The Alternating Knot 10₁₀₄

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Acknowledgement

KnotPlot

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

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

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

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

Reversible 1 4 3 / NotAvailable 1

Alexander Polynomial: t^-4 - 4t^-3 + 9t^-2 - 15t^-1 + 19 - 15t + 9t² - 4t³ + t⁴

Conway Polynomial: 1 + z² + 5z⁴ + 4z⁶ + z⁸

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

Determinant and Signature: {77, 0}

Jones Polynomial: - q^-5 + 3q^-4 - 6q^-3 + 10q^-2 - 12q^-1 + 13 - 12q + 10q² - 6q³ + 3q⁴ - q⁵

Other knots (up to mirrors) with the same Jones Polynomial: {10₇₁, ...}

A2 (sl(3)) Invariant: - q^-14 + q^-12 - 2q^-10 + 2q^-8 + q^-6 - q^-4 + 3q^-2 - 3 + 3q² - q⁴ + q⁶ + 2q⁸ - 2q¹⁰ + q¹² - q¹⁴

HOMFLY-PT Polynomial: - a^-2 - 5a^-2z² - 4a^-2z⁴ - a^-2z⁶ + 3 + 11z² + 13z⁴ + 6z⁶ + z⁸ - a² - 5a²z² - 4a²z⁴ - a²z⁶

Kauffman Polynomial: - 2a^-5z³ + a^-5z⁵ + 2a^-4z² - 6a^-4z⁴ + 3a^-4z⁶ - 2a^-3z + 8a^-3z³ - 11a^-3z⁵ + 5a^-3z⁷ + a^-2 - 6a^-2z² + 12a^-2z⁴ - 11a^-2z⁶ + 5a^-2z⁸ - 4a^-1z + 13a^-1z³ - 12a^-1z⁵ + 3a^-1z⁷ + 2a^-1z⁹ + 3 - 15z² + 27z⁴ - 22z⁶ + 9z⁸ - 2az + 4az³ - 6az⁵ + 2az⁷ + 2az⁹ + a² - 4a²z² + 3a²z⁴ - 5a²z⁶ + 4a²z⁸ + a³z - a³z³ - 5a³z⁵ + 4a³z⁷ + 3a⁴z² - 6a⁴z⁴ + 3a⁴z⁶ + a⁵z - 2a⁵z³ + a⁵z⁵

V₂ and V₃, the type 2 and 3 Vassiliev invariants: {1, 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 10₁₀₄. 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 4 1

j = 5 6 2

j = 3 6 4

j = 1 7 6

j = -1 6 7

j = -3 4 6

j = -5 2 6

j = -7 1 4

j = -9 2

j = -11 1

n Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q^-15 - 3q^-14 + q^-13 + 8q^-12 - 16q^-11 + 3q^-10 + 33q^-9 - 46q^-8 - 7q^-7 + 83q^-6 - 76q^-5 - 37q^-4 + 133q^-3 - 84q^-2 - 70q^-1 + 154 - 68q - 86q² + 134q³ - 36q⁴ - 78q⁵ + 84q⁶ - 5q⁷ - 49q⁸ + 33q⁹ + 6q¹⁰ - 18q¹¹ + 7q¹² + 2q¹³ - 3q¹⁴ + q¹⁵

3 - q^-30 + 3q^-29 - q^-28 - 3q^-27 - 2q^-26 + 10q^-25 - 20q^-23 + q^-22 + 42q^-21 + 2q^-20 - 82q^-19 - 20q^-18 + 143q^-17 + 65q^-16 - 213q^-15 - 147q^-14 + 270q^-13 + 277q^-12 - 317q^-11 - 414q^-10 + 310q^-9 + 571q^-8 - 281q^-7 - 700q^-6 + 214q^-5 + 809q^-4 - 141q^-3 - 867q^-2 + 51q^-1 + 894 + 32q - 872q² - 122q³ + 819q⁴ + 196q⁵ - 717q⁶ - 268q⁷ + 594q⁸ + 307q⁹ - 438q¹⁰ - 328q¹¹ + 297q¹² + 293q¹³ - 156q¹⁴ - 244q¹⁵ + 63q¹⁶ + 173q¹⁷ - 8q¹⁸ - 104q¹⁹ - 15q²⁰ + 55q²¹ + 14q²² - 25q²³ - 8q²⁴ + 11q²⁵ + 2q²⁶ - 3q²⁷ - 2q²⁸ + 3q²⁹ - q³⁰

4 q^-50 - 3q^-49 + q^-48 + 3q^-47 - 3q^-46 + 8q^-45 - 13q^-44 + 4q^-43 + 10q^-42 - 22q^-41 + 24q^-40 - 33q^-39 + 35q^-38 + 55q^-37 - 83q^-36 - 11q^-35 - 135q^-34 + 135q^-33 + 295q^-32 - 68q^-31 - 159q^-30 - 607q^-29 + 84q^-28 + 849q^-27 + 452q^-26 - 66q^-25 - 1605q^-24 - 712q^-23 + 1223q^-22 + 1648q^-21 + 999q^-20 - 2510q^-19 - 2391q^-18 + 599q^-17 + 2809q^-16 + 3073q^-15 - 2457q^-14 - 4122q^-13 - 1034q^-12 + 3109q^-11 + 5234q^-10 - 1459q^-9 - 5061q^-8 - 2818q^-7 + 2562q^-6 + 6625q^-5 - 193q^-4 - 5106q^-3 - 4092q^-2 + 1639q^-1 + 7099 + 931q - 4519q² - 4776q³ + 540q⁴ + 6757q⁵ + 1925q⁶ - 3339q⁷ - 4892q⁸ - 767q⁹ + 5538q¹⁰ + 2672q¹¹ - 1581q¹² - 4204q¹³ - 1976q¹⁴ + 3480q¹⁵ + 2700q¹⁶ + 226q¹⁷ - 2656q¹⁸ - 2382q¹⁹ + 1288q²⁰ + 1785q²¹ + 1167q²² - 920q²³ - 1726q²⁴ - 32q²⁵ + 583q²⁶ + 989q²⁷ + 62q²⁸ - 728q²⁹ - 261q³⁰ - 60q³¹ + 410q³² + 197q³³ - 163q³⁴ - 83q³⁵ - 117q³⁶ + 91q³⁷ + 72q³⁸ - 28q³⁹ + 3q⁴⁰ - 38q⁴¹ + 15q⁴² + 14q⁴³ - 10q⁴⁴ + 5q⁴⁵ - 6q⁴⁶ + 3q⁴⁷ + 2q⁴⁸ - 3q⁴⁹ + q⁵⁰

