + 3240q³¹ + 3772q³² + 1389q³³ - 1527q³⁴ - 2759q³⁵ - 1752q³⁶ + 371q³⁷ + 1710q³⁸ + 1510q³⁹ + 331q⁴⁰ - 802q⁴¹ - 1137q⁴² - 531q⁴³ + 264q⁴⁴ + 623q⁴⁵ + 496q⁴⁶ + 55q⁴⁷ - 320q⁴⁸ - 321q⁴⁹ - 116q⁵⁰ + 82q⁵¹ + 184q⁵² + 115q⁵³ - 26q⁵⁴ - 65q⁵⁵ - 58q⁵⁶ - 30q⁵⁷ + 33q⁵⁸ + 36q⁵⁹ - q⁶¹ - 2q⁶² - 19q⁶³ + 3q⁶⁴ + 10q⁶⁵ - 4q⁶⁶ + 5q⁶⁸ - 5q⁶⁹ - q⁷⁰ + 3q⁷¹ - q⁷² - q⁷³ + 2q⁷⁴ - q⁷⁵

6 q^-105 - 2q^-104 + q^-103 + q^-102 - 3q^-101 + q^-100 + q^-99 + 7q^-98 - 10q^-97 - 2q^-96 + 9q^-95 - 11q^-94 + 2q^-93 + 7q^-92 + 28q^-91 - 24q^-90 - 25q^-89 + 15q^-88 - 34q^-87 + 35q^-85 + 112q^-84 - 13q^-83 - 74q^-82 - 26q^-81 - 167q^-80 - 88q^-79 + 64q^-78 + 381q^-77 + 228q^-76 + 32q^-75 - 53q^-74 - 602q^-73 - 662q^-72 - 338q^-71 + 671q^-70 + 970q^-69 + 1015q^-68 + 792q^-67 - 858q^-66 - 2037q^-65 - 2377q^-64 - 558q^-63 + 1092q^-62 + 3075q^-61 + 4281q^-60 + 1713q^-59 - 2051q^-58 - 5863q^-57 - 5626q^-56 - 3482q^-55 + 2486q^-54 + 9325q^-53 + 9897q^-52 + 4979q^-51 - 5026q^-50 - 12051q^-49 - 15616q^-48 - 7985q^-47 + 7805q^-46 + 19382q^-45 + 21416q^-44 + 8559q^-43 - 9501q^-42 - 28770q^-41 - 29573q^-40 - 9214q^-39 + 18063q^-38 + 38427q^-37 + 34429q^-36 + 10439q^-35 - 29717q^-34 - 51274q^-33 - 39049q^-32 - 1253q^-31 + 42472q^-30 + 59898q^-29 + 42619q^-28 - 13042q^-27 - 59892q^-26 - 67753q^-25 - 31400q^-24 + 30032q^-23 + 72651q^-22 + 72550q^-21 + 12770q^-20 - 53480q^-19 - 84155q^-18 - 58447q^-17 + 9989q^-16 + 71898q^-15 + 90602q^-14 + 35273q^-13 - 40401q^-12 - 88276q^-11 - 74856q^-10 - 7328q^-9 + 64968q^-8 + 97593q^-7 + 49272q^-6 - 28571q^-5 - 86374q^-4 - 82476q^-3 - 18968q^-2 + 57604q^-1 + 99005 + 57604q - 18968q² - 82476q³ - 86374q⁴ - 28571q⁵ + 49272q⁶ + 97593q⁷ + 64968q⁸ - 7328q⁹ - 74856q¹⁰ - 88276q¹¹ - 40401q¹² + 35273q¹³ + 90602q¹⁴ + 71898q¹⁵ + 9989q¹⁶ - 58447q¹⁷ - 84155q¹⁸ - 53480q¹⁹ + 12770q²⁰ + 72550q²¹ + 72651q²² + 30032q²³ - 31400q²⁴ - 67753q²⁵ - 59892q²⁶ - 13042q²⁷ + 42619q²⁸ + 59898q²⁹ + 42472q³⁰ - 1253q³¹ - 39049q³² - 51274q³³ - 29717q³⁴ + 10439q³⁵ + 34429q³⁶ + 38427q³⁷ + 18063q³⁸ - 9214q³⁹ - 29573q⁴⁰ - 28770q⁴¹ - 9501q⁴² + 8559q⁴³ + 21416q⁴⁴ + 19382q⁴⁵ + 7805q⁴⁶ - 7985q⁴⁷ - 15616q⁴⁸ - 12051q⁴⁹ - 5026q⁵⁰ + 4979q⁵¹ + 9897q⁵² + 9325q⁵³ + 2486q⁵⁴ - 3482q⁵⁵ - 5626q⁵⁶ - 5863q⁵⁷ - 2051q⁵⁸ + 1713q⁵⁹ + 4281q⁶⁰ + 3075q⁶¹ + 1092q⁶² - 558q⁶³ - 2377q⁶⁴ - 2037q⁶⁵ - 858q⁶⁶ + 792q⁶⁷ + 1015q⁶⁸ + 970q⁶⁹ + 671q⁷⁰ - 338q⁷¹ - 662q⁷² - 602q⁷³ - 53q⁷⁴ + 32q⁷⁵ + 228q⁷⁶ + 381q⁷⁷ + 64q⁷⁸ - 88q⁷⁹ - 167q⁸⁰ - 26q⁸¹ - 74q⁸² - 13q⁸³ + 112q⁸⁴ + 35q⁸⁵ - 34q⁸⁷ + 15q⁸⁸ - 25q⁸⁹ - 24q⁹⁰ + 28q⁹¹ + 7q⁹² + 2q⁹³ - 11q⁹⁴ + 9q⁹⁵ - 2q⁹⁶ - 10q⁹⁷ + 7q⁹⁸ + q⁹⁹ + q¹⁰⁰ - 3q¹⁰¹ + q¹⁰² + q¹⁰³ - 2q¹⁰⁴ + q¹⁰⁵

