The Alternating Knot 9₂₉

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Acknowledgement

KnotPlot

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

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

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

Minimum Braid Representative:

Length is 9, width is 4
Braid index is 4

A Morse Link Presentation:

3D Invariants:

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

Reversible 2 3 3 / 4--7 1

Alexander Polynomial: t^-3 - 5t^-2 + 12t^-1 - 15 + 12t - 5t² + t³

Conway Polynomial: 1 + z² + z⁴ + z⁶

Other knots with the same Alexander/Conway Polynomial: {9₂₈, 10₁₆₃, K11n87, ...}

Determinant and Signature: {51, -2}

Jones Polynomial: - q^-6 + 3q^-5 - 6q^-4 + 8q^-3 - 8q^-2 + 9q^-1 - 7 + 5q - 3q² + q³

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

A2 (sl(3)) Invariant: - q^-18 + q^-16 - 2q^-14 - q^-12 + 2q^-10 + 4q^-6 + q^-2 - 2q² + q⁴ - q⁶ + q¹⁰

HOMFLY-PT Polynomial: a^-2 + a^-2z² - 3 - 5z² - 2z⁴ + 5a² + 7a²z² + 4a²z⁴ + a²z⁶ - 2a⁴ - 2a⁴z² - a⁴z⁴

Kauffman Polynomial: - a^-2 + 3a^-2z² - 3a^-2z⁴ + a^-2z⁶ - a^-1z + 9a^-1z³ - 10a^-1z⁵ + 3a^-1z⁷ - 3 + 12z² - 11z⁴ - z⁶ + 2z⁸ - az + 14az³ - 24az⁵ + 9az⁷ - 5a² + 17a²z² - 24a²z⁴ + 6a²z⁶ + 2a²z⁸ + 2a³z - a³z³ - 8a³z⁵ + 6a³z⁷ - 2a⁴ + 8a⁴z² - 13a⁴z⁴ + 8a⁴z⁶ + 2a⁵z - 5a⁵z³ + 6a⁵z⁵ + 3a⁶z⁴ + a⁷z³

V₂ and V₃, the type 2 and 3 Vassiliev invariants: {1, -2}

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=-2 is the signature of 9₂₉. 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

j = 7 1

j = 5 2

j = 3 3 1

j = 1 4 2

j = -1 5 3

j = -3 4 5

j = -5 4 4

j = -7 2 4

j = -9 1 4

j = -11 2

j = -13 1

n Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q^-17 - 3q^-16 + 3q^-15 + 4q^-14 - 16q^-13 + 15q^-12 + 12q^-11 - 42q^-10 + 29q^-9 + 27q^-8 - 65q^-7 + 32q^-6 + 43q^-5 - 71q^-4 + 20q^-3 + 51q^-2 - 60q^-1 + 4 + 46q - 37q² - 8q³ + 30q⁴ - 13q⁵ - 9q⁶ + 11q⁷ - q⁸ - 3q⁹ + q¹⁰

3 - q^-33 + 3q^-32 - 3q^-31 - q^-30 + 4q^-29 + 3q^-28 - 12q^-27 - 3q^-26 + 29q^-25 + 2q^-24 - 58q^-23 - 2q^-22 + 94q^-21 + 16q^-20 - 144q^-19 - 37q^-18 + 190q^-17 + 69q^-16 - 222q^-15 - 113q^-14 + 243q^-13 + 147q^-12 - 232q^-11 - 190q^-10 + 222q^-9 + 206q^-8 - 180q^-7 - 231q^-6 + 149q^-5 + 230q^-4 - 96q^-3 - 237q^-2 + 57q^-1 + 220 - 8q - 199q² - 32q³ + 165q⁴ + 62q⁵ - 121q⁶ - 78q⁷ + 76q⁸ + 78q⁹ - 36q¹⁰ - 64q¹¹ + 8q¹² + 42q¹³ + 7q¹⁴ - 23q¹⁵ - 9q¹⁶ + 9q¹⁷ + 5q¹⁸ - q¹⁹ - 3q²⁰ + q²¹

