The Alternating Knot 10₆₂

Visit 10₆₂'s page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 10₆₂'s page at Knotilus!

Acknowledgement

KnotPlot

PD Presentation: X₁₄₂₅ X_3,10,4,11 X_11,19,12,18 X_5,15,6,14 X_7,17,8,16 X_15,7,16,6 X_17,9,18,8 X_13,1,14,20 X_19,13,20,12 X_9,2,10,3

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

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

Reversible 2 4 3 / NotAvailable 1

Alexander Polynomial: t^-4 - 3t^-3 + 6t^-2 - 8t^-1 + 9 - 8t + 6t² - 3t³ + t⁴

Conway Polynomial: 1 + 5z² + 8z⁴ + 5z⁶ + z⁸

Other knots with the same Alexander/Conway Polynomial: {K11n76, K11n78, ...}

Determinant and Signature: {45, 4}

Jones Polynomial: - q^-1 + 2 - 3q + 6q² - 6q³ + 7q⁴ - 7q⁵ + 6q⁶ - 4q⁷ + 2q⁸ - q⁹

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

A2 (sl(3)) Invariant: - q^-2 - q² + q⁴ + 2q⁶ + q⁸ + 3q¹⁰ - q¹² + 2q¹⁴ - 2q²² - q²⁶

HOMFLY-PT Polynomial: - 4a^-6 - 8a^-6z² - 5a^-6z⁴ - a^-6z⁶ + 7a^-4 + 20a^-4z² + 18a^-4z⁴ + 7a^-4z⁶ + a^-4z⁸ - 2a^-2 - 7a^-2z² - 5a^-2z⁴ - a^-2z⁶

Kauffman Polynomial: - a^-11z + a^-11z³ - a^-10z² + 2a^-10z⁴ + a^-9z - 2a^-9z³ + 3a^-9z⁵ + 4a^-8z² - 6a^-8z⁴ + 4a^-8z⁶ - a^-7z + 5a^-7z³ - 8a^-7z⁵ + 4a^-7z⁷ + 4a^-6 - 8a^-6z² + 6a^-6z⁴ - 7a^-6z⁶ + 3a^-6z⁸ - 6a^-5z + 16a^-5z³ - 15a^-5z⁵ + 2a^-5z⁷ + a^-5z⁹ + 7a^-4 - 23a^-4z² + 30a^-4z⁴ - 21a^-4z⁶ + 5a^-4z⁸ - 5a^-3z + 15a^-3z³ - 9a^-3z⁵ - a^-3z⁷ + a^-3z⁹ + 2a^-2 - 10a^-2z² + 16a^-2z⁴ - 10a^-2z⁶ + 2a^-2z⁸ - 2a^-1z + 7a^-1z³ - 5a^-1z⁵ + a^-1z⁷

V₂ and V₃, the type 2 and 3 Vassiliev invariants: {5, 9}

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=4 is the signature of 10₆₂. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

t^rq^j r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7

j = 19 1

j = 17 1

j = 15 3 1

j = 13 3 1

j = 11 4 3

j = 9 3 3

j = 7 3 4

j = 5 3 3

j = 3 1 4

j = 1 1 2

j = -1 1

j = -3 1

n Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q^-5 - 2q^-4 - q^-3 + 6q^-2 - 5q^-1 - 7 + 15q - 3q² - 18q³ + 23q⁴ + 3q⁵ - 29q⁶ + 24q⁷ + 11q⁸ - 36q⁹ + 19q¹⁰ + 19q¹¹ - 34q¹² + 12q¹³ + 20q¹⁴ - 27q¹⁵ + 7q¹⁶ + 13q¹⁷ - 16q¹⁸ + 5q¹⁹ + 5q²⁰ - 7q²¹ + 3q²² + q²³ - 2q²⁴ + q²⁵

3 - q^-12 + 2q^-11 + q^-10 - 2q^-9 - 5q^-8 + 4q^-7 + 9q^-6 - 2q^-5 - 17q^-4 - 2q^-3 + 22q^-2 + 12q^-1 - 26 - 25q + 25q² + 36q³ - 14q⁴ - 49q⁵ + 9q⁶ + 48q⁷ + 12q⁸ - 54q⁹ - 16q¹⁰ + 41q¹¹ + 34q¹² - 41q¹³ - 34q¹⁴ + 21q¹⁵ + 49q¹⁶ - 16q¹⁷ - 47q¹⁸ - 4q¹⁹ + 55q²⁰ + 15q²¹ - 52q²² - 31q²³ + 49q²⁴ + 37q²⁵ - 38q²⁶ - 41q²⁷ + 28q²⁸ + 35q²⁹ - 15q³⁰ - 27q³¹ + 7q³² + 16q³³ - 3q³⁴ - 4q³⁵ - 2q³⁶ + q³⁷ - q³⁸ + 4q³⁹ - q⁴⁰ - 2q⁴¹ - q⁴² + 4q⁴³ - 2q⁴⁴ - q⁴⁶ + 2q⁴⁷ - q⁴⁸

4 q^-22 - 2q^-21 - q^-20 + 2q^-19 + q^-18 + 6q^-17 - 8q^-16 - 7q^-15 + 2q^-14 + 2q^-13 + 26q^-12 - 9q^-11 - 21q^-10 - 13q^-9 - 13q^-8 + 59q^-7 + 14q^-6 - 14q^-5 - 36q^-4 - 70q^-3 + 64q^-2 + 45q^-1 + 42 - 15q - 135q² + 16q³ + 18q⁴ + 100q⁵ + 71q⁶ - 136q⁷ - 25q⁸ - 77q⁹ + 81q¹⁰ + 150q¹¹ - 67q¹² + 11q¹³ - 162q¹⁴ - 13q¹⁵ + 149q¹⁶ + 2q¹⁷ + 118q¹⁸ - 175q¹⁹ - 117q²⁰ + 79q²¹ + 30q²² + 234q²³ - 134q²⁴ - 189q²⁵ - 11q²⁶ + 30q²⁷ + 332q²⁸ - 79q²⁹ - 249q³⁰ - 102q³¹ + 32q³² + 418q³³ - 15q³⁴ - 297q³⁵ - 196q³⁶ + 12q³⁷ + 476q³⁸ + 77q³⁹ - 289q⁴⁰ - 272q⁴¹ - 58q⁴² + 448q⁴³ + 161q⁴⁴ - 191q⁴⁵ - 262q⁴⁶ - 139q⁴⁷ + 315q⁴⁸ + 169q⁴⁹ - 64q⁵⁰ - 160q⁵¹ - 155q⁵² + 155q⁵³ + 100q⁵⁴ + 8q⁵⁵ - 51q⁵⁶ - 106q⁵⁷ + 58q⁵⁸ + 30q⁵⁹ + 15q⁶⁰ + 2q⁶¹ - 50q⁶² + 22q⁶³ - q⁶⁴ + 6q⁶⁵ + 10q⁶⁶ - 21q⁶⁷ + 11q⁶⁸ - 3q⁶⁹ + q⁷⁰ + 4q⁷¹ - 8q⁷² + 5q⁷³ - q⁷⁴ + q⁷⁶ - 2q⁷⁷ + q⁷⁸