7 - q^-140 + 2q^-139 - q^-138 - q^-137 + 3q^-136 - q^-135 - q^-134 - 3q^-133 - 2q^-132 + 12q^-131 - 3q^-130 - 8q^-129 + 7q^-128 - 3q^-127 - q^-126 - 12q^-125 - 10q^-124 + 45q^-123 + 11q^-122 - 21q^-121 - q^-120 - 27q^-119 - 6q^-118 - 43q^-117 - 39q^-116 + 124q^-115 + 102q^-114 + 33q^-113 + 4q^-112 - 129q^-111 - 116q^-110 - 214q^-109 - 212q^-108 + 219q^-107 + 398q^-106 + 465q^-105 + 379q^-104 - 129q^-103 - 434q^-102 - 963q^-101 - 1195q^-100 - 334q^-99 + 576q^-98 + 1633q^-97 + 2262q^-96 + 1491q^-95 + 275q^-94 - 1996q^-93 - 4083q^-92 - 3891q^-91 - 2340q^-90 + 1273q^-89 + 5489q^-88 + 7372q^-87 + 6927q^-86 + 2127q^-85 - 5346q^-84 - 10976q^-83 - 13790q^-82 - 9783q^-81 + 675q^-80 + 12280q^-79 + 22019q^-78 + 22128q^-77 + 10764q^-76 - 7003q^-75 - 27320q^-74 - 37935q^-73 - 31183q^-72 - 8944q^-71 + 24729q^-70 + 52041q^-69 + 58145q^-68 + 38820q^-67 - 6602q^-66 - 56918q^-65 - 87316q^-64 - 81709q^-63 - 30662q^-62 + 43870q^-61 + 108006q^-60 + 131582q^-59 + 88820q^-58 - 5449q^-57 - 110563q^-56 - 178746q^-55 - 161426q^-54 - 59774q^-53 + 85085q^-52 + 209955q^-51 + 238590q^-50 + 148394q^-49 - 28036q^-48 - 215295q^-47 - 307035q^-46 - 249368q^-45 - 57689q^-44 + 188369q^-43 + 354197q^-42 + 349946q^-41 + 163581q^-40 - 130500q^-39 - 373356q^-38 - 436960q^-37 - 275780q^-36 + 48810q^-35 + 362085q^-34 + 501111q^-33 + 382490q^-32 + 45578q^-31 - 326281q^-30 - 538813q^-29 - 472292q^-28 - 140463q^-27 + 273206q^-26 + 551729q^-25 + 540897q^-24 + 226027q^-23 - 214004q^-22 - 545455q^-21 - 586355q^-20 - 296298q^-19 + 155968q^-18 + 527435q^-17 + 613307q^-16 + 349033q^-15 - 106339q^-14 - 504324q^-13 - 625272q^-12 - 386210q^-11 + 65682q^-10 + 481641q^-9 + 629916q^-8 + 411290q^-7 - 35563q^-6 - 461732q^-5 - 629573q^-4 - 429427q^-3 + 10614q^-2 + 445007q^-1 + 630069 + 445007q + 10614q² - 429427q³ - 629573q⁴ - 461732q⁵ - 35563q⁶ + 411290q⁷ + 629916q⁸ + 481641q⁹ + 65682q¹⁰ - 386210q¹¹ - 625272q¹² - 504324q¹³ - 106339q¹⁴ + 349033q¹⁵ + 613307q¹⁶ + 527435q¹⁷ + 155968q¹⁸ - 296298q¹⁹ - 586355q²⁰ - 545455q²¹ - 214004q²² + 226027q²³ + 540897q²⁴ + 551729q²⁵ + 273206q²⁶ - 140463q²⁷ - 472292q²⁸ - 538813q²⁹ - 326281q³⁰ + 45578q³¹ + 382490q³² + 501111q³³ + 362085q³⁴ + 48810q³⁵ - 275780q³⁶ - 436960q³⁷ - 373356q³⁸ - 130500q³⁹ + 163581q⁴⁰ + 349946q⁴¹ + 354197q⁴² + 188369q⁴³ - 57689q⁴⁴ - 249368q⁴⁵ - 307035q⁴⁶ - 215295q⁴⁷ - 28036q⁴⁸ + 148394q⁴⁹ + 238590q⁵⁰ + 209955q⁵¹ + 85085q⁵² - 59774q⁵³ - 161426q⁵⁴ - 178746q⁵⁵ - 110563q⁵⁶ - 5449q⁵⁷ + 88820q⁵⁸ + 131582q⁵⁹ + 108006q⁶⁰ + 43870q⁶¹ - 30662q⁶² - 81709q⁶³ - 87316q⁶⁴ - 56918q⁶⁵ - 6602q⁶⁶ + 38820q⁶⁷ + 58145q⁶⁸ + 52041q⁶⁹ + 24729q⁷⁰ - 8944q⁷¹ - 31183q⁷² - 37935q⁷³ - 27320q⁷⁴ - 7003q⁷⁵ + 10764q⁷⁶ + 22128q⁷⁷ + 22019q⁷⁸ + 12280q⁷⁹ + 675q⁸⁰ - 9783q⁸¹ - 13790q⁸² - 10976q⁸³ - 5346q⁸⁴ + 2127q⁸⁵ + 6927q⁸⁶ + 7372q⁸⁷ + 5489q⁸⁸ + 1273q⁸⁹ - 2340q⁹⁰ - 3891q⁹¹ - 4083q⁹² - 1996q⁹³ + 275q⁹⁴ + 1491q⁹⁵ + 2262q⁹⁶ + 1633q⁹⁷ + 576q⁹⁸ - 334q⁹⁹ - 1195q¹⁰⁰ - 963q¹⁰¹ - 434q¹⁰² - 129q¹⁰³ + 379q¹⁰⁴ + 465q¹⁰⁵ + 398q¹⁰⁶ + 219q¹⁰⁷ - 212q¹⁰⁸ - 214q¹⁰⁹ - 116q¹¹⁰ - 129q¹¹¹ + 4q¹¹² + 33q¹¹³ + 102q¹¹⁴ + 124q¹¹⁵ - 39q¹¹⁶ - 43q¹¹⁷ - 6q¹¹⁸ - 27q¹¹⁹ - q¹²⁰ - 21q¹²¹ + 11q¹²² + 45q¹²³ - 10q¹²⁴ - 12q¹²⁵ - q¹²⁶ - 3q¹²⁷ + 7q¹²⁸ - 8q¹²⁹ - 3q¹³⁰ + 12q¹³¹ - 2q¹³² - 3q¹³³ - q¹³⁴ - q¹³⁵ + 3q¹³⁶ - q¹³⁷ - q¹³⁸ + 2q¹³⁹ - 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, 79]]