5 - q^-75 + 3q^-74 - q^-73 - 3q^-72 + 3q^-71 - 3q^-70 - 5q^-69 + 9q^-68 + 6q^-67 - 6q^-66 + 11q^-65 - 5q^-64 - 32q^-63 - 10q^-62 + 10q^-61 + 23q^-60 + 70q^-59 + 55q^-58 - 61q^-57 - 163q^-56 - 164q^-55 - 21q^-54 + 281q^-53 + 476q^-52 + 271q^-51 - 349q^-50 - 945q^-49 - 890q^-48 + 102q^-47 + 1495q^-46 + 2034q^-45 + 792q^-44 - 1803q^-43 - 3652q^-42 - 2658q^-41 + 1267q^-40 + 5374q^-39 + 5695q^-38 + 697q^-37 - 6494q^-36 - 9614q^-35 - 4509q^-34 + 6140q^-33 + 13631q^-32 + 10157q^-31 - 3525q^-30 - 16860q^-29 - 17023q^-28 - 1386q^-27 + 18129q^-26 + 24030q^-25 + 8540q^-24 - 17145q^-23 - 30208q^-22 - 16666q^-21 + 13766q^-20 + 34502q^-19 + 25065q^-18 - 8697q^-17 - 36861q^-16 - 32386q^-15 + 2706q^-14 + 37200q^-13 + 38333q^-12 + 3267q^-11 - 36155q^-10 - 42512q^-9 - 8711q^-8 + 34174q^-7 + 45291q^-6 + 13266q^-5 - 31774q^-4 - 46784q^-3 - 17132q^-2 + 29068q^-1 + 47526 + 20413q - 26095q² - 47429q³ - 23499q⁴ + 22493q⁵ + 46666q⁶ + 26441q⁷ - 18138q⁸ - 44748q⁹ - 29227q¹⁰ + 12732q¹¹ + 41546q¹² + 31418q¹³ - 6498q¹⁴ - 36589q¹⁵ - 32564q¹⁶ - 191q¹⁷ + 30076q¹⁸ + 31939q¹⁹ + 6459q²⁰ - 22091q²¹ - 29398q²² - 11486q²³ + 13819q²⁴ + 24701q²⁵ + 14355q²⁶ - 5907q²⁷ - 18664q²⁸ - 14905q²⁹ - 185q³⁰ + 12099q³¹ + 13104q³² + 4158q³³ - 6179q³⁴ - 9957q³⁵ - 5691q³⁶ + 1767q³⁷ + 6313q³⁸ + 5369q³⁹ + 855q⁴⁰ - 3165q⁴¹ - 4015q⁴² - 1854q⁴³ + 1034q⁴⁴ + 2416q⁴⁵ + 1768q⁴⁶ + 95q⁴⁷ - 1154q⁴⁸ - 1238q⁴⁹ - 446q⁵⁰ + 397q⁵¹ + 687q⁵² + 398q⁵³ - 67q⁵⁴ - 289q⁵⁵ - 251q⁵⁶ - 47q⁵⁷ + 123q⁵⁸ + 119q⁵⁹ + 22q⁶⁰ - 25q⁶¹ - 37q⁶² - 33q⁶³ + 14q⁶⁴ + 24q⁶⁵ - 5q⁶⁶ - 3q⁶⁷ + 4q⁶⁸ - 6q⁶⁹ - q⁷⁰ + 6q⁷¹ - 3q⁷² - 2q⁷³ + 3q⁷⁴ - q⁷⁵