4 q^-54 - 3q^-53 + 3q^-52 + q^-51 - 7q^-50 + 9q^-49 - 6q^-48 + 9q^-47 - 8q^-46 - 30q^-45 + 48q^-44 + 8q^-43 + 13q^-42 - 67q^-41 - 110q^-40 + 156q^-39 + 124q^-38 + 49q^-37 - 257q^-36 - 364q^-35 + 307q^-34 + 444q^-33 + 260q^-32 - 522q^-31 - 889q^-30 + 298q^-29 + 862q^-28 + 743q^-27 - 608q^-26 - 1495q^-25 + 10q^-24 + 1060q^-23 + 1284q^-22 - 383q^-21 - 1830q^-20 - 375q^-19 + 909q^-18 + 1583q^-17 - 18q^-16 - 1782q^-15 - 631q^-14 + 552q^-13 + 1593q^-12 + 314q^-11 - 1498q^-10 - 747q^-9 + 153q^-8 + 1429q^-7 + 588q^-6 - 1091q^-5 - 785q^-4 - 258q^-3 + 1139q^-2 + 809q^-1 - 585 - 713q - 625q² + 697q³ + 861q⁴ - 53q⁵ - 440q⁶ - 792q⁷ + 177q⁸ + 632q⁹ + 294q¹⁰ - 40q¹¹ - 630q¹² - 179q¹³ + 231q¹⁴ + 300q¹⁵ + 221q¹⁶ - 275q¹⁷ - 210q¹⁸ - 46q¹⁹ + 104q²⁰ + 203q²¹ - 27q²² - 72q²³ - 78q²⁴ - 18q²⁵ + 73q²⁶ + 19q²⁷ + 4q²⁸ - 21q²⁹ - 19q³⁰ + 9q³¹ + 3q³² + 5q³³ - q³⁴ - 3q³⁵ + q³⁶

5 - q^-80 + 3q^-79 - 3q^-78 - q^-77 + 7q^-76 - 6q^-75 - 6q^-74 + 9q^-73 + 2q^-72 + 3q^-71 + 4q^-70 - 28q^-69 - 28q^-68 + 37q^-67 + 71q^-66 + 33q^-65 - 71q^-64 - 172q^-63 - 98q^-62 + 170q^-61 + 392q^-60 + 220q^-59 - 323q^-58 - 764q^-57 - 511q^-56 + 490q^-55 + 1371q^-54 + 1066q^-53 - 579q^-52 - 2219q^-51 - 1978q^-50 + 473q^-49 + 3159q^-48 + 3291q^-47 + 42q^-46 - 4076q^-45 - 4922q^-44 - 964q^-43 + 4672q^-42 + 6628q^-41 + 2348q^-40 - 4816q^-39 - 8171q^-38 - 3979q^-37 + 4462q^-36 + 9284q^-35 + 5569q^-34 - 3624q^-33 - 9832q^-32 - 6985q^-31 + 2584q^-30 + 9875q^-29 + 7895q^-28 - 1445q^-27 - 9411q^-26 - 8492q^-25 + 451q^-24 + 8763q^-23 + 8574q^-22 + 434q^-21 - 7887q^-20 - 8550q^-19 - 1117q^-18 + 7044q^-17 + 8239q^-16 + 1787q^-15 - 6080q^-14 - 8012q^-13 - 2392q^-12 + 5147q^-11 + 7581q^-10 + 3081q^-9 - 4013q^-8 - 7193q^-7 - 3748q^-6 + 2836q^-5 + 6521q^-4 + 4371q^-3 - 1432q^-2 - 5707q^-1 - 4825 + 62q + 4547q² + 4966q³ + 1288q⁴ - 3172q⁵ - 4724q⁶ - 2353q⁷ + 1672q⁸ + 4016q⁹ + 3013q¹⁰ - 221q¹¹ - 2961q¹² - 3154q¹³ - 939q¹⁴ + 1718q¹⁵ + 2778q¹⁶ + 1651q¹⁷ - 532q¹⁸ - 2026q¹⁹ - 1852q²⁰ - 372q²¹ + 1140q²² + 1603q²³ + 854q²⁴ - 342q²⁵ - 1087q²⁶ - 946q²⁷ - 187q²⁸ + 547q²⁹ + 745q³⁰ + 397q³¹ - 119q³² - 436q³³ - 393q³⁴ - 97q³⁵ + 183q³⁶ + 254q³⁷ + 141q³⁸ - 14q³⁹ - 121q⁴⁰ - 116q⁴¹ - 30q⁴² + 40q⁴³ + 49q⁴⁴ + 31q⁴⁵ + 6q⁴⁶ - 24q⁴⁷ - 17q⁴⁸ - q⁴⁹ + 3q⁵⁰ + 3q⁵¹ + 5q⁵² - q⁵³ - 3q⁵⁴ + q⁵⁵