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

In[3]:=
GaussCode[Knot[10, 79]]

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

In[4]:=
DTCode[Knot[10, 79]]

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

In[5]:=
br = BR[Knot[10, 79]]

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

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

Out[6]=
{3, 10}

In[7]:=
BraidIndex[Knot[10, 79]]

Out[7]=
3

In[8]:=
Show[DrawMorseLink[Knot[10, 79]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[10, 79]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

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

In[10]:=
alex = Alexander[Knot[10, 79]][t]

Out[10]=
-4 3 7 12 2 3 4 15 + t - -- + -- - -- - 12 t + 7 t - 3 t + t 3 2 t t t

In[11]:=
Conway[Knot[10, 79]][z]

Out[11]=
2 4 6 8 1 + 5 z + 9 z + 5 z + z

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

Out[12]=
{Knot[10, 79]}

In[13]:=
{KnotDet[Knot[10, 79]], KnotSignature[Knot[10, 79]]}

Out[13]=
{61, 0}

In[14]:=
Jones[Knot[10, 79]][q]

Out[14]=
-5 2 5 8 9 2 3 4 5 11 - q + -- - -- + -- - - - 9 q + 8 q - 5 q + 2 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, 79]}

In[16]:=
A2Invariant[Knot[10, 79]][q]

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

In[17]:=
HOMFLYPT[Knot[10, 79]][a, z]

Out[17]=
2 4 6 5 2 2 9 z 2 2 4 5 z 2 4 6 z 11 - -- - 5 a + 23 z - ---- - 9 a z + 19 z - ---- - 5 a z + 7 z - -- - 2 2 2 2 a a a a 2 6 8 > a z + z

In[18]:=
Kauffman[Knot[10, 79]][a, z]

Out[18]=
2 5 2 2 z 2 z 11 z 3 5 2 z 11 + -- + 5 a + --- - --- - ---- - 11 a z - 2 a z + 2 a z - 28 z + -- - 2 5 3 a 4 a a a a 2 3 3 3 13 z 2 2 4 2 3 z 4 z 22 z 3 3 3 > ----- - 13 a z + a z - ---- + ---- + ----- + 22 a z + 4 a z - 2 5 3 a a a a 4 4 5 5 5 5 3 4 4 z 12 z 2 4 4 4 z 6 z 15 z > 3 a z + 32 z - ---- + ----- + 12 a z - 4 a z + -- - ---- - ----- - 4 2 5 3 a a a a a 6 6 5 3 5 5 5 6 2 z 7 z 2 6 4 6 > 15 a z - 6 a z + a z - 18 z + ---- - ---- - 7 a z + 2 a z + 4 2 a a 7 7 8 9 3 z 4 z 7 3 7 8 3 z 2 8 z 9 > ---- + ---- + 4 a z + 3 a z + 6 z + ---- + 3 a z + -- + a z 3 a 2 a a a