6 q^-105 - 3q^-104 + q^-103 + 3q^-102 - 3q^-101 + 3q^-100 + 9q^-98 - 19q^-97 - 10q^-96 + 17q^-95 - 13q^-94 + 15q^-93 + 19q^-92 + 55q^-91 - 51q^-90 - 72q^-89 + q^-88 - 77q^-87 + 18q^-86 + 110q^-85 + 292q^-84 + 16q^-83 - 169q^-82 - 168q^-81 - 501q^-80 - 299q^-79 + 166q^-78 + 1075q^-77 + 861q^-76 + 375q^-75 - 222q^-74 - 1916q^-73 - 2378q^-72 - 1408q^-71 + 1786q^-70 + 3579q^-69 + 4343q^-68 + 3085q^-67 - 2672q^-66 - 7826q^-65 - 9619q^-64 - 3668q^-63 + 4379q^-62 + 13580q^-61 + 17739q^-60 + 7980q^-59 - 9048q^-58 - 25751q^-57 - 26746q^-56 - 13934q^-55 + 14474q^-54 + 42740q^-53 + 44982q^-52 + 18929q^-51 - 27811q^-50 - 61998q^-49 - 68586q^-48 - 26042q^-47 + 45884q^-46 + 96047q^-45 + 93195q^-44 + 24314q^-43 - 66608q^-42 - 138131q^-41 - 121818q^-40 - 17260q^-39 + 108811q^-38 + 181754q^-37 + 138022q^-36 + 4030q^-35 - 162967q^-34 - 230111q^-33 - 144794q^-32 + 42212q^-31 + 221127q^-30 + 261231q^-29 + 137189q^-28 - 108523q^-27 - 286996q^-26 - 277579q^-25 - 81286q^-24 + 184750q^-23 + 332789q^-22 + 270203q^-21 - 4313q^-20 - 273672q^-19 - 359716q^-18 - 200408q^-17 + 105598q^-16 + 340790q^-15 + 354812q^-14 + 93967q^-13 - 222905q^-12 - 385092q^-11 - 276884q^-10 + 30850q^-9 + 315542q^-8 + 390377q^-7 + 158867q^-6 - 172088q^-5 - 380833q^-4 - 315639q^-3 - 22359q^-2 + 285289q^-1 + 401436 + 200275q - 129417q² - 367677q³ - 339461q⁴ - 67949q⁵ + 251066q⁶ + 402942q⁷ + 239507q⁸ - 77759q⁹ - 340910q¹⁰ - 358439q¹¹ - 125667q¹² + 193064q¹³ + 384935q¹⁴ + 281535q¹⁵ + 743q¹⁶ - 278055q¹⁷ - 356551q¹⁸ - 194325q¹⁹ + 95656q²⁰ + 321579q²¹ + 302526q²² + 97371q²³ - 165941q²⁴ - 303184q²⁵ - 240500q²⁶ - 23862q²⁷ + 202106q²⁸ + 265940q²⁹ + 168175q³⁰ - 30849q³¹ - 189632q³² - 222060q³³ - 111824q³⁴ + 61451q³⁵ + 165780q³⁶ + 167522q³⁷ + 66062q³⁸ - 58278q³⁹ - 137731q⁴⁰ - 122927q⁴¹ - 35000q⁴² + 50999q⁴³ + 101148q⁴⁴ + 83921q⁴⁵ + 25482q⁴⁶ - 42123q⁴⁷ - 71828q⁴⁸ - 54412q⁴⁹ - 16452q⁵⁰ + 27242q⁵¹ + 46398q⁵² + 39409q⁵³ + 9192q⁵⁴ - 17245q⁵⁵ - 27807q⁵⁶ - 25137q⁵⁷ - 7749q⁵⁸ + 8912q⁵⁹ + 18860q⁶⁰ + 14235q⁶¹ + 5092q⁶² - 3800q⁶³ - 10515q⁶⁴ - 9094q⁶⁵ - 3795q⁶⁶ + 3061q⁶⁷ + 4964q⁶⁸ + 4816q⁶⁹ + 2585q⁷⁰ - 1278q⁷¹ - 2915q⁷² - 2728q⁷³ - 636q⁷⁴ + 287q⁷⁵ + 1246q⁷⁶ + 1444q⁷⁷ + 397q⁷⁸ - 339q⁷⁹ - 702q⁸⁰ - 288q⁸¹ - 249q⁸² + 75q⁸³ + 350q⁸⁴ + 164q⁸⁵ + 6q⁸⁶ - 117q⁸⁷ - 4q⁸⁸ - 76q⁸⁹ - 32q⁹⁰ + 64q⁹¹ + 25q⁹² + 3q⁹³ - 27q⁹⁴ + 18q⁹⁵ - 7q⁹⁶ - 15q⁹⁷ + 12q⁹⁸ + 2q⁹⁹ + q¹⁰⁰ - 6q¹⁰¹ + 3q¹⁰² + 2q¹⁰³ - 3q¹⁰⁴ + q¹⁰⁵

7 - q^-140 + 3q^-139 - q^-138 - 3q^-137 + 3q^-136 - 3q^-135 - 4q^-133 + q^-132 + 23q^-131 - q^-130 - 15q^-129 + 3q^-128 - 19q^-127 - 9q^-126 - 26q^-125 - 10q^-124 + 104q^-123 + 52q^-122 - 6q^-121 - 4q^-120 - 96q^-119 - 79q^-118 - 152q^-117 - 116q^-116 + 276q^-115 + 325q^-114 + 262q^-113 + 158q^-112 - 287q^-111 - 460q^-110 - 800q^-109 - 816q^-108 + 293q^-107 + 1116q^-106 + 1776q^-105 + 1796q^-104 + 305q^-103 - 1268q^-102 - 3439q^-101 - 4731q^-100 - 2678q^-99 + 883q^-98 + 5832q^-97 + 9575q^-96 + 8249q^-95 + 3149q^-94 - 6682q^-93 - 16983q^-92 - 19937q^-91 - 14353q^-90 + 2286q^-89 + 23892q^-88 + 37772q^-87 + 37795q^-86 + 15943q^-85 - 22473q^-84 - 58416q^-83 - 76629q^-82 - 57380q^-81 - 611q^-80 + 69607q^-79 + 125778q^-78 + 128743q^-77 + 63240q^-76 - 49228q^-75 - 168627q^-74 - 226960q^-73 - 178510q^-72 - 29634q^-71 + 173728q^-70 + 329721q^-69 + 345947q^-68 + 191169q^-67 - 100941q^-66 - 396890q^-65 - 542380q^-64 - 440527q^-63 - 84864q^-62 + 375651q^-61 + 718631q^-60 + 754927q^-59 + 399165q^-58 - 216793q^-57 - 811349q^-56 - 1082708q^-55 - 823276q^-54 - 104316q^-53 + 759909q^-52 + 1350661q^-51 + 1304139q^-50 + 578157q^-49 - 526603q^-48 - 1489739q^-47 - 1767730q^-46 - 1154315q^-45 + 114596q^-44 + 1451632q^-43 + 2134505q^-42 + 1758761q^-41 + 437263q^-40 - 1226882q^-39 - 2350485q^-38 - 2311609q^-37 - 1056474q^-36 + 845475q^-35 + 2390772q^-34 + 2748218q^-33 + 1668352q^-32 - 364936q^-31 - 2272868q^-30 - 3036274q^-29 - 2204755q^-28 - 144474q^-27 + 2037596q^-26 + 3174791q^-25 + 2626569q^-24 + 622231q^-23 - 1741650q^-22 - 3190950q^-21 - 2921481q^-20 - 1026220q^-19 + 1436714q^-18 + 3124901q^-17 + 3103245q^-16 + 1339105q^-15 - 1161189q^-14 - 3018262q^-13 - 3199562q^-12 - 1565745q^-11 + 933672q^-10 + 2904738q^-9 + 3244072q^-8 + 1725179q^-7 - 755930q^-6 - 2803510q^-5 - 3265298q^-4 - 1844767q^-3 + 613441q^-2 + 2720250q^-1 + 3285566 + 1951561q - 485486q² - 2646880q³ - 3312851q⁴ - 2068490q⁵ + 344210q⁶ + 2565024q⁷ + 3346285q⁸ + 2210037q⁹ - 165925q¹⁰ - 2449564q¹¹ - 3369639q¹² - 2378474q¹³ - 70093q¹⁴ + 2271671q¹⁵ + 3358549q¹⁶ + 2562865q¹⁷ + 370991q¹⁸ - 2006306q¹⁹ - 3279440q²⁰ - 2736128q²¹ - 729246q²² + 1637881q²³ + 3098896q²⁴ + 2859077q²⁵ + 1116697q²⁶ - 1169337q²⁷ - 2790521q²⁸ - 2886590q²⁹ - 1487424q³⁰ + 627580q³¹ + 2348100q³² + 2777646q³³ + 1782232q³⁴ - 62409q³⁵ - 1790364q³⁶ - 2511320q³⁷ - 1946542q³⁸ - 457096q³⁹ + 1168781q⁴⁰ + 2094478q⁴¹ + 1939263q⁴² + 861955q⁴³ - 552151q⁴⁴ - 1570252q⁴⁵ - 1756628q⁴⁶ - 1098365q⁴⁷ + 21565q⁴⁸ + 1007808q⁴⁹ + 1427166q⁵⁰ + 1146031q⁵¹ + 362231q⁵² - 486598q⁵³ - 1016784q⁵⁴ - 1025922q⁵⁵ - 566306q⁵⁶ + 76731q⁵⁷ + 601389q⁵⁸ + 791184q⁵⁹ + 601362q⁶⁰ + 184864q⁶¹ - 251172q⁶² - 514971q⁶³ - 511236q⁶⁴ - 296859q⁶⁵ + 9059q⁶⁶ + 262171q⁶⁷ + 357405q⁶⁸ + 292893q⁶⁹ + 117808q⁷⁰ - 76206q⁷¹ - 199339q⁷² - 222562q⁷³ - 150480q⁷⁴ - 29758q⁷⁵ + 75698q⁷⁶ + 133437q⁷⁷ + 126227q⁷⁸ + 68789q⁷⁹ - 1221q⁸⁰ - 59083q⁸¹ - 81143q⁸² - 65205q⁸³ - 29499q⁸⁴ + 12071q⁸⁵ + 39575q⁸⁶ + 44420q⁸⁷ + 32659q⁸⁸ + 9142q⁸⁹ - 12545q⁹⁰ - 23275q⁹¹ - 23673q⁹² - 13353q⁹³ - 463q⁹⁴ + 8636q⁹⁵ + 13220q⁹⁶ + 10556q⁹⁷ + 4304q⁹⁸ - 1635q⁹⁹ - 6031q¹⁰⁰ - 6008q¹⁰¹ - 3601q¹⁰² - 968q¹⁰³ + 1930q¹⁰⁴ + 2882q¹⁰⁵ + 2364q¹⁰⁶ + 1155q¹⁰⁷ - 610q¹⁰⁸ - 1121q¹⁰⁹ - 969q¹¹⁰ - 740q¹¹¹ - 49q¹¹² + 318q¹¹³ + 491q¹¹⁴ + 451q¹¹⁵ - 23q¹¹⁶ - 149q¹¹⁷ - 103q¹¹⁸ - 137q¹¹⁹ - 31q¹²⁰ - 21q¹²¹ + 50q¹²² + 113q¹²³ - 8q¹²⁴ - 34q¹²⁵ - 8q¹²⁶ - 9q¹²⁷ + 11q¹²⁸ - 15q¹²⁹ - 6q¹³⁰ + 25q¹³¹ - q¹³² - 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, 104]]