5 - q^-35 + 2q^-34 + q^-33 - 2q^-32 - q^-31 - 2q^-30 - 2q^-29 + 6q^-28 + 9q^-27 - 2q^-26 - 7q^-25 - 10q^-24 - 12q^-23 + 6q^-22 + 27q^-21 + 21q^-20 - q^-19 - 25q^-18 - 44q^-17 - 28q^-16 + 27q^-15 + 62q^-14 + 58q^-13 + 7q^-12 - 63q^-11 - 101q^-10 - 58q^-9 + 34q^-8 + 117q^-7 + 124q^-6 + 35q^-5 - 98q^-4 - 169q^-3 - 124q^-2 + 17q^-1 + 168 + 212q + 89q² - 102q³ - 227q⁴ - 218q⁵ - 34q⁶ + 200q⁷ + 285q⁸ + 168q⁹ - 45q¹⁰ - 279q¹¹ - 321q¹² - 108q¹³ + 182q¹⁴ + 343q¹⁵ + 318q¹⁶ + 18q¹⁷ - 336q¹⁸ - 444q¹⁹ - 230q²⁰ + 157q²¹ + 529q²² + 485q²³ + 19q²⁴ - 493q²⁵ - 659q²⁶ - 311q²⁷ + 405q²⁸ + 820q²⁹ + 532q³⁰ - 230q³¹ - 872q³² - 805q³³ + 57q³⁴ + 921q³⁵ + 965q³⁶ + 144q³⁷ - 893q³⁸ - 1167q³⁹ - 294q⁴⁰ + 899q⁴¹ + 1272q⁴² + 450q⁴³ - 881q⁴⁴ - 1429q⁴⁵ - 565q⁴⁶ + 902q⁴⁷ + 1547q⁴⁸ + 703q⁴⁹ - 908q⁵⁰ - 1706q⁵¹ - 835q⁵² + 892q⁵³ + 1828q⁵⁴ + 1021q⁵⁵ - 828q⁵⁶ - 1931q⁵⁷ - 1187q⁵⁸ + 683q⁵⁹ + 1922q⁶⁰ + 1372q⁶¹ - 469q⁶² - 1835q⁶³ - 1464q⁶⁴ + 217q⁶⁵ + 1605q⁶⁶ + 1474q⁶⁷ + 40q⁶⁸ - 1306q⁶⁹ - 1373q⁷⁰ - 225q⁷¹ + 965q⁷² + 1163q⁷³ + 339q⁷⁴ - 639q⁷⁵ - 917q⁷⁶ - 361q⁷⁷ + 390q⁷⁸ + 650q⁷⁹ + 311q⁸⁰ - 194q⁸¹ - 430q⁸² - 245q⁸³ + 96q⁸⁴ + 253q⁸⁵ + 162q⁸⁶ - 25q⁸⁷ - 143q⁸⁸ - 104q⁸⁹ + 8q⁹⁰ + 65q⁹¹ + 58q⁹² + 14q⁹³ - 38q⁹⁴ - 32q⁹⁵ + 7q⁹⁷ + 12q⁹⁸ + 13q⁹⁹ - 9q¹⁰⁰ - 11q¹⁰¹ + 7q¹⁰² - 2q¹⁰³ - 2q¹⁰⁴ + 6q¹⁰⁵ - 3q¹⁰⁶ - 3q¹⁰⁷ + 5q¹⁰⁸ - q¹⁰⁹ - 2q¹¹⁰ + q¹¹¹ - q¹¹³ + 2q¹¹⁴ - q¹¹⁵

6 q^-51 - 2q^-50 - q^-49 + 2q^-48 + q^-47 + 2q^-46 - 2q^-45 + 4q^-44 - 8q^-43 - 9q^-42 + 5q^-41 + 6q^-40 + 12q^-39 - q^-38 + 16q^-37 - 19q^-36 - 33q^-35 - 10q^-34 + q^-33 + 31q^-32 + 15q^-31 + 70q^-30 - 4q^-29 - 59q^-28 - 62q^-27 - 61q^-26 - q^-25 + q^-24 + 169q^-23 + 102q^-22 + 23q^-21 - 60q^-20 - 143q^-19 - 147q^-18 - 191q^-17 + 134q^-16 + 181q^-15 + 242q^-14 + 171q^-13 + 21q^-12 - 173q^-11 - 480q^-10 - 191q^-9 - 109q^-8 + 210q^-7 + 391q^-6 + 487q^-5 + 278q^-4 - 326q^-3 - 353q^-2 - 630q^-1 - 367 - 31q + 568q² + 804q³ + 387q⁴ + 281q⁵ - 489q⁶ - 784q⁷ - 935q⁸ - 267q⁹ + 407q¹⁰ + 654q¹¹ + 1209q¹² + 627q¹³ - 30q¹⁴ - 1081q¹⁵ - 1180q¹⁶ - 922q¹⁷ - 431q¹⁸ + 1053q¹⁹ + 1570q²⁰ + 1581q²¹ + 247q²² - 786q²³ - 1818q²⁴ - 2242q²⁵ - 642q²⁶ + 1028q²⁷ + 2569q²⁸ + 2183q²⁹ + 1108q³⁰ - 1132q³¹ - 3283q³² - 2838q³³ - 968q³⁴ + 1982q³⁵ + 3321q³⁶ + 3399q³⁷ + 909q³⁸ - 2817q³⁹ - 4255q⁴⁰ - 3313q⁴¹ + 161q⁴² + 3107q⁴³ + 4977q⁴⁴ + 3253q⁴⁵ - 1270q⁴⁶ - 4495q⁴⁷ - 5045q⁴⁸ - 1927q⁴⁹ + 1999q⁵⁰ + 5565q⁵¹ + 5072q⁵² + 466q⁵³ - 4008q⁵⁴ - 5957q⁵⁵ - 3560q⁵⁶ + 810q⁵⁷ + 5602q⁵⁸ + 6207q⁵⁹ + 1751q⁶⁰ - 3536q⁶¹ - 6472q⁶² - 4618q⁶³ + 91q⁶⁴ + 5735q⁶⁵ + 7063q⁶⁶ + 2566q⁶⁷ - 3480q⁶⁸ - 7159q⁶⁹ - 5556q⁷⁰ - 313q⁷¹ + 6180q⁷² + 8137q⁷³ + 3494q⁷⁴ - 3420q⁷⁵ - 8014q⁷⁶ - 6839q⁷⁷ - 1184q⁷⁸ + 6239q⁷⁹ + 9202q⁸⁰ + 4977q⁸¹ - 2407q⁸² - 8082q⁸³ - 8034q⁸⁴ - 2883q⁸⁵ + 4911q⁸⁶ + 9113q⁸⁷ + 6338q⁸⁸ - 287q⁸⁹ - 6423q⁹⁰ - 7880q⁹¹ - 4436q⁹² + 2375q⁹³ + 7113q⁹⁴ + 6235q⁹⁵ + 1650q⁹⁶ - 3585q⁹⁷ - 5873q⁹⁸ - 4498q⁹⁹ + 158q¹⁰⁰ + 4113q¹⁰¹ + 4445q¹⁰² + 2137q¹⁰³ - 1218q¹⁰⁴ - 3169q¹⁰⁵ - 3115q¹⁰⁶ - 668q¹⁰⁷ + 1742q¹⁰⁸ + 2251q¹⁰⁹ + 1440q¹¹⁰ - 158q¹¹¹ - 1222q¹¹² - 1544q¹¹³ - 507q¹¹⁴ + 593q¹¹⁵ + 818q¹¹⁶ + 627q¹¹⁷ + 46q¹¹⁸ - 340q¹¹⁹ - 605q¹²⁰ - 204q¹²¹ + 204q¹²² + 220q¹²³ + 208q¹²⁴ + 30q¹²⁵ - 62q¹²⁶ - 223q¹²⁷ - 63q¹²⁸ + 80q¹²⁹ + 37q¹³⁰ + 68q¹³¹ + 14q¹³² + 5q¹³³ - 85q¹³⁴ - 20q¹³⁵ + 32q¹³⁶ - 6q¹³⁷ + 25q¹³⁸ + 4q¹³⁹ + 12q¹⁴⁰ - 31q¹⁴¹ - 4q¹⁴² + 13q¹⁴³ - 10q¹⁴⁴ + 10q¹⁴⁵ - 2q¹⁴⁶ + 7q¹⁴⁷ - 9q¹⁴⁸ + 5q¹⁵⁰ - 6q¹⁵¹ + 4q¹⁵² - 2q¹⁵³ + 2q¹⁵⁴ - q¹⁵⁵ + q¹⁵⁷ - 2q¹⁵⁸ + q¹⁵⁹