In[19]:=
{Vassiliev[2][Knot[10, 79]], Vassiliev[3][Knot[10, 79]]}

Out[19]=
{5, 0}

In[20]:=
Kh[Knot[10, 79]][q, t]

Out[20]=
6 1 1 1 4 1 4 4 5 4 - + 6 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 > 4 q t + 5 q t + 4 q t + 4 q t + q t + 4 q t + q t + q t + 11 5 > q t

In[21]:=
ColouredJones[Knot[10, 79], 2][q]

Out[21]=
-15 2 -13 5 11 2 21 30 5 53 48 24 85 99 + q - --- + q + --- - --- + --- + -- - -- - -- + -- - -- - -- + -- - 14 12 11 10 9 8 7 6 5 4 3 q q q q q q q q q q q 53 44 2 3 4 5 6 7 8 > -- - -- - 44 q - 53 q + 85 q - 24 q - 48 q + 53 q - 5 q - 30 q + 2 q q 9 10 11 12 13 14 15 > 21 q + 2 q - 11 q + 5 q + q - 2 q + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 10₇₉

10₇₈
10₈₀

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-15 - 2q^-14 + q^-13 + 5q^-12 - 11q^-11 + 2q^-10 + 21q^-9 - 30q^-8 - 5q^-7 + 53q^-6 - 48q^-5 - 24q^-4 + 85q^-3 - 53q^-2 - 44q^-1 + 99 - 44q - 53q² + 85q³ - 24q⁴ - 48q⁵ + 53q⁶ - 5q⁷ - 30q⁸ + 21q⁹ + 2q¹⁰ - 11q¹¹ + 5q¹² + q¹³ - 2q¹⁴ + q¹⁵
3	- q^-30 + 2q^-29 - q^-28 - q^-27 - q^-26 + 7q^-25 - 3q^-24 - 10q^-23 + q^-22 + 27q^-21 - 4q^-20 - 43q^-19 - 13q^-18 + 80q^-17 + 31q^-16 - 107q^-15 - 80q^-14 + 135q^-13 + 143q^-12 - 152q^-11 - 212q^-10 + 143q^-9 + 291q^-8 - 131q^-7 - 348q^-6 + 90q^-5 + 412q^-4 - 67q^-3 - 427q^-2 + 12q^-1 + 455 + 12q - 427q² - 67q³ + 412q⁴ + 90q⁵ - 348q⁶ - 131q⁷ + 291q⁸ + 143q⁹ - 212q¹⁰ - 152q¹¹ + 143q¹² + 135q¹³ - 80q¹⁴ - 107q¹⁵ + 31q¹⁶ + 80q¹⁷ - 13q¹⁸ - 43q¹⁹ - 4q²⁰ + 27q²¹ + q²² - 10q²³ - 3q²⁴ + 7q²⁵ - q²⁶ - q²⁷ - q²⁸ + 2q²⁹ - q³⁰
4	q^-50 - 2q^-49 + q^-48 + q^-47 - 3q^-46 + 5q^-45 - 7q^-44 + 5q^-43 + 6q^-42 - 16q^-41 + 11q^-40 - 18q^-39 + 23q^-38 + 33q^-37 - 47q^-36 - 5q^-35 - 70q^-34 + 66q^-33 + 141q^-32 - 41q^-31 - 50q^-30 - 274q^-29 + 32q^-28 + 353q^-27 + 159q^-26 + 37q^-25 - 655q^-24 - 296q^-23 + 451q^-22 + 578q^-21 + 521q^-20 - 952q^-19 - 932q^-18 + 150q^-17 + 937q^-16 + 1362q^-15 - 873q^-14 - 1548q^-13 - 507q^-12 + 973q^-11 + 2200q^-10 - 474q^-9 - 1867q^-8 - 1182q^-7 + 733q^-6 + 2731q^-5 - 11q^-4 - 1872q^-3 - 1645q^-2 + 386q^-1 + 2903 + 386q - 1645q² - 1872q³ - 11q⁴ + 2731q⁵ + 733q⁶ - 1182q⁷ - 1867q⁸ - 474q⁹ + 2200q¹⁰ + 973q¹¹ - 507q¹² - 1548q¹³ - 873q¹⁴ + 1362q¹⁵ + 937q¹⁶ + 150q¹⁷ - 932q¹⁸ - 952q¹⁹ + 521q²⁰ + 578q²¹ + 451q²² - 296q²³ - 655q²⁴ + 37q²⁵ + 159q²⁶ + 353q²⁷ + 32q²⁸ - 274q²⁹ - 50q³⁰ - 41q³¹ + 141q³² + 66q³³ - 70q³⁴ - 5q³⁵ - 47q³⁶ + 33q³⁷ + 23q³⁸ - 18q³⁹ + 11q⁴⁰ - 16q⁴¹ + 6q⁴² + 5q⁴³ - 7q⁴⁴ + 5q⁴⁵ - 3q⁴⁶ + q⁴⁷ + q⁴⁸ - 2q⁴⁹ + q⁵⁰