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

In[3]:=
GaussCode[Knot[10, 104]]

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

In[4]:=
DTCode[Knot[10, 104]]

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

In[5]:=
br = BR[Knot[10, 104]]

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

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

Out[6]=
{3, 10}

In[7]:=
BraidIndex[Knot[10, 104]]

Out[7]=
3

In[8]:=
Show[DrawMorseLink[Knot[10, 104]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[10, 104]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

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

In[10]:=
alex = Alexander[Knot[10, 104]][t]

Out[10]=
-4 4 9 15 2 3 4 19 + t - -- + -- - -- - 15 t + 9 t - 4 t + t 3 2 t t t

In[11]:=
Conway[Knot[10, 104]][z]

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

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

Out[12]=
{Knot[10, 104]}

In[13]:=
{KnotDet[Knot[10, 104]], KnotSignature[Knot[10, 104]]}

Out[13]=
{77, 0}

In[14]:=
Jones[Knot[10, 104]][q]

Out[14]=
-5 3 6 10 12 2 3 4 5 13 - q + -- - -- + -- - -- - 12 q + 10 q - 6 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, 71], Knot[10, 104]}

In[16]:=
A2Invariant[Knot[10, 104]][q]

Out[16]=
-14 -12 2 2 -6 -4 3 2 4 6 8 10 -3 - q + q - --- + -- + q - q + -- + 3 q - q + q + 2 q - 2 q + 10 8 2 q q q 12 14 > q - q

In[17]:=
HOMFLYPT[Knot[10, 104]][a, z]

Out[17]=
2 4 6 -2 2 2 5 z 2 2 4 4 z 2 4 6 z 3 - a - a + 11 z - ---- - 5 a z + 13 z - ---- - 4 a z + 6 z - -- - 2 2 2 a a a 2 6 8 > a z + z

In[18]:=
Kauffman[Knot[10, 104]][a, z]

Out[18]=
2 2 -2 2 2 z 4 z 3 5 2 2 z 6 z 3 + a + a - --- - --- - 2 a z + a z + a z - 15 z + ---- - ---- - 3 a 4 2 a a a 3 3 3 2 2 4 2 2 z 8 z 13 z 3 3 3 5 3 > 4 a z + 3 a z - ---- + ---- + ----- + 4 a z - a z - 2 a z + 5 3 a a a 4 4 5 5 5 4 6 z 12 z 2 4 4 4 z 11 z 12 z 5 > 27 z - ---- + ----- + 3 a z - 6 a z + -- - ----- - ----- - 6 a z - 4 2 5 3 a a a a a 6 6 7 7 3 5 5 5 6 3 z 11 z 2 6 4 6 5 z 3 z > 5 a z + a z - 22 z + ---- - ----- - 5 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 > 2 a z + 4 a z + 9 z + ---- + 4 a z + ---- + 2 a z 2 a a