7 - q^-70 + 2q^-69 + q^-68 - 2q^-67 - q^-66 - 2q^-65 + 2q^-64 - 2q^-62 + 8q^-61 + 6q^-60 - 4q^-59 - 6q^-58 - 14q^-57 - 2q^-56 + 4q^-55 - 6q^-54 + 22q^-53 + 26q^-52 + 11q^-51 - q^-50 - 44q^-49 - 33q^-48 - 17q^-47 - 34q^-46 + 27q^-45 + 65q^-44 + 73q^-43 + 83q^-42 - 27q^-41 - 69q^-40 - 83q^-39 - 156q^-38 - 76q^-37 + 6q^-36 + 109q^-35 + 260q^-34 + 176q^-33 + 94q^-32 + 3q^-31 - 253q^-30 - 302q^-29 - 323q^-28 - 222q^-27 + 164q^-26 + 319q^-25 + 467q^-24 + 524q^-23 + 173q^-22 - 90q^-21 - 493q^-20 - 824q^-19 - 582q^-18 - 344q^-17 + 168q^-16 + 815q^-15 + 918q^-14 + 973q^-13 + 485q^-12 - 423q^-11 - 896q^-10 - 1394q^-9 - 1279q^-8 - 426q^-7 + 288q^-6 + 1337q^-5 + 1839q^-4 + 1402q^-3 + 836q^-2 - 535q^-1 - 1742 - 2042q - 2072q² - 882q³ + 681q⁴ + 1801q⁵ + 2914q⁶ + 2491q⁷ + 1059q⁸ - 493q⁹ - 2603q¹⁰ - 3469q¹¹ - 3091q¹² - 1853q¹³ + 1039q¹⁴ + 3318q¹⁵ + 4440q¹⁶ + 4361q¹⁷ + 1756q¹⁸ - 1444q¹⁹ - 4438q²⁰ - 6443q²¹ - 5031q²² - 1701q²³ + 2682q²⁴ + 6953q²⁵ + 7818q²⁶ + 5722q²⁷ + 854q²⁸ - 5658q²⁹ - 9364q³⁰ - 9514q³¹ - 5339q³² + 2383q³³ + 8872q³⁴ + 12219q³⁵ + 10193q³⁶ + 2322q³⁷ - 6480q³⁸ - 13219q³⁹ - 14222q⁴⁰ - 7648q⁴¹ + 2269q⁴² + 12176q⁴³ + 16985q⁴⁴ + 12872q⁴⁵ + 2938q⁴⁶ - 9405q⁴⁷ - 17944q⁴⁸ - 17174q⁴⁹ - 8564q⁵⁰ + 5184q⁵¹ + 17270q⁵² + 20294q⁵³ + 13847q⁵⁴ - 268q⁵⁵ - 15106q⁵⁶ - 21951q⁵⁷ - 18488q⁵⁸ - 4936q⁵⁹ + 12033q⁶⁰ + 22420q⁶¹ + 22076q⁶² + 9833q⁶³ - 8322q⁶⁴ - 21836q⁶⁵ - 24789q⁶⁶ - 14238q⁶⁷ + 4632q⁶⁸ + 20649q⁶⁹ + 26487q⁷⁰ + 17871q⁷¹ - 1077q⁷² - 19111q⁷³ - 27598q⁷⁴ - 20759q⁷⁵ - 1859q⁷⁶ + 17653q⁷⁷ + 28170q⁷⁸ + 22870q⁷⁹ + 4185q⁸⁰ - 16500q⁸¹ - 28643q⁸² - 24446q⁸³ - 5748q⁸⁴ + 15940q⁸⁵ + 29252q⁸⁶ + 25707q⁸⁷ + 6713q⁸⁸ - 15978q⁸⁹ - 30265q⁹⁰ - 27039q⁹¹ - 7457q⁹² + 16497q⁹³ + 31846q⁹⁴ + 28806q⁹⁵ + 8351q⁹⁶ - 17115q⁹⁷ - 33734q⁹⁸ - 31135q⁹⁹ - 10013q¹⁰⁰ + 17123q¹⁰¹ + 35577q¹⁰² + 34080q¹⁰³ + 12651q¹⁰⁴ - 16061q¹⁰⁵ - 36609q¹⁰⁶ - 36921q¹⁰⁷ - 16293q¹⁰⁸ + 13260q¹⁰⁹ + 36099q¹¹⁰ + 39162q¹¹¹ + 20451q¹¹² - 8984q¹¹³ - 33577q¹¹⁴ - 39726q¹¹⁵ - 24199q¹¹⁶ + 3555q¹¹⁷ + 28907q¹¹⁸ + 38148q¹¹⁹ + 26703q¹²⁰ + 2046q¹²¹ - 22748q¹²² - 34318q¹²³ - 27132q¹²⁴ - 6705q¹²⁵ + 15912q¹²⁶ + 28569q¹²⁷ + 25394q¹²⁸ + 9805q¹²⁹ - 9529q¹³⁰ - 22019q¹³¹ - 21805q¹³² - 10831q¹³³ + 4444q¹³⁴ + 15480q¹³⁵ + 17118q¹³⁶ + 10196q¹³⁷ - 999q¹³⁸ - 9926q¹³⁹ - 12332q¹⁴⁰ - 8391q¹⁴¹ - 778q¹⁴² + 5760q¹⁴³ + 8069q¹⁴⁴ + 6115q¹⁴⁵ + 1390q¹⁴⁶ - 2971q¹⁴⁷ - 4819q¹⁴⁸ - 4030q¹⁴⁹ - 1266q¹⁵⁰ + 1401q¹⁵¹ + 2602q¹⁵² + 2354q¹⁵³ + 866q¹⁵⁴ - 584q¹⁵⁵ - 1263q¹⁵⁶ - 1239q¹⁵⁷ - 481q¹⁵⁸ + 238q¹⁵⁹ + 559q¹⁶⁰ + 599q¹⁶¹ + 196q¹⁶² - 140q¹⁶³ - 220q¹⁶⁴ - 242q¹⁶⁵ - 39q¹⁶⁶ + 72q¹⁶⁷ + 76q¹⁶⁸ + 122q¹⁶⁹ - 12q¹⁷⁰ - 76q¹⁷¹ - 40q¹⁷² - 53q¹⁷³ + 47q¹⁷⁴ + 49q¹⁷⁵ + 36q¹⁷⁷ - 16q¹⁷⁸ - 31q¹⁷⁹ - 18q¹⁸⁰ - 35q¹⁸¹ + 33q¹⁸² + 29q¹⁸³ - 10q¹⁸⁴ + 16q¹⁸⁵ - 5q¹⁸⁶ - 9q¹⁸⁷ - 3q¹⁸⁸ - 22q¹⁸⁹ + 14q¹⁹⁰ + 12q¹⁹¹ - 6q¹⁹² + 9q¹⁹³ - 6q¹⁹⁴ - 2q¹⁹⁵ + 3q¹⁹⁶ - 9q¹⁹⁷ + 4q¹⁹⁸ + 3q¹⁹⁹ - 2q²⁰⁰ + 4q²⁰¹ - 3q²⁰² - q²⁰³ + 2q²⁰⁴ - 2q²⁰⁵ + q²⁰⁶ - q²⁰⁸ + 2q²⁰⁹ - q²¹⁰