5	- q^-75 + 2q^-74 - q^-73 - q^-72 + 3q^-71 - q^-70 - 5q^-69 + 5q^-68 - 4q^-66 + 10q^-65 + 3q^-64 - 19q^-63 - 2q^-62 - q^-61 + 36q^-59 + 33q^-58 - 30q^-57 - 58q^-56 - 65q^-55 - 26q^-54 + 115q^-53 + 184q^-52 + 82q^-51 - 116q^-50 - 321q^-49 - 320q^-48 + 55q^-47 + 496q^-46 + 623q^-45 + 264q^-44 - 531q^-43 - 1137q^-42 - 802q^-41 + 331q^-40 + 1510q^-39 + 1710q^-38 + 371q^-37 - 1752q^-36 - 2759q^-35 - 1527q^-34 + 1389q^-33 + 3772q^-32 + 3240q^-31 - 423q^-30 - 4456q^-29 - 5166q^-28 - 1245q^-27 + 4500q^-26 + 7083q^-25 + 3498q^-24 - 3903q^-23 - 8681q^-22 - 5933q^-21 + 2616q^-20 + 9688q^-19 + 8440q^-18 - 956q^-17 - 10187q^-16 - 10475q^-15 - 962q^-14 + 10093q^-13 + 12223q^-12 + 2709q^-11 - 9719q^-10 - 13272q^-9 - 4355q^-8 + 9034q^-7 + 14136q^-6 + 5574q^-5 - 8372q^-4 - 14366q^-3 - 6711q^-2 + 7511q^-1 + 14645 + 7511q - 6711q² - 14366q³ - 8372q⁴ + 5574q⁵ + 14136q⁶ + 9034q⁷ - 4355q⁸ - 13272q⁹ - 9719q¹⁰ + 2709q¹¹ + 12223q¹² + 10093q¹³ - 962q¹⁴ - 10475q¹⁵ - 10187q¹⁶ - 956q¹⁷ + 8440q¹⁸ + 9688q¹⁹ + 2616q²⁰ - 5933q²¹ - 8681q²² - 3903q²³ + 3498q²⁴ + 7083q²⁵ + 4500q²⁶ - 1245q²⁷ - 5166q²⁸ - 4456q²⁹ - 423q³⁰ + 3240q³¹ + 3772q³² + 1389q³³ - 1527q³⁴ - 2759q³⁵ - 1752q³⁶ + 371q³⁷ + 1710q³⁸ + 1510q³⁹ + 331q⁴⁰ - 802q⁴¹ - 1137q⁴² - 531q⁴³ + 264q⁴⁴ + 623q⁴⁵ + 496q⁴⁶ + 55q⁴⁷ - 320q⁴⁸ - 321q⁴⁹ - 116q⁵⁰ + 82q⁵¹ + 184q⁵² + 115q⁵³ - 26q⁵⁴ - 65q⁵⁵ - 58q⁵⁶ - 30q⁵⁷ + 33q⁵⁸ + 36q⁵⁹ - q⁶¹ - 2q⁶² - 19q⁶³ + 3q⁶⁴ + 10q⁶⁵ - 4q⁶⁶ + 5q⁶⁸ - 5q⁶⁹ - q⁷⁰ + 3q⁷¹ - q⁷² - q⁷³ + 2q⁷⁴ - q⁷⁵
6	q^-105 - 2q^-104 + q^-103 + q^-102 - 3q^-101 + q^-100 + q^-99 + 7q^-98 - 10q^-97 - 2q^-96 + 9q^-95 - 11q^-94 + 2q^-93 + 7q^-92 + 28q^-91 - 24q^-90 - 25q^-89 + 15q^-88 - 34q^-87 + 35q^-85 + 112q^-84 - 13q^-83 - 74q^-82 - 26q^-81 - 167q^-80 - 88q^-79 + 64q^-78 + 381q^-77 + 228q^-76 + 32q^-75 - 53q^-74 - 602q^-73 - 662q^-72 - 338q^-71 + 671q^-70 + 970q^-69 + 1015q^-68 + 792q^-67 - 858q^-66 - 2037q^-65 - 2377q^-64 - 558q^-63 + 1092q^-62 + 3075q^-61 + 4281q^-60 + 1713q^-59 - 2051q^-58 - 5863q^-57 - 5626q^-56 - 3482q^-55 + 2486q^-54 + 9325q^-53 + 9897q^-52 + 4979q^-51 - 5026q^-50 - 12051q^-49 - 15616q^-48 - 7985q^-47 + 7805q^-46 + 19382q^-45 + 21416q^-44 + 8559q^-43 - 9501q^-42 - 28770q^-41 - 29573q^-40 - 9214q^-39 + 18063q^-38 + 38427q^-37 + 34429q^-36 + 10439q^-35 - 29717q^-34 - 51274q^-33 - 39049q^-32 - 1253q^-31 + 42472q^-30 + 59898q^-29 + 42619q^-28 - 13042q^-27 - 59892q^-26 - 67753q^-25 - 31400q^-24 + 30032q^-23 + 72651q^-22 + 72550q^-21 + 12770q^-20 - 53480q^-19 - 84155q^-18 - 58447q^-17 + 9989q^-16 + 71898q^-15 + 90602q^-14 + 35273q^-13 - 40401q^-12 - 88276q^-11 - 74856q^-10 - 7328q^-9 + 64968q^-8 + 97593q^-7 + 49272q^-6 - 28571q^-5 - 86374q^-4 - 82476q^-3 - 18968q^-2 + 57604q^-1 + 99005 + 57604q - 18968q² - 82476q³ - 86374q⁴ - 28571q⁵ + 49272q⁶ + 97593q⁷ + 64968q⁸ - 7328q⁹ - 74856q¹⁰ - 88276q¹¹ - 40401q¹² + 35273q¹³ + 90602q¹⁴ + 71898q¹⁵ + 9989q¹⁶ - 58447q¹⁷ - 84155q¹⁸ - 53480q¹⁹ + 12770q²⁰ + 72550q²¹ + 72651q²² + 30032q²³ - 31400q²⁴ - 67753q²⁵ - 59892q²⁶ - 13042q²⁷ + 42619q²⁸ + 59898q²⁹ + 42472q³⁰ - 1253q³¹ - 39049q³² - 51274q³³ - 29717q³⁴ + 10439q³⁵ + 34429q³⁶ + 38427q³⁷ + 18063q³⁸ - 9214q³⁹ - 29573q⁴⁰ - 28770q⁴¹ - 9501q⁴² + 8559q⁴³ + 21416q⁴⁴ + 19382q⁴⁵ + 7805q⁴⁶ - 7985q⁴⁷ - 15616q⁴⁸ - 12051q⁴⁹ - 5026q⁵⁰ + 4979q⁵¹ + 9897q⁵² + 9325q⁵³ + 2486q⁵⁴ - 3482q⁵⁵ - 5626q⁵⁶ - 5863q⁵⁷ - 2051q⁵⁸ + 1713q⁵⁹ + 4281q⁶⁰ + 3075q⁶¹ + 1092q⁶² - 558q⁶³ - 2377q⁶⁴ - 2037q⁶⁵ - 858q⁶⁶ + 792q⁶⁷ + 1015q⁶⁸ + 970q⁶⁹ + 671q⁷⁰ - 338q⁷¹ - 662q⁷² - 602q⁷³ - 53q⁷⁴ + 32q⁷⁵ + 228q⁷⁶ + 381q⁷⁷ + 64q⁷⁸ - 88q⁷⁹ - 167q⁸⁰ - 26q⁸¹ - 74q⁸² - 13q⁸³ + 112q⁸⁴ + 35q⁸⁵ - 34q⁸⁷ + 15q⁸⁸ - 25q⁸⁹ - 24q⁹⁰ + 28q⁹¹ + 7q⁹² + 2q⁹³ - 11q⁹⁴ + 9q⁹⁵ - 2q⁹⁶ - 10q⁹⁷ + 7q⁹⁸ + q⁹⁹ + q¹⁰⁰ - 3q¹⁰¹ + q¹⁰² + q¹⁰³ - 2q¹⁰⁴ + q¹⁰⁵
7	- q^-140 + 2q^-139 - q^-138 - q^-137 + 3q^-136 - q^-135 - q^-134 - 3q^-133 - 2q^-132 + 12q^-131 - 3q^-130 - 8q^-129 + 7q^-128 - 3q^-127 - q^-126 - 12q^-125 - 10q^-124 + 45q^-123 + 11q^-122 - 21q^-121 - q^-120 - 27q^-119 - 6q^-118 - 43q^-117 - 39q^-116 + 124q^-115 + 102q^-114 + 33q^-113 + 4q^-112 - 129q^-111 - 116q^-110 - 214q^-109 - 212q^-108 + 219q^-107 + 398q^-106 + 465q^-105 + 379q^-104 - 129q^-103 - 434q^-102 - 963q^-101 - 1195q^-100 - 334q^-99 + 576q^-98 + 1633q^-97 + 2262q^-96 + 1491q^-95 + 275q^-94 - 1996q^-93 - 4083q^-92 - 3891q^-91 - 2340q^-90 + 1273q^-89 + 5489q^-88 + 7372q^-87 + 6927q^-86 + 2127q^-85 - 5346q^-84 - 10976q^-83 - 13790q^-82 - 9783q^-81 + 675q^-80 + 12280q^-79 + 22019q^-78 + 22128q^-77 + 10764q^-76 - 7003q^-75 - 27320q^-74 - 37935q^-73 - 31183q^-72 - 8944q^-71 + 24729q^-70 + 52041q^-69 + 58145q^-68 + 38820q^-67 - 6602q^-66 - 56918q^-65 - 87316q^-64 - 81709q^-63 - 30662q^-62 + 43870q^-61 + 108006q^-60 + 131582q^-59 + 88820q^-58 - 5449q^-57 - 110563q^-56 - 178746q^-55 - 161426q^-54 - 59774q^-53 + 85085q^-52 + 209955q^-51 + 238590q^-50 + 148394q^-49 - 28036q^-48 - 215295q^-47 - 307035q^-46 - 249368q^-45 - 57689q^-44 + 188369q^-43 + 354197q^-42 + 349946q^-41 + 163581q^-40 - 130500q^-39 - 373356q^-38 - 436960q^-37 - 275780q^-36 + 48810q^-35 + 362085q^-34 + 501111q^-33 + 382490q^-32 + 45578q^-31 - 326281q^-30 - 538813q^-29 - 472292q^-28 - 140463q^-27 + 273206q^-26 + 551729q^-25 + 540897q^-24 + 226027q^-23 - 214004q^-22 - 545455q^-21 - 586355q^-20 - 296298q^-19 + 155968q^-18 + 527435q^-17 + 613307q^-16 + 349033q^-15 - 106339q^-14 - 504324q^-13 - 625272q^-12 - 386210q^-11 + 65682q^-10 + 481641q^-9 + 629916q^-8 + 411290q^-7 - 35563q^-6 - 461732q^-5 - 629573q^-4 - 429427q^-3 + 10614q^-2 + 445007q^-1 + 630069 + 445007q + 10614q² - 429427q³ - 629573q⁴ - 461732q⁵ - 35563q⁶ + 411290q⁷ + 629916q⁸ + 481641q⁹ + 65682q¹⁰ - 386210q¹¹ - 625272q¹² - 504324q¹³ - 106339q¹⁴ + 349033q¹⁵ + 613307q¹⁶ + 527435q¹⁷ + 155968q¹⁸ - 296298q¹⁹ - 586355q²⁰ - 545455q²¹ - 214004q²² + 226027q²³ + 540897q²⁴ + 551729q²⁵ + 273206q²⁶ - 140463q²⁷ - 472292q²⁸ - 538813q²⁹ - 326281q³⁰ + 45578q³¹ + 382490q³² + 501111q³³ + 362085q³⁴ + 48810q³⁵ - 275780q³⁶ - 436960q³⁷ - 373356q³⁸ - 130500q³⁹ + 163581q⁴⁰ + 349946q⁴¹ + 354197q⁴² + 188369q⁴³ - 57689q⁴⁴ - 249368q⁴⁵ - 307035q⁴⁶ - 215295q⁴⁷ - 28036q⁴⁸ + 148394q⁴⁹ + 238590q⁵⁰ + 209955q⁵¹ + 85085q⁵² - 59774q⁵³ - 161426q⁵⁴ - 178746q⁵⁵ - 110563q⁵⁶ - 5449q⁵⁷ + 88820q⁵⁸ + 131582q⁵⁹ + 108006q⁶⁰ + 43870q⁶¹ - 30662q⁶² - 81709q⁶³ - 87316q⁶⁴ - 56918q⁶⁵ - 6602q⁶⁶ + 38820q⁶⁷ + 58145q⁶⁸ + 52041q⁶⁹ + 24729q⁷⁰ - 8944q⁷¹ - 31183q⁷² - 37935q⁷³ - 27320q⁷⁴ - 7003q⁷⁵ + 10764q⁷⁶ + 22128q⁷⁷ + 22019q⁷⁸ + 12280q⁷⁹ + 675q⁸⁰ - 9783q⁸¹ - 13790q⁸² - 10976q⁸³ - 5346q⁸⁴ + 2127q⁸⁵ + 6927q⁸⁶ + 7372q⁸⁷ + 5489q⁸⁸ + 1273q⁸⁹ - 2340q⁹⁰ - 3891q⁹¹ - 4083q⁹² - 1996q⁹³ + 275q⁹⁴ + 1491q⁹⁵ + 2262q⁹⁶ + 1633q⁹⁷ + 576q⁹⁸ - 334q⁹⁹ - 1195q¹⁰⁰ - 963q¹⁰¹ - 434q¹⁰² - 129q¹⁰³ + 379q¹⁰⁴ + 465q¹⁰⁵ + 398q¹⁰⁶ + 219q¹⁰⁷ - 212q¹⁰⁸ - 214q¹⁰⁹ - 116q¹¹⁰ - 129q¹¹¹ + 4q¹¹² + 33q¹¹³ + 102q¹¹⁴ + 124q¹¹⁵ - 39q¹¹⁶ - 43q¹¹⁷ - 6q¹¹⁸ - 27q¹¹⁹ - q¹²⁰ - 21q¹²¹ + 11q¹²² + 45q¹²³ - 10q¹²⁴ - 12q¹²⁵ - q¹²⁶ - 3q¹²⁷ + 7q¹²⁸ - 8q¹²⁹ - 3q¹³⁰ + 12q¹³¹ - 2q¹³² - 3q¹³³ - q¹³⁴ - q¹³⁵ + 3q¹³⁶ - q¹³⁷ - q¹³⁸ + 2q¹³⁹ - q¹⁴⁰