6 q^-111 - 3q^-110 + 3q^-109 + q^-108 - 7q^-107 + 6q^-106 + 3q^-105 + 3q^-104 - 20q^-103 + 3q^-102 + 23q^-101 - 18q^-100 + 24q^-99 + 8q^-98 - 32q^-97 - 81q^-96 + 12q^-95 + 137q^-94 + 43q^-93 + 68q^-92 - 77q^-91 - 294q^-90 - 337q^-89 + 117q^-88 + 695q^-87 + 558q^-86 + 282q^-85 - 611q^-84 - 1551q^-83 - 1472q^-82 + 272q^-81 + 2617q^-80 + 2954q^-79 + 1708q^-78 - 1818q^-77 - 5525q^-76 - 5801q^-75 - 987q^-74 + 6473q^-73 + 9869q^-72 + 7705q^-71 - 1642q^-70 - 12912q^-69 - 16791q^-68 - 8177q^-67 + 9222q^-66 + 21648q^-65 + 22233q^-64 + 5635q^-63 - 19131q^-62 - 33676q^-61 - 25265q^-60 + 3796q^-59 + 31684q^-58 + 42812q^-57 + 23543q^-56 - 15894q^-55 - 47621q^-54 - 47747q^-53 - 12630q^-52 + 30939q^-51 + 58777q^-50 + 45690q^-49 - 1443q^-48 - 49510q^-47 - 63985q^-46 - 32354q^-45 + 19047q^-44 + 61882q^-43 + 60548q^-42 + 15693q^-41 - 40453q^-40 - 67352q^-39 - 44936q^-38 + 4746q^-37 + 54687q^-36 + 63623q^-35 + 26676q^-34 - 28849q^-33 - 61616q^-32 - 48015q^-31 - 5020q^-30 + 44895q^-29 + 59547q^-28 + 30914q^-27 - 19825q^-26 - 53611q^-25 - 46555q^-24 - 10914q^-23 + 36146q^-22 + 54119q^-21 + 33152q^-20 - 12014q^-19 - 45769q^-18 - 45092q^-17 - 17057q^-16 + 26735q^-15 + 48571q^-14 + 36528q^-13 - 2035q^-12 - 36056q^-11 - 43540q^-10 - 25216q^-9 + 13842q^-8 + 40200q^-7 + 39573q^-6 + 10767q^-5 - 21762q^-4 - 38225q^-3 - 32653q^-2 - 2422q^-1 + 26018 + 37520q + 22509q² - 3472q³ - 25683q⁴ - 33580q⁵ - 17140q⁶ + 7049q⁷ + 26530q⁸ + 26434q⁹ + 12872q¹⁰ - 7497q¹¹ - 24071q¹² - 22746q¹³ - 9630q¹⁴ + 9107q¹⁵ + 18793q¹⁶ + 19072q¹⁷ + 8110q¹⁸ - 7864q¹⁹ - 16099q²⁰ - 15491q²¹ - 5335q²² + 4715q²³ + 12976q²⁴ + 12713q²⁵ + 4702q²⁶ - 3787q²⁷ - 9751q²⁸ - 8939q²⁹ - 5071q³⁰ + 2474q³¹ + 7131q³² + 6942q³³ + 3704q³⁴ - 1195q³⁵ - 4131q³⁶ - 5620q³⁷ - 2883q³⁸ + 412q³⁹ + 2741q⁴⁰ + 3431q⁴¹ + 2230q⁴² + 492q⁴³ - 1870q⁴⁴ - 2121q⁴⁵ - 1558q⁴⁶ - 376q⁴⁷ + 702q⁴⁸ + 1242q⁴⁹ + 1217q⁵⁰ + 207q⁵¹ - 271q⁵² - 647q⁵³ - 581q⁵⁴ - 324q⁵⁵ + 74q⁵⁶ + 393q⁵⁷ + 241q⁵⁸ + 180q⁵⁹ - 6q⁶⁰ - 101q⁶¹ - 164q⁶² - 90q⁶³ + 24q⁶⁴ + 18q⁶⁵ + 54q⁶⁶ + 32q⁶⁷ + 18q⁶⁸ - 22q⁶⁹ - 20q⁷⁰ + q⁷¹ - 7q⁷² + 3q⁷³ + 3q⁷⁴ + 5q⁷⁵ - q⁷⁶ - 3q⁷⁷ + q⁷⁸