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, 62]]

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

In[3]:=
GaussCode[Knot[10, 62]]

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

In[4]:=
DTCode[Knot[10, 62]]

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

In[5]:=
br = BR[Knot[10, 62]]

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

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

Out[6]=
{3, 10}

In[7]:=
BraidIndex[Knot[10, 62]]

Out[7]=
3

In[8]:=
Show[DrawMorseLink[Knot[10, 62]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[10, 62]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

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

In[10]:=
alex = Alexander[Knot[10, 62]][t]

Out[10]=
-4 3 6 8 2 3 4 9 + t - -- + -- - - - 8 t + 6 t - 3 t + t 3 2 t t t

In[11]:=
Conway[Knot[10, 62]][z]

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

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

Out[12]=
{Knot[10, 62], Knot[11, NonAlternating, 76], Knot[11, NonAlternating, 78]}

In[13]:=
{KnotDet[Knot[10, 62]], KnotSignature[Knot[10, 62]]}

Out[13]=
{45, 4}

In[14]:=
Jones[Knot[10, 62]][q]

Out[14]=
1 2 3 4 5 6 7 8 9 2 - - - 3 q + 6 q - 6 q + 7 q - 7 q + 6 q - 4 q + 2 q - q q

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

Out[15]=
{Knot[10, 62]}

In[16]:=
A2Invariant[Knot[10, 62]][q]

Out[16]=
-2 2 4 6 8 10 12 14 22 26 -q - q + q + 2 q + q + 3 q - q + 2 q - 2 q - q

In[17]:=
HOMFLYPT[Knot[10, 62]][a, z]

Out[17]=
2 2 2 4 4 4 6 6 6 8 -4 7 2 8 z 20 z 7 z 5 z 18 z 5 z z 7 z z z -- + -- - -- - ---- + ----- - ---- - ---- + ----- - ---- - -- + ---- - -- + -- 6 4 2 6 4 2 6 4 2 6 4 2 4 a a a a a a a a a a a a a

In[18]:=
Kauffman[Knot[10, 62]][a, z]

Out[18]=
2 2 2 2 4 7 2 z z z 6 z 5 z 2 z z 4 z 8 z 23 z -- + -- + -- - --- + -- - -- - --- - --- - --- - --- + ---- - ---- - ----- - 6 4 2 11 9 7 5 3 a 10 8 6 4 a a a a a a a a a a a a 2 3 3 3 3 3 3 4 4 4 10 z z 2 z 5 z 16 z 15 z 7 z 2 z 6 z 6 z > ----- + --- - ---- + ---- + ----- + ----- + ---- + ---- - ---- + ---- + 2 11 9 7 5 3 a 10 8 6 a a a a a a a a a 4 4 5 5 5 5 5 6 6 6 30 z 16 z 3 z 8 z 15 z 9 z 5 z 4 z 7 z 21 z > ----- + ----- + ---- - ---- - ----- - ---- - ---- + ---- - ---- - ----- - 4 2 9 7 5 3 a 8 6 4 a a a a a a a a a 6 7 7 7 7 8 8 8 9 9 10 z 4 z 2 z z z 3 z 5 z 2 z z z > ----- + ---- + ---- - -- + -- + ---- + ---- + ---- + -- + -- 2 7 5 3 a 6 4 2 5 3 a a a a a a a a a