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	PD[Knot[10, 79]]
Out[2]=	PD[X[6, 2, 7, 1], X[8, 4, 9, 3], X[12, 6, 13, 5], X[18, 13, 19, 14], > X[16, 9, 17, 10], X[10, 17, 11, 18], X[20, 15, 1, 16], X[14, 19, 15, 20], > X[2, 8, 3, 7], X[4, 12, 5, 11]]
In[3]:=	GaussCode[Knot[10, 79]]
Out[3]=	GaussCode[1, -9, 2, -10, 3, -1, 9, -2, 5, -6, 10, -3, 4, -8, 7, -5, 6, -4, 8, > -7]
In[4]:=	DTCode[Knot[10, 79]]
Out[4]=	DTCode[6, 8, 12, 2, 16, 4, 18, 20, 10, 14]
In[5]:=	br = BR[Knot[10, 79]]
Out[5]=	BR[3, {-1, -1, -1, 2, 2, -1, -1, 2, 2, 2}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{3, 10}
In[7]:=	BraidIndex[Knot[10, 79]]
Out[7]=	3
In[8]:=	Show[DrawMorseLink[Knot[10, 79]]]

Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 79]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{NegativeAmphicheiral, {2, 3}, 4, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 79]][t]
Out[10]=	-4 3 7 12 2 3 4 15 + t - -- + -- - -- - 12 t + 7 t - 3 t + t 3 2 t t t
In[11]:=	Conway[Knot[10, 79]][z]
Out[11]=	2 4 6 8 1 + 5 z + 9 z + 5 z + z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 79]}
In[13]:=	{KnotDet[Knot[10, 79]], KnotSignature[Knot[10, 79]]}
Out[13]=	{61, 0}
In[14]:=	Jones[Knot[10, 79]][q]
Out[14]=	-5 2 5 8 9 2 3 4 5 11 - q + -- - -- + -- - - - 9 q + 8 q - 5 q + 2 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, 79]}
In[16]:=	A2Invariant[Knot[10, 79]][q]
Out[16]=	-14 3 5 2 10 14 1 - q - --- + -- + 5 q - 3 q - q 10 2 q q
In[17]:=	HOMFLYPT[Knot[10, 79]][a, z]
Out[17]=	2 4 6 5 2 2 9 z 2 2 4 5 z 2 4 6 z 11 - -- - 5 a + 23 z - ---- - 9 a z + 19 z - ---- - 5 a z + 7 z - -- - 2 2 2 2 a a a a 2 6 8 > a z + z
In[18]:=	Kauffman[Knot[10, 79]][a, z]
Out[18]=	2 5 2 2 z 2 z 11 z 3 5 2 z 11 + -- + 5 a + --- - --- - ---- - 11 a z - 2 a z + 2 a z - 28 z + -- - 2 5 3 a 4 a a a a 2 3 3 3 13 z 2 2 4 2 3 z 4 z 22 z 3 3 3 > ----- - 13 a z + a z - ---- + ---- + ----- + 22 a z + 4 a z - 2 5 3 a a a a 4 4 5 5 5 5 3 4 4 z 12 z 2 4 4 4 z 6 z 15 z > 3 a z + 32 z - ---- + ----- + 12 a z - 4 a z + -- - ---- - ----- - 4 2 5 3 a a a a a 6 6 5 3 5 5 5 6 2 z 7 z 2 6 4 6 > 15 a z - 6 a z + a z - 18 z + ---- - ---- - 7 a z + 2 a z + 4 2 a a 7 7 8 9 3 z 4 z 7 3 7 8 3 z 2 8 z 9 > ---- + ---- + 4 a z + 3 a z + 6 z + ---- + 3 a z + -- + a z 3 a 2 a a a
In[19]:=	{Vassiliev[2][Knot[10, 79]], Vassiliev[3][Knot[10, 79]]}
Out[19]=	{5, 0}
In[20]:=	Kh[Knot[10, 79]][q, t]
Out[20]=	6 1 1 1 4 1 4 4 5 4 - + 6 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 > 4 q t + 5 q t + 4 q t + 4 q t + q t + 4 q t + q t + q t + 11 5 > q t
In[21]:=	ColouredJones[Knot[10, 79], 2][q]
Out[21]=	-15 2 -13 5 11 2 21 30 5 53 48 24 85 99 + q - --- + q + --- - --- + --- + -- - -- - -- + -- - -- - -- + -- - 14 12 11 10 9 8 7 6 5 4 3 q q q q q q q q q q q 53 44 2 3 4 5 6 7 8 > -- - -- - 44 q - 53 q + 85 q - 24 q - 48 q + 53 q - 5 q - 30 q + 2 q q 9 10 11 12 13 14 15 > 21 q + 2 q - 11 q + 5 q + q - 2 q + q

The Alternating Knot 1079

The Alternating Knot 10₇₉