In[19]:=
{Vassiliev[2][Knot[10, 104]], Vassiliev[3][Knot[10, 104]]}

Out[19]=
{1, 0}

In[20]:=
Kh[Knot[10, 104]][q, t]

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

In[21]:=
ColouredJones[Knot[10, 104], 2][q]

Out[21]=
-15 3 -13 8 16 3 33 46 7 83 76 37 133 154 + q - --- + q + --- - --- + --- + -- - -- - -- + -- - -- - -- + --- - 14 12 11 10 9 8 7 6 5 4 3 q q q q q q q q q q q 84 70 2 3 4 5 6 7 8 > -- - -- - 68 q - 86 q + 134 q - 36 q - 78 q + 84 q - 5 q - 49 q + 2 q q 9 10 11 12 13 14 15 > 33 q + 6 q - 18 q + 7 q + 2 q - 3 q + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 10₁₀₄

10₁₀₃
10₁₀₅

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-15 - 3q^-14 + q^-13 + 8q^-12 - 16q^-11 + 3q^-10 + 33q^-9 - 46q^-8 - 7q^-7 + 83q^-6 - 76q^-5 - 37q^-4 + 133q^-3 - 84q^-2 - 70q^-1 + 154 - 68q - 86q² + 134q³ - 36q⁴ - 78q⁵ + 84q⁶ - 5q⁷ - 49q⁸ + 33q⁹ + 6q¹⁰ - 18q¹¹ + 7q¹² + 2q¹³ - 3q¹⁴ + q¹⁵
3	- q^-30 + 3q^-29 - q^-28 - 3q^-27 - 2q^-26 + 10q^-25 - 20q^-23 + q^-22 + 42q^-21 + 2q^-20 - 82q^-19 - 20q^-18 + 143q^-17 + 65q^-16 - 213q^-15 - 147q^-14 + 270q^-13 + 277q^-12 - 317q^-11 - 414q^-10 + 310q^-9 + 571q^-8 - 281q^-7 - 700q^-6 + 214q^-5 + 809q^-4 - 141q^-3 - 867q^-2 + 51q^-1 + 894 + 32q - 872q² - 122q³ + 819q⁴ + 196q⁵ - 717q⁶ - 268q⁷ + 594q⁸ + 307q⁹ - 438q¹⁰ - 328q¹¹ + 297q¹² + 293q¹³ - 156q¹⁴ - 244q¹⁵ + 63q¹⁶ + 173q¹⁷ - 8q¹⁸ - 104q¹⁹ - 15q²⁰ + 55q²¹ + 14q²² - 25q²³ - 8q²⁴ + 11q²⁵ + 2q²⁶ - 3q²⁷ - 2q²⁸ + 3q²⁹ - q³⁰
4	q^-50 - 3q^-49 + q^-48 + 3q^-47 - 3q^-46 + 8q^-45 - 13q^-44 + 4q^-43 + 10q^-42 - 22q^-41 + 24q^-40 - 33q^-39 + 35q^-38 + 55q^-37 - 83q^-36 - 11q^-35 - 135q^-34 + 135q^-33 + 295q^-32 - 68q^-31 - 159q^-30 - 607q^-29 + 84q^-28 + 849q^-27 + 452q^-26 - 66q^-25 - 1605q^-24 - 712q^-23 + 1223q^-22 + 1648q^-21 + 999q^-20 - 2510q^-19 - 2391q^-18 + 599q^-17 + 2809q^-16 + 3073q^-15 - 2457q^-14 - 4122q^-13 - 1034q^-12 + 3109q^-11 + 5234q^-10 - 1459q^-9 - 5061q^-8 - 2818q^-7 + 2562q^-6 + 6625q^-5 - 193q^-4 - 5106q^-3 - 4092q^-2 + 1639q^-1 + 7099 + 931q - 4519q² - 4776q³ + 540q⁴ + 6757q⁵ + 1925q⁶ - 3339q⁷ - 4892q⁸ - 767q⁹ + 5538q¹⁰ + 2672q¹¹ - 1581q¹² - 4204q¹³ - 1976q¹⁴ + 3480q¹⁵ + 2700q¹⁶ + 226q¹⁷ - 2656q¹⁸ - 2382q¹⁹ + 1288q²⁰ + 1785q²¹ + 1167q²² - 920q²³ - 1726q²⁴ - 32q²⁵ + 583q²⁶ + 989q²⁷ + 62q²⁸ - 728q²⁹ - 261q³⁰ - 60q³¹ + 410q³² + 197q³³ - 163q³⁴ - 83q³⁵ - 117q³⁶ + 91q³⁷ + 72q³⁸ - 28q³⁹ + 3q⁴⁰ - 38q⁴¹ + 15q⁴² + 14q⁴³ - 10q⁴⁴ + 5q⁴⁵ - 6q⁴⁶ + 3q⁴⁷ + 2q⁴⁸ - 3q⁴⁹ + q⁵⁰
5	- q^-75 + 3q^-74 - q^-73 - 3q^-72 + 3q^-71 - 3q^-70 - 5q^-69 + 9q^-68 + 6q^-67 - 6q^-66 + 11q^-65 - 5q^-64 - 32q^-63 - 10q^-62 + 10q^-61 + 23q^-60 + 70q^-59 + 55q^-58 - 61q^-57 - 163q^-56 - 164q^-55 - 21q^-54 + 281q^-53 + 476q^-52 + 271q^-51 - 349q^-50 - 945q^-49 - 890q^-48 + 102q^-47 + 1495q^-46 + 2034q^-45 + 792q^-44 - 1803q^-43 - 3652q^-42 - 2658q^-41 + 1267q^-40 + 5374q^-39 + 5695q^-38 + 697q^-37 - 6494q^-36 - 9614q^-35 - 4509q^-34 + 6140q^-33 + 13631q^-32 + 10157q^-31 - 3525q^-30 - 16860q^-29 - 17023q^-28 - 1386q^-27 + 18129q^-26 + 24030q^-25 + 8540q^-24 - 17145q^-23 - 30208q^-22 - 16666q^-21 + 13766q^-20 + 34502q^-19 + 25065q^-18 - 8697q^-17 - 36861q^-16 - 32386q^-15 + 2706q^-14 + 37200q^-13 + 38333q^-12 + 3267q^-11 - 36155q^-10 - 42512q^-9 - 8711q^-8 + 34174q^-7 + 45291q^-6 + 13266q^-5 - 31774q^-4 - 46784q^-3 - 17132q^-2 + 29068q^-1 + 47526 + 20413q - 26095q² - 47429q³ - 23499q⁴ + 22493q⁵ + 46666q⁶ + 26441q⁷ - 18138q⁸ - 44748q⁹ - 29227q¹⁰ + 12732q¹¹ + 41546q¹² + 31418q¹³ - 6498q¹⁴ - 36589q¹⁵ - 32564q¹⁶ - 191q¹⁷ + 