In[19]:=
{Vassiliev[2][Knot[10, 62]], Vassiliev[3][Knot[10, 62]]}

Out[19]=
{5, 9}

In[20]:=
Kh[Knot[10, 62]][q, t]

Out[20]=
3 3 5 1 1 q 2 q q 5 7 7 2 4 q + 3 q + ----- + ---- + -- + --- + -- + 3 q t + 3 q t + 4 q t + 3 3 2 2 t t q t q t t 9 2 9 3 11 3 11 4 13 4 13 5 15 5 > 3 q t + 3 q t + 4 q t + 3 q t + 3 q t + q t + 3 q t + 15 6 17 6 19 7 > q t + q t + q t

In[21]:=
ColouredJones[Knot[10, 62], 2][q]

Out[21]=
-5 2 -3 6 5 2 3 4 5 6 -7 + q - -- - q + -- - - + 15 q - 3 q - 18 q + 23 q + 3 q - 29 q + 4 2 q q q 7 8 9 10 11 12 13 14 > 24 q + 11 q - 36 q + 19 q + 19 q - 34 q + 12 q + 20 q - 15 16 17 18 19 20 21 22 23 > 27 q + 7 q + 13 q - 16 q + 5 q + 5 q - 7 q + 3 q + q - 24 25 > 2 q + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 10₆₂