7 - q^-147 + 3q^-146 - 3q^-145 - q^-144 + 7q^-143 - 6q^-142 - 3q^-141 + 8q^-139 + 15q^-138 - 29q^-137 - 9q^-136 + 22q^-135 - 10q^-134 + 11q^-133 + 12q^-132 + 27q^-131 + 9q^-130 - 128q^-129 - 71q^-128 + 64q^-127 + 102q^-126 + 201q^-125 + 97q^-124 - 74q^-123 - 283q^-122 - 623q^-121 - 325q^-120 + 359q^-119 + 983q^-118 + 1386q^-117 + 627q^-116 - 809q^-115 - 2289q^-114 - 3180q^-113 - 1701q^-112 + 1609q^-111 + 5121q^-110 + 6942q^-109 + 4013q^-108 - 2628q^-107 - 9910q^-106 - 13916q^-105 - 9348q^-104 + 2775q^-103 + 17297q^-102 + 26152q^-101 + 20036q^-100 - 477q^-99 - 26780q^-98 - 44848q^-97 - 39001q^-96 - 7869q^-95 + 36131q^-94 + 70437q^-93 + 69152q^-92 + 26537q^-91 - 41494q^-90 - 100865q^-89 - 111331q^-88 - 59298q^-87 + 36969q^-86 + 130712q^-85 + 163644q^-84 + 108723q^-83 - 16964q^-82 - 153536q^-81 - 220234q^-80 - 172351q^-79 - 21770q^-78 + 161037q^-77 + 272303q^-76 + 244798q^-75 + 78719q^-74 - 148664q^-73 - 311118q^-72 - 316152q^-71 - 147897q^-70 + 115480q^-69 + 329393q^-68 + 376410q^-67 + 220557q^-66 - 65768q^-65 - 325264q^-64 - 418086q^-63 - 286250q^-62 + 8157q^-61 + 301531q^-60 + 437136q^-59 + 336921q^-58 + 48726q^-57 - 264871q^-56 - 435963q^-55 - 368829q^-54 - 96262q^-53 + 223868q^-52 + 419089q^-51 + 381977q^-50 + 131056q^-49 - 184955q^-48 - 394208q^-47 - 381130q^-46 - 152125q^-45 + 153100q^-44 + 366896q^-43 + 371166q^-42 + 163029q^-41 - 128398q^-40 - 341928q^-39 - 358376q^-38 - 167583q^-37 + 109928q^-36 + 320128q^-35 + 345916q^-34 + 171077q^-33 - 93702q^-32 - 301601q^-31 - 336443q^-30 - 176163q^-29 + 76834q^-28 + 283380q^-27 + 329272q^-26 + 185744q^-25 - 55747q^-24 - 263598q^-23 - 323576q^-22 - 199228q^-21 + 29174q^-20 + 238621q^-19 + 316237q^-18 + 216215q^-17 + 4237q^-16 - 206976q^-15 - 304972q^-14 - 233495q^-13 - 42882q^-12 + 166696q^-11 + 285755q^-10 + 248069q^-9 + 85277q^-8 - 118052q^-7 - 256846q^-6 - 255411q^-5 - 126646q^-4 + 62457q^-3 + 215796q^-2 + 251327q^-1 + 162847 - 3501q - 163734q² - 232727q³ - 187532q⁴ - 53039q⁵ + 102902q⁶ + 198060q⁷ + 196107q⁸ + 100893q⁹ - 39161q¹⁰ - 149479q¹¹ - 185483q¹² - 132996q¹³ - 20006q¹⁴ + 91657q¹⁵ + 155949q¹⁶ + 144984q¹⁷ + 66891q¹⁸ - 32813q¹⁹ - 112112q²⁰ - 135477q²¹ - 94844q²² - 18257q²³ + 61475q²⁴ + 107869q²⁵ + 101325q²⁶ + 53920q²⁷ - 13652q²⁸ - 69424q²⁹ - 88369q³⁰ - 69988q³¹ - 22796q³² + 29443q³³ + 62149q³⁴ + 67283q³⁵ + 43079q³⁶ + 3323q³⁷ - 31706q³⁸ - 51133q³⁹ - 46602q⁴⁰ - 23262q⁴¹ + 5120q⁴² + 29101q⁴³ + 37723q⁴⁴ + 29687q⁴⁵ + 12056q⁴⁶ - 8982q⁴⁷ - 22994q⁴⁸ - 25401q⁴⁹ - 18514q⁵⁰ - 4633q⁵¹ + 8580q⁵² + 15988q⁵³ + 16875q⁵⁴ + 10330q⁵⁵ + 1127q⁵⁶ - 6485q⁵⁷ - 10883q⁵⁸ - 9858q⁵⁹ - 5479q⁶⁰ - 155q⁶¹ + 4805q⁶² + 6556q⁶³ + 5492q⁶⁴ + 2878q⁶⁵ - 617q⁶⁶ - 2798q⁶⁷ - 3567q⁶⁸ - 3091q⁶⁹ - 1225q⁷⁰ + 493q⁷¹ + 1583q⁷² + 1944q⁷³ + 1308q⁷⁴ + 540q⁷⁵ - 221q⁷⁶ - 884q⁷⁷ - 875q⁷⁸ - 602q⁷⁹ - 187q⁸⁰ + 226q⁸¹ + 314q⁸² + 342q⁸³ + 268q⁸⁴ + 42q⁸⁵ - 89q⁸⁶ - 152q⁸⁷ - 133q⁸⁸ - 35q⁸⁹ - 20q⁹⁰ + 19q⁹¹ + 57q⁹² + 37q⁹³ + 19q⁹⁴ - 10q⁹⁵ - 18q⁹⁶ - 2q⁹⁷ - 5q⁹⁸ - 7q⁹⁹ + 3q¹⁰⁰ + 3q¹⁰¹ + 5q¹⁰² - q¹⁰³ - 3q¹⁰⁴ + q¹⁰⁵