30076q¹⁸ + 31939q¹⁹ + 6459q²⁰ - 22091q²¹ - 29398q²² - 11486q²³ + 13819q²⁴ + 24701q²⁵ + 14355q²⁶ - 5907q²⁷ - 18664q²⁸ - 14905q²⁹ - 185q³⁰ + 12099q³¹ + 13104q³² + 4158q³³ - 6179q³⁴ - 9957q³⁵ - 5691q³⁶ + 1767q³⁷ + 6313q³⁸ + 5369q³⁹ + 855q⁴⁰ - 3165q⁴¹ - 4015q⁴² - 1854q⁴³ + 1034q⁴⁴ + 2416q⁴⁵ + 1768q⁴⁶ + 95q⁴⁷ - 1154q⁴⁸ - 1238q⁴⁹ - 446q⁵⁰ + 397q⁵¹ + 687q⁵² + 398q⁵³ - 67q⁵⁴ - 289q⁵⁵ - 251q⁵⁶ - 47q⁵⁷ + 123q⁵⁸ + 119q⁵⁹ + 22q⁶⁰ - 25q⁶¹ - 37q⁶² - 33q⁶³ + 14q⁶⁴ + 24q⁶⁵ - 5q⁶⁶ - 3q⁶⁷ + 4q⁶⁸ - 6q⁶⁹ - q⁷⁰ + 6q⁷¹ - 3q⁷² - 2q⁷³ + 3q⁷⁴ - q⁷⁵
6	q^-105 - 3q^-104 + q^-103 + 3q^-102 - 3q^-101 + 3q^-100 + 9q^-98 - 19q^-97 - 10q^-96 + 17q^-95 - 13q^-94 + 15q^-93 + 19q^-92 + 55q^-91 - 51q^-90 - 72q^-89 + q^-88 - 77q^-87 + 18q^-86 + 110q^-85 + 292q^-84 + 16q^-83 - 169q^-82 - 168q^-81 - 501q^-80 - 299q^-79 + 166q^-78 + 1075q^-77 + 861q^-76 + 375q^-75 - 222q^-74 - 1916q^-73 - 2378q^-72 - 1408q^-71 + 1786q^-70 + 3579q^-69 + 4343q^-68 + 3085q^-67 - 2672q^-66 - 7826q^-65 - 9619q^-64 - 3668q^-63 + 4379q^-62 + 13580q^-61 + 17739q^-60 + 7980q^-59 - 9048q^-58 - 25751q^-57 - 26746q^-56 - 13934q^-55 + 14474q^-54 + 42740q^-53 + 44982q^-52 + 18929q^-51 - 27811q^-50 - 61998q^-49 - 68586q^-48 - 26042q^-47 + 45884q^-46 + 96047q^-45 + 93195q^-44 + 24314q^-43 - 66608q^-42 - 138131q^-41 - 121818q^-40 - 17260q^-39 + 108811q^-38 + 181754q^-37 + 138022q^-36 + 4030q^-35 - 162967q^-34 - 230111q^-33 - 144794q^-32 + 42212q^-31 + 221127q^-30 + 261231q^-29 + 137189q^-28 - 108523q^-27 - 286996q^-26 - 277579q^-25 - 81286q^-24 + 184750q^-23 + 332789q^-22 + 270203q^-21 - 4313q^-20 - 273672q^-19 - 359716q^-18 - 200408q^-17 + 105598q^-16 + 340790q^-15 + 354812q^-14 + 93967q^-13 - 222905q^-12 - 385092q^-11 - 276884q^-10 + 30850q^-9 + 315542q^-8 + 390377q^-7 + 158867q^-6 - 172088q^-5 - 380833q^-4 - 315639q^-3 - 22359q^-2 + 285289q^-1 + 401436 + 200275q - 129417q² - 367677q³ - 339461q⁴ - 67949q⁵ + 251066q⁶ + 402942q⁷ + 239507q⁸ - 77759q⁹ - 340910q¹⁰ - 358439q¹¹ - 125667q¹² + 193064q¹³ + 384935q¹⁴ + 281535q¹⁵ + 743q¹⁶ - 278055q¹⁷ - 356551q¹⁸ - 194325q¹⁹ + 95656q²⁰ + 321579q²¹ + 302526q²² + 97371q²³ - 165941q²⁴ - 303184q²⁵ - 240500q²⁶ - 23862q²⁷ + 202106q²⁸ + 265940q²⁹ + 168175q³⁰ - 30849q³¹ - 189632q³² - 222060q³³ - 111824q³⁴ + 61451q³⁵ + 165780q³⁶ + 167522q³⁷ + 66062q³⁸ - 58278q³⁹ - 137731q⁴⁰ - 122927q⁴¹ - 35000q⁴² + 50999q⁴³ + 101148q⁴⁴ + 83921q⁴⁵ + 25482q⁴⁶ - 42123q⁴⁷ - 71828q⁴⁸ - 54412q⁴⁹ - 16452q⁵⁰ + 27242q⁵¹ + 46398q⁵² + 39409q⁵³ + 9192q⁵⁴ - 17245q⁵⁵ - 27807q⁵⁶ - 25137q⁵⁷ - 7749q⁵⁸ + 8912q⁵⁹ + 18860q⁶⁰ + 14235q⁶¹ + 5092q⁶² - 3800q⁶³ - 10515q⁶⁴ - 9094q⁶⁵ - 3795q⁶⁶ + 3061q⁶⁷ + 4964q⁶⁸ + 4816q⁶⁹ + 2585q⁷⁰ - 1278q⁷¹ - 2915q⁷² - 2728q⁷³ - 636q⁷⁴ + 287q⁷⁵ + 1246q⁷⁶ + 1444q⁷⁷ + 397q⁷⁸ - 339q⁷⁹ - 702q⁸⁰ - 288q⁸¹ - 249q⁸² + 75q⁸³ + 350q⁸⁴ + 164q⁸⁵ + 6q⁸⁶ - 117q⁸⁷ - 4q⁸⁸ - 76q⁸⁹ - 32q⁹⁰ + 64q⁹¹ + 25q⁹² + 3q⁹³ - 27q⁹⁴ + 18q⁹⁵ - 7q⁹⁶ - 15q⁹⁷ + 12q⁹⁸ + 2q⁹⁹ + q¹⁰⁰ - 6q¹⁰¹ + 3q¹⁰² + 2q¹⁰³ - 3q¹⁰⁴ + q¹⁰⁵
7	- q^-140 + 3q^-139 - q^-138 - 3q^-137 + 3q^-136 - 3q^-135 - 4q^-133 + q^-132 + 23q^-131 - q^-130 - 15q^-129 + 3q^-128 - 19q^-127 - 9q^-126 - 26q^-125 - 10q^-124 + 104q^-123 + 52q^-122 - 6q^-121 - 4q^-120 - 96q^-119 - 79q^-118 - 152q^-117 - 116q^-116 + 276q^-115 + 325q^-114 + 262q^-113 + 158q^-112 - 287q^-111 - 460q^-110 - 800q^-109 - 816q^-108 + 293q^-107 + 1116q^-106 + 1776q^-105 + 1796q^-104 + 305q^-103 - 1268q^-102 - 3439q^-101 - 4731q^-100 - 2678q^-99 + 883q^-98 + 5832q^-97 + 9575q^-96 + 8249q^-95 + 3149q^-94 - 6682q^-93 - 16983q^-92 - 19937q^-91 - 14353q^-90 + 2286q^-89 + 23892q^-88 + 37772q^-87 + 37795q^-86 + 15943q^-85 - 22473q^-84 - 58416q^-83 - 76629q^-82 - 57380q^-81 - 611q^-80 + 69607q^-79 + 125778q^-78 + 128743q^-77 + 63240q^-76 - 49228q^-75 - 168627q^-74 - 226960q^-73 - 178510q^-72 - 29634q^-71 + 173728q^-70 + 329721q^-69 + 345947q^-68 + 191169q^-67 - 100941q^-66 - 396890q^-65 - 542380q^-64 - 440527q^-63 - 84864q^-62 + 375651q^-61 + 718631q^-60 + 754927q^-59 + 399165q^-58 - 216793q^-57 - 811349q^-56 - 1082708q^-55 - 823276q^-54 - 104316q^-53 + 759909q^-52 + 1350661q^-51 + 1304139q^-50 + 578157q^-49 - 526603q^-48 - 1489739q^-47 - 1767730q^-46 - 1154315q^-45 + 114596q^-44 + 1451632q^-43 + 2134505q^-42 + 1758761q^-41 + 437263q^-40 - 1226882q^-39 - 2350485q^-38 - 2311609q^-37 - 1056474q^-36 + 845475q^-35 + 2390772q^-34 + 2748218q^-33 + 1668352q^-32 - 364936q^-31 - 2272868q^-30 - 3036274q^-29 - 2204755q^-28 - 144474q^-27 + 2037596q^-26 + 3174791q^-25 + 2626569q^-24 + 622231q^-23 - 1741650q^-22 - 3190950q^-21 - 2921481q^-20 - 1026220q^-19 + 1436714q^-18 + 3124901q^-17 + 3103245q^-16 + 1339105q^-15 - 1161189q^-14 - 3018262q^-13 - 3199562q^-12 - 1565745q^-11 + 933672q^-10 + 2904738q^-9 + 3244072q^-8 + 1725179q^-7 - 755930q^-6 - 2803510q^-5 - 3265298q^-4 - 1844767q^-3 + 613441q^-2 + 2720250q^-1 + 3285566 + 1951561q - 485486q² - 2646880q³ - 3312851q⁴ - 2068490q⁵ + 344210q⁶ + 2565024q⁷ + 3346285q⁸ + 2210037q⁹ - 165925q¹⁰ - 2449564q¹¹ - 3369639q¹² - 2378474q¹³ - 70093q¹⁴ + 2271671q¹⁵ + 3358549q¹⁶ + 2562865q¹⁷ + 370991q¹⁸ - 2006306q¹⁹ - 3279440q²⁰ - 2736128q²¹ - 729246q²² + 1637881q²³ + 3098896q²⁴ + 2859077q²⁵ + 1116697q²⁶ - 1169337q²⁷ - 2790521q²⁸ - 2886590q²⁹ - 1487424q³⁰ + 627580q³¹ + 2348100q³² + 2777646q³³ + 1782232q³⁴ - 62409q³⁵ - 1790364q³⁶ - 2511320q³⁷ - 1946542q³⁸ - 457096q³⁹ + 1168781q⁴⁰ + 2094478q⁴¹ + 1939263q⁴² + 861955q⁴³ - 552151q⁴⁴ - 1570252q⁴⁵ - 1756628q⁴⁶ - 1098365q⁴⁷ + 21565q⁴⁸ + 1007808q⁴⁹ + 1427166q⁵⁰ + 1146031q⁵¹ + 362231q⁵² - 486598q⁵³ - 1016784q⁵⁴ - 1025922q⁵⁵ - 566306q⁵⁶ + 76731q⁵⁷ + 601389q⁵⁸ + 791184q⁵⁹ + 601362q⁶⁰ + 184864q⁶¹ - 251172q⁶² - 514971q⁶³ - 511236q⁶⁴ - 296859q⁶⁵ + 9059q⁶⁶ + 262171q⁶⁷ + 357405q⁶⁸ + 292893q⁶⁹ + 117808q⁷⁰ - 76206q⁷¹ - 199339q⁷² - 222562q⁷³ - 150480q⁷⁴ - 29758q⁷⁵ + 75698q⁷⁶ + 133437q⁷⁷ + 126227q⁷⁸ + 68789q⁷⁹ - 1221q⁸⁰ - 59083q⁸¹ - 81143q⁸² - 65205q⁸³ - 29499q⁸⁴ + 12071q⁸⁵ + 39575q⁸⁶ + 44420q⁸⁷ + 32659q⁸⁸ + 9142q⁸⁹ - 12545q⁹⁰ - 23275q⁹¹ - 23673q⁹² - 13353q⁹³ - 463q⁹⁴ + 8636q⁹⁵ + 13220q⁹⁶ + 10556q⁹⁷ + 4304q⁹⁸ - 1635q⁹⁹ - 6031q¹⁰⁰ - 6008q¹⁰¹ - 3601q¹⁰² - 968q¹⁰³ + 1930q¹⁰⁴ + 2882q¹⁰⁵ + 2364q¹⁰⁶ + 1155q¹⁰⁷ - 610q¹⁰⁸ - 1121q¹⁰⁹ - 969q¹¹⁰ - 740q¹¹¹ - 49q¹¹² + 318q¹¹³ + 491q¹¹⁴ + 451q¹¹⁵ - 23q¹¹⁶ - 149q¹¹⁷ - 103q¹¹⁸ - 137q¹¹⁹ - 31q¹²⁰ - 21q¹²¹ + 50q¹²² + 113q¹²³ - 8q¹²⁴ - 34q¹²⁵ - 8q¹²⁶ - 9q¹²⁷ + 11q¹²⁸ - 15q¹²⁹ - 6q¹³⁰ + 25q¹³¹ - q¹³² - 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, 104]]
Out[2]=	PD[X[6, 2, 7, 1], X[16, 4, 17, 3], X[18, 9, 19, 10], X[14, 7, 15, 8], > X[20, 13, 1, 14], X[8, 17, 9, 18], X[10, 19, 11, 20], X[12, 6, 13, 5], > X[4, 12, 5, 11], X[2, 16, 3, 15]]
In[3]:=	GaussCode[Knot[10, 104]]
Out[3]=	GaussCode[1, -10, 2, -9, 8, -1, 4, -6, 3, -7, 9, -8, 5, -4, 10, -2, 6, -3, 7, > -5]
In[4]:=	DTCode[Knot[10, 104]]
Out[4]=	DTCode[6, 16, 12, 14, 18, 4, 20, 2, 8, 10]
In[5]:=	br = BR[Knot[10, 104]]
Out[5]=	BR[3, {-1, -1, -1, 2, 2, -1, 2, -1, 2, 2}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{3, 10}
In[7]:=	BraidIndex[Knot[10, 104]]
Out[7]=	3
In[8]:=	Show[DrawMorseLink[Knot[10, 104]]]

Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 104]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 1, 4, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 104]][t]
Out[10]=	-4 4 9 15 2 3 4 19 + t - -- + -- - -- - 15 t + 9 t - 4 t + t 3 2 t t t
In[11]:=	Conway[Knot[10, 104]][z]
Out[11]=	2 4 6 8 1 + z + 5 z + 4 z + z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 104]}
In[13]:=	{KnotDet[Knot[10, 104]], KnotSignature[Knot[10, 104]]}
Out[13]=	{77, 0}
In[14]:=	Jones[Knot[10, 104]][q]
Out[14]=	-5 3 6 10 12 2 3 4 5 13 - q + -- - -- + -- - -- - 12 q + 10 q - 6 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, 71], Knot[10, 104]}
In[16]:=	A2Invariant[Knot[10, 104]][q]
Out[16]=	-14 -12 2 2 -6 -4 3 2 4 6 8 10 -3 - q + q - --- + -- + q - q + -- + 3 q - q + q + 2 q - 2 q + 10 8 2 q q q 12 14 > q - q
In[17]:=	HOMFLYPT[Knot[10, 104]][a, z]
Out[17]=	2 4 6 -2 2 2 5 z 2 2 4 4 z 2 4 6 z 3 - a - a + 11 z - ---- - 5 a z + 13 z - ---- - 4 a z + 6 z - -- - 2 2 2 a a a 2 6 8 > a z + z
In[18]:=	Kauffman[Knot[10, 104]][a, z]
Out[18]=	2 2 -2 2 2 z 4 z 3 5 2 2 z 6 z 3 + a + a - --- - --- - 2 a z + a z + a z - 15 z + ---- - ---- - 3 a 4 2 a a a 3 3 3 2 2 4 2 2 z 8 z 13 z 3 3 3 5 3 > 4 a z + 3 a z - ---- + ---- + ----- + 4 a z - a z - 2 a z + 5 3 a a a 4 4 5 5 5 4 6 z 12 z 2 4 4 4 z 11 z 12 z 5 > 27 z - ---- + ----- + 3 a z - 6 a z + -- - ----- - ----- - 6 a z - 4 2 5 3 a a a a a 6 6 7 7 3 5 5 5 6 3 z 11 z 2 6 4 6 5 z 3 z > 5 a z + a z - 22 z + ---- - ----- - 5 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 > 2 a z + 4 a z + 9 z + ---- + 4 a z + ---- + 2 a z 2 a a
In[19]:=	{Vassiliev[2][Knot[10, 104]], Vassiliev[3][Knot[10, 104]]}
Out[19]=	{1, 0}
In[20]:=	Kh[Knot[10, 104]][q, t]
Out[20]=	7 1 2 1 4 2 6 4 6 6 - + 7 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 + 6 q t + 4 q t + 6 q t + 2 q t + 4 q t + q t + 2 q t + 11 5 > q t
In[21]:=	ColouredJones[Knot[10, 104], 2][q]
Out[21]=	-15 3 -13 8 16 3 33 46 7 83 76 37 133 154 + q - --- + q + --- - --- + --- + -- - -- - -- + -- - -- - -- + --- - 14 12 11 10 9 8 7 6 5 4 3 q q q q q q q q q q q 84 70 2 3 4 5 6 7 8 > -- - -- - 68 q - 86 q + 134 q - 36 q - 78 q + 84 q - 5 q - 49 q + 2 q q 9 10 11 12 13 14 15 > 33 q + 6 q - 18 q + 7 q + 2 q - 3 q + q

The Alternating Knot 10104

The Alternating Knot 10₁₀₄