10₆₁
10₆₃

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-5 - 2q^-4 - q^-3 + 6q^-2 - 5q^-1 - 7 + 15q - 3q² - 18q³ + 23q⁴ + 3q⁵ - 29q⁶ + 24q⁷ + 11q⁸ - 36q⁹ + 19q¹⁰ + 19q¹¹ - 34q¹² + 12q¹³ + 20q¹⁴ - 27q¹⁵ + 7q¹⁶ + 13q¹⁷ - 16q¹⁸ + 5q¹⁹ + 5q²⁰ - 7q²¹ + 3q²² + q²³ - 2q²⁴ + q²⁵
3	- q^-12 + 2q^-11 + q^-10 - 2q^-9 - 5q^-8 + 4q^-7 + 9q^-6 - 2q^-5 - 17q^-4 - 2q^-3 + 22q^-2 + 12q^-1 - 26 - 25q + 25q² + 36q³ - 14q⁴ - 49q⁵ + 9q⁶ + 48q⁷ + 12q⁸ - 54q⁹ - 16q¹⁰ + 41q¹¹ + 34q¹² - 41q¹³ - 34q¹⁴ + 21q¹⁵ + 49q¹⁶ - 16q¹⁷ - 47q¹⁸ - 4q¹⁹ + 55q²⁰ + 15q²¹ - 52q²² - 31q²³ + 49q²⁴ + 37q²⁵ - 38q²⁶ - 41q²⁷ + 28q²⁸ + 35q²⁹ - 15q³⁰ - 27q³¹ + 7q³² + 16q³³ - 3q³⁴ - 4q³⁵ - 2q³⁶ + q³⁷ - q³⁸ + 4q³⁹ - q⁴⁰ - 2q⁴¹ - q⁴² + 4q⁴³ - 2q⁴⁴ - q⁴⁶ + 2q⁴⁷ - q⁴⁸
4	q^-22 - 2q^-21 - q^-20 + 2q^-19 + q^-18 + 6q^-17 - 8q^-16 - 7q^-15 + 2q^-14 + 2q^-13 + 26q^-12 - 9q^-11 - 21q^-10 - 13q^-9 - 13q^-8 + 59q^-7 + 14q^-6 - 14q^-5 - 36q^-4 - 70q^-3 + 64q^-2 + 45q^-1 + 42 - 15q - 135q² + 16q³ + 18q⁴ + 100q⁵ + 71q⁶ - 136q⁷ - 25q⁸ - 77q⁹ + 81q¹⁰ + 150q¹¹ - 67q¹² + 11q¹³ - 162q¹⁴ - 13q¹⁵ + 149q¹⁶ + 2q¹⁷ + 118q¹⁸ - 175q¹⁹ - 117q²⁰ + 79q²¹ + 30q²² + 234q²³ - 134q²⁴ - 189q²⁵ - 11q²⁶ + 30q²⁷ + 332q²⁸ - 79q²⁹ - 249q³⁰ - 102q³¹ + 32q³² + 418q³³ - 15q³⁴ - 297q³⁵ - 196q³⁶ + 12q³⁷ + 476q³⁸ + 77q³⁹ - 289q⁴⁰ - 272q⁴¹ - 58q⁴² + 448q⁴³ + 161q⁴⁴ - 191q⁴⁵ - 262q⁴⁶ - 139q⁴⁷ + 315q⁴⁸ + 169q⁴⁹ - 64q⁵⁰ - 160q⁵¹ - 155q⁵² + 155q⁵³ + 100q⁵⁴ + 8q⁵⁵ - 51q⁵⁶ - 106q⁵⁷ + 58q⁵⁸ + 30q⁵⁹ + 15q⁶⁰ + 2q⁶¹ - 50q⁶² + 22q⁶³ - q⁶⁴ + 6q⁶⁵ + 10q⁶⁶ - 21q⁶⁷ + 11q⁶⁸ - 3q⁶⁹ + q⁷⁰ + 4q⁷¹ - 8q⁷² + 5q⁷³ - q⁷⁴ + q⁷⁶ - 2q⁷⁷ + q⁷⁸
5	- q^-35 + 2q^-34 + q^-33 - 2q^-32 - q^-31 - 2q^-30 - 2q^-29 + 6q^-28 + 9q^-27 - 2q^-26 - 7q^-25 - 10q^-24 - 12q^-23 + 6q^-22 + 27q^-21 + 21q^-20 - q^-19 - 25q^-18 - 44q^-17 - 28q^-16 + 27q^-15 + 62q^-14 + 58q^-13 + 7q^-12 - 63q^-11 - 101q^-10 - 58q^-9 + 34q^-8 + 117q^-7 + 124q^-6 + 35q^-5 - 98q^-4 - 169q^-3 - 124q^-2 + 17q^-1 + 168 + 212q + 89q² - 102q³ - 227q⁴ - 218q⁵ - 34q⁶ + 200q⁷ + 285q⁸ + 168q⁹ - 45q¹⁰ - 279q¹¹ - 321q¹² - 108q¹³ + 182q¹⁴ + 343q¹⁵ + 318q¹⁶ + 18q¹⁷ - 336q¹⁸ - 444q¹⁹ - 230q²⁰ + 157q²¹ + 529q²² + 485q²³ + 19q²⁴ - 493q²⁵ - 659q²⁶ - 311q²⁷ + 405q²⁸ + 820q²⁹ + 532q³⁰ - 230q³¹ - 872q³² - 805q³³ + 57q³⁴ + 921q³⁵ + 965q³⁶ + 144q³⁷ - 893q³⁸ - 1167q³⁹ - 294q⁴⁰ + 899q⁴¹ + 1272q⁴² + 450q⁴³ - 881q⁴⁴ - 1429q⁴⁵ - 565q⁴⁶ + 902q⁴⁷ + 1547q⁴⁸ + 703q⁴⁹ - 908q⁵⁰ - 1706q⁵¹ - 835q⁵² + 892q⁵³ + 1828q⁵⁴ + 1021q⁵⁵ - 828q⁵⁶ - 1931q⁵⁷ - 1187q⁵⁸ + 683q⁵⁹ + 1922q⁶⁰ + 1372q⁶¹ - 469q⁶² - 1835q⁶³ - 1464q⁶⁴ + 217q⁶⁵ + 1605q⁶⁶ + 1474q⁶⁷ + 40q⁶⁸ - 1306q⁶⁹ - 1373q⁷⁰ - 225q⁷¹ + 965q⁷² + 1163q⁷³ + 339q⁷⁴ - 639q⁷⁵ - 917q⁷⁶ - 361q⁷⁷ + 390q⁷⁸ + 650q⁷⁹ + 311q⁸⁰ - 194q⁸¹ - 430q⁸² - 245q⁸³ + 96q⁸⁴ + 253q⁸⁵ + 162q⁸⁶ - 25q⁸⁷ - 143q⁸⁸ - 104q⁸⁹ + 8q⁹⁰ + 65q⁹¹ + 58q⁹² + 14q⁹³ - 38q⁹⁴ - 32q⁹⁵ + 7q⁹⁷ + 12q⁹⁸ + 13q⁹⁹ - 9q¹⁰⁰ - 11q¹⁰¹ + 7q¹⁰² - 2q¹⁰³ - 2q¹⁰⁴ + 6q¹⁰⁵ - 3q¹⁰⁶ - 3q¹⁰⁷ + 5q¹⁰⁸ - q¹⁰⁹ - 2q¹¹⁰ + q¹¹¹ - q¹¹³ + 2q¹¹⁴ - q¹¹⁵
6	q^-51 - 2q^-50 - q^-49 + 2q^-48 + q^-47 + 2q^-46 - 2q^-45 + 4q^-44 - 8q^-43 - 9q^-42 + 5q^-41 + 6q^-40 + 12q^-39 - q^-38 + 16q^-37 - 19q^-36 - 33q^-35 - 10q^-34 + q^-33 + 31q^-32 + 15q^-31 + 70q^-30 - 4q^-29 - 59q^-28 - 62q^-27 - 61q^-26 - q^-25 + q^-24 + 169q^-23 + 102q^-22 + 23q^-21 - 60q^-20 - 143q^-19 - 147q^-18 - 191q^-17 + 134q^-16 + 181q^-15 + 242q^-14 + 171q^-13 + 21q^-12 - 173q^-11 - 480q^-10 - 191q^-9 - 109q^-8 + 210q^-7 + 391q^-6 + 487q^-5 + 278q^-4 - 326q^-3 - 353q^-2 - 630q^-1 - 367 - 31q + 568q² + 804q³ + 387q⁴ + 281q⁵ - 489q⁶ - 784q⁷ - 935q⁸ - 267q⁹ + 407q¹⁰ + 654q¹¹ + 1209q¹² + 627q¹³ - 30q¹⁴ - 1081q¹⁵ - 1180q¹⁶ - 922q¹⁷ - 431q¹⁸ + 1053q¹⁹ + 1570q²⁰ + 1581q²¹ + 247q²² - 786q²³ - 1818q²⁴ - 2242q²⁵ - 642q²⁶ + 1028q²⁷ + 2569q²⁸ + 2183q²⁹ + 1108q³⁰ - 1132q³¹ - 3283q³² - 2838q³³ - 968q³⁴ + 1982q³⁵ + 3321q³⁶ + 3399q³⁷ + 909q³⁸ - 2817q³⁹ - 4255q⁴⁰ - 3313q⁴¹ + 161q⁴² + 3107q⁴³ + 4977q⁴⁴ + 3253q⁴⁵ - 1270q⁴⁶ - 4495q⁴⁷ - 5045q⁴⁸ - 1927q⁴⁹ + 1999q⁵⁰ + 5565q⁵¹ + 5072q⁵² + 466q⁵³ - 4008q⁵⁴ - 5957q⁵⁵ - 3560q⁵⁶ + 810q⁵⁷ + 5602q⁵⁸ + 6207q⁵⁹ + 1751q⁶⁰ - 3536q⁶¹ - 6472q⁶² - 4618q⁶³ + 91q⁶⁴ + 5735q⁶⁵ + 7063q⁶⁶ + 2566q⁶⁷ - 3480q⁶⁸ - 7159q⁶⁹ - 5556q⁷⁰ - 313q⁷¹ + 6180q⁷² + 8137q⁷³ + 3494q⁷⁴ - 3420q⁷⁵ - 8014q⁷⁶ - 6839q⁷⁷ - 1184q⁷⁸ + 6239q⁷⁹ + 9202q⁸⁰ + 4977q⁸¹ - 2407q⁸² - 8082q⁸³ - 8034q⁸⁴ - 2883q⁸⁵ + 4911q⁸⁶ + 9113q⁸⁷ + 6338q⁸⁸ - 287q⁸⁹ - 6423q⁹⁰ - 7880q⁹¹ - 4436q⁹² + 2375q⁹³ + 7113q⁹⁴ + 6235q⁹⁵ + 1650q⁹⁶ - 3585q⁹⁷ - 5873q⁹⁸ - 4498q⁹⁹ + 158q¹⁰⁰ + 4113q¹⁰¹ + 4445q¹⁰² + 2137q¹⁰³ - 1218q¹⁰⁴ - 3169q¹⁰⁵ - 3115q¹⁰⁶ - 668q¹⁰⁷ + 1742q¹⁰⁸ + 2251q¹⁰⁹ + 1440q¹¹⁰ - 158q¹¹¹ - 1222q¹¹² - 1544q¹¹³ - 507q¹¹⁴ + 593q¹¹⁵ + 818q¹¹⁶ + 627q¹¹⁷ + 46q¹¹⁸ - 340q¹¹⁹ - 605q¹²⁰ - 204q¹²¹ + 204q¹²² + 220q¹²³ + 208q¹²⁴ + 30q¹²⁵ - 62q¹²⁶ - 223q¹²⁷ - 63q¹²⁸ + 80q¹²⁹ + 37q¹³⁰ + 68q¹³¹ + 14q¹³² + 5q¹³³ - 85q¹³⁴ - 20q¹³⁵ + 32q¹³⁶ - 6q¹³⁷ + 25q¹³⁸ + 4q¹³⁹ + 12q¹⁴⁰ - 31q¹⁴¹ - 4q¹⁴² + 13q¹⁴³ - 10q¹⁴⁴ + 10q¹⁴⁵ - 2q¹⁴⁶ + 7q¹⁴⁷ - 9q¹⁴⁸ + 5q¹⁵⁰ - 6q¹⁵¹ + 4q¹⁵² - 2q¹⁵³ + 2q¹⁵⁴ - q¹⁵⁵ + q¹⁵⁷ - 2q¹⁵⁸ + q¹⁵⁹
7	- q^-70 + 2q^-69 + q^-68 - 2q^-67 - q^-66 - 2q^-65 + 2q^-64 - 2q^-62 + 8q^-61 + 6q^-60 - 4q^-59 - 6q^-58 - 14q^-57 - 2q^-56 + 4q^-55 - 6q^-54 + 22q^-53 + 26q^-52 + 11q^-51 - q^-50 - 44q^-49 - 33q^-48 - 17q^-47 - 34q^-46 + 27q^-45 + 65q^-44 + 73q^-43 + 83q^-42 - 27q^-41 - 69q^-40 - 83q^-39 - 156q^-38 - 76q^-37 + 6q^-36 + 109q^-35 + 260q^-34 + 176q^-33 + 94q^-32 + 3q^-31 - 253q^-30 - 302q^-29 - 323q^-28 - 222q^-27 + 164q^-26 + 319q^-25 + 467q^-24 + 524q^-23 + 173q^-22 - 90q^-21 - 493q^-20 - 824q^-19 - 582q^-18 - 344q^-17 + 168q^-16 + 815q^-15 + 918q^-14 + 973q^-13 + 485q^-12 - 423q^-11 - 896q^-10 - 1394q^-9 - 1279q^-8 - 426q^-7 + 288q^-6 + 1337q^-5 + 1839q^-4 + 1402q^-3 + 836q^-2 - 535q^-1 - 1742 - 2042q - 2072q² - 882q³ + 681q⁴ + 1801q⁵ + 2914q⁶ + 2491q⁷ + 1059q⁸ - 493q⁹ - 2603q¹⁰ - 3469q¹¹ - 3091q¹² - 1853q¹³ + 1039q¹⁴ + 3318q¹⁵ + 4440q¹⁶ + 4361q¹⁷ + 1756q¹⁸ - 1444q¹⁹ - 4438q²⁰ - 6443q²¹ - 5031q²² - 1701q²³ + 2682q²⁴ + 6953q²⁵ + 7818q²⁶ + 5722q²⁷ + 854q²⁸ - 5658q²⁹ - 9364q³⁰ - 9514q³¹ - 5339q³² + 2383q³³ + 8872q³⁴ + 12219q³⁵ + 10193q³⁶ + 2322q³⁷ - 6480q³⁸ - 13219q³⁹ - 14222q⁴⁰ - 7648q⁴¹ + 2269q⁴² + 12176q⁴³ + 16985q⁴⁴ + 12872q⁴⁵ + 2938q⁴⁶ - 9405q⁴⁷ - 17944q⁴⁸ - 17174q⁴⁹ - 8564q⁵⁰ + 5184q⁵¹ + 17270q⁵² + 20294q⁵³ + 13847q⁵⁴ - 268q⁵⁵ - 15106q⁵⁶ - 21951q⁵⁷ - 18488q⁵⁸ - 4936q⁵⁹ + 12033q⁶⁰ + 22420q⁶¹ + 22076q⁶² + 9833q⁶³ - 8322q⁶⁴ - 21836q⁶⁵ - 24789q⁶⁶ - 14238q⁶⁷ + 4632q⁶⁸ + 20649q⁶⁹ + 26487q⁷⁰ + 17871q⁷¹ - 1077q⁷² - 19111q⁷³ - 27598q⁷⁴ - 20759q⁷⁵ - 1859q⁷⁶ + 17653q⁷⁷ + 28170q⁷⁸ + 22870q⁷⁹ + 4185q⁸⁰ - 16500q⁸¹ - 28643q⁸² - 24446q⁸³ - 5748q⁸⁴ + 15940q⁸⁵ + 29252q⁸⁶ + 25707q⁸⁷ + 6713q⁸⁸ - 15978q⁸⁹ - 30265q⁹⁰ - 27039q⁹¹ - 7457q⁹² + 16497q⁹³ + 31846q⁹⁴ + 28806q⁹⁵ + 8351q⁹⁶ - 17115q⁹⁷ - 33734q⁹⁸ - 31135q⁹⁹ - 10013q¹⁰⁰ + 17123q¹⁰¹ + 35577q¹⁰² + 34080q¹⁰³ + 12651q¹⁰⁴ - 16061q¹⁰⁵ - 36609q¹⁰⁶ - 36921q¹⁰⁷ - 16293q¹⁰⁸ + 13260q¹⁰⁹ + 36099q¹¹⁰ + 39162q¹¹¹ + 20451q¹¹² - 8984q¹¹³ - 33577q¹¹⁴ - 39726q¹¹⁵ - 24199q¹¹⁶ + 3555q¹¹⁷ + 28907q¹¹⁸ + 38148q¹¹⁹ + 26703q¹²⁰ + 2046q¹²¹ - 22748q¹²² - 34318q¹²³ - 27132q¹²⁴ - 6705q¹²⁵ + 15912q¹²⁶ + 28569q¹²⁷ + 25394q¹²⁸ + 9805q¹²⁹ - 9529q¹³⁰ - 22019q¹³¹ - 21805q¹³² - 10831q¹³³ + 4444q¹³⁴ + 15480q¹³⁵ + 17118q¹³⁶ + 10196q¹³⁷ - 999q¹³⁸ - 9926q¹³⁹ - 12332q¹⁴⁰ - 8391q¹⁴¹ - 778q¹⁴² + 5760q¹⁴³ + 8069q¹⁴⁴ + 6115q¹⁴⁵ + 1390q¹⁴⁶ - 2971q¹⁴⁷ - 4819q¹⁴⁸ - 4030q¹⁴⁹ - 1266q¹⁵⁰ + 1401q¹⁵¹ + 2602q¹⁵² + 2354q¹⁵³ + 866q¹⁵⁴ - 584q¹⁵⁵ - 1263q¹⁵⁶ - 1239q¹⁵⁷ - 481q¹⁵⁸ + 238q¹⁵⁹ + 559q¹⁶⁰ + 599q¹⁶¹ + 196q¹⁶² - 140q¹⁶³ - 220q¹⁶⁴ - 242q¹⁶⁵ - 39q¹⁶⁶ + 72q¹⁶⁷ + 76q¹⁶⁸ + 122q¹⁶⁹ - 12q¹⁷⁰ - 76q¹⁷¹ - 40q¹⁷² - 53q¹⁷³ + 47q¹⁷⁴ + 49q¹⁷⁵ + 36q¹⁷⁷ - 16q¹⁷⁸ - 31q¹⁷⁹ - 18q¹⁸⁰ - 35q¹⁸¹ + 33q¹⁸² + 29q¹⁸³ - 10q¹⁸⁴ + 16q¹⁸⁵ - 5q¹⁸⁶ - 9q¹⁸⁷ - 3q¹⁸⁸ - 22q¹⁸⁹ + 14q¹⁹⁰ + 12q¹⁹¹ - 6q¹⁹² + 9q¹⁹³ - 6q¹⁹⁴ - 2q¹⁹⁵ + 3q¹⁹⁶ - 9q¹⁹⁷ + 4q¹⁹⁸ + 3q¹⁹⁹ - 2q²⁰⁰ + 4q²⁰¹ - 3q²⁰² - q²⁰³ + 2q²⁰⁴ - 2q²⁰⁵ + q²⁰⁶ - q²⁰⁸ + 2q²⁰⁹ - q²¹⁰

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	PD[Knot[10, 62]]
Out[2]=	PD[X[1, 4, 2, 5], X[3, 10, 4, 11], X[11, 19, 12, 18], X[5, 15, 6, 14], > X[7, 17, 8, 16], X[15, 7, 16, 6], X[17, 9, 18, 8], X[13, 1, 14, 20], > X[19, 13, 20, 12], X[9, 2, 10, 3]]
In[3]:=	GaussCode[Knot[10, 62]]
Out[3]=	GaussCode[-1, 10, -2, 1, -4, 6, -5, 7, -10, 2, -3, 9, -8, 4, -6, 5, -7, 3, -9, > 8]
In[4]:=	DTCode[Knot[10, 62]]
Out[4]=	DTCode[4, 10, 14, 16, 2, 18, 20, 6, 8, 12]
In[5]:=	br = BR[Knot[10, 62]]
Out[5]=	BR[3, {1, 1, 1, 1, -2, 1, 1, 1, -2, -2}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{3, 10}
In[7]:=	BraidIndex[Knot[10, 62]]
Out[7]=	3
In[8]:=	Show[DrawMorseLink[Knot[10, 62]]]

Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 62]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 4, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 62]][t]
Out[10]=	-4 3 6 8 2 3 4 9 + t - -- + -- - - - 8 t + 6 t - 3 t + t 3 2 t t t
In[11]:=	Conway[Knot[10, 62]][z]
Out[11]=	2 4 6 8 1 + 5 z + 8 z + 5 z + z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 62], Knot[11, NonAlternating, 76], Knot[11, NonAlternating, 78]}
In[13]:=	{KnotDet[Knot[10, 62]], KnotSignature[Knot[10, 62]]}
Out[13]=	{45, 4}
In[14]:=	Jones[Knot[10, 62]][q]
Out[14]=	1 2 3 4 5 6 7 8 9 2 - - - 3 q + 6 q - 6 q + 7 q - 7 q + 6 q - 4 q + 2 q - q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 62]}
In[16]:=	A2Invariant[Knot[10, 62]][q]
Out[16]=	-2 2 4 6 8 10 12 14 22 26 -q - q + q + 2 q + q + 3 q - q + 2 q - 2 q - q
In[17]:=	HOMFLYPT[Knot[10, 62]][a, z]
Out[17]=	2 2 2 4 4 4 6 6 6 8 -4 7 2 8 z 20 z 7 z 5 z 18 z 5 z z 7 z z z -- + -- - -- - ---- + ----- - ---- - ---- + ----- - ---- - -- + ---- - -- + -- 6 4 2 6 4 2 6 4 2 6 4 2 4 a a a a a a a a a a a a a
In[18]:=	Kauffman[Knot[10, 62]][a, z]
Out[18]=	2 2 2 2 4 7 2 z z z 6 z 5 z 2 z z 4 z 8 z 23 z -- + -- + -- - --- + -- - -- - --- - --- - --- - --- + ---- - ---- - ----- - 6 4 2 11 9 7 5 3 a 10 8 6 4 a a a a a a a a a a a a 2 3 3 3 3 3 3 4 4 4 10 z z 2 z 5 z 16 z 15 z 7 z 2 z 6 z 6 z > ----- + --- - ---- + ---- + ----- + ----- + ---- + ---- - ---- + ---- + 2 11 9 7 5 3 a 10 8 6 a a a a a a a a a 4 4 5 5 5 5 5 6 6 6 30 z 16 z 3 z 8 z 15 z 9 z 5 z 4 z 7 z 21 z > ----- + ----- + ---- - ---- - ----- - ---- - ---- + ---- - ---- - ----- - 4 2 9 7 5 3 a 8 6 4 a a a a a a a a a 6 7 7 7 7 8 8 8 9 9 10 z 4 z 2 z z z 3 z 5 z 2 z z z > ----- + ---- + ---- - -- + -- + ---- + ---- + ---- + -- + -- 2 7 5 3 a 6 4 2 5 3 a a a a a a a a a
In[19]:=	{Vassiliev[2][Knot[10, 62]], Vassiliev[3][Knot[10, 62]]}
Out[19]=	{5, 9}
In[20]:=	Kh[Knot[10, 62]][q, t]
Out[20]=	3 3 5 1 1 q 2 q q 5 7 7 2 4 q + 3 q + ----- + ---- + -- + --- + -- + 3 q t + 3 q t + 4 q t + 3 3 2 2 t t q t q t t 9 2 9 3 11 3 11 4 13 4 13 5 15 5 > 3 q t + 3 q t + 4 q t + 3 q t + 3 q t + q t + 3 q t + 15 6 17 6 19 7 > q t + q t + q t
In[21]:=	ColouredJones[Knot[10, 62], 2][q]
Out[21]=	-5 2 -3 6 5 2 3 4 5 6 -7 + q - -- - q + -- - - + 15 q - 3 q - 18 q + 23 q + 3 q - 29 q + 4 2 q q q 7 8 9 10 11 12 13 14 > 24 q + 11 q - 36 q + 19 q + 19 q - 34 q + 12 q + 20 q - 15 16 17 18 19 20 21 22 23 > 27 q + 7 q + 13 q - 16 q + 5 q + 5 q - 7 q + 3 q + q - 24 25 > 2 q + q

The Alternating Knot 1062

The Alternating Knot 10₆₂