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[9, 29]]

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

In[3]:=
GaussCode[Knot[9, 29]]

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

In[4]:=
DTCode[Knot[9, 29]]

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

In[5]:=
br = BR[Knot[9, 29]]

Out[5]=
BR[4, {1, -2, -2, 3, -2, 1, -2, 3, -2}]

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

Out[6]=
{4, 9}

In[7]:=
BraidIndex[Knot[9, 29]]

Out[7]=
4

In[8]:=
Show[DrawMorseLink[Knot[9, 29]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[9, 29]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

Out[9]=
{Reversible, 2, 3, 3, {4, 7}, 1}

In[10]:=
alex = Alexander[Knot[9, 29]][t]

Out[10]=
-3 5 12 2 3 -15 + t - -- + -- + 12 t - 5 t + t 2 t t

In[11]:=
Conway[Knot[9, 29]][z]

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

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

Out[12]=
{Knot[9, 28], Knot[9, 29], Knot[10, 163], Knot[11, NonAlternating, 87]}

In[13]:=
{KnotDet[Knot[9, 29]], KnotSignature[Knot[9, 29]]}

Out[13]=
{51, -2}

In[14]:=
Jones[Knot[9, 29]][q]

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

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

Out[15]=
{Knot[9, 29]}

In[16]:=
A2Invariant[Knot[9, 29]][q]

Out[16]=
-18 -16 2 -12 2 4 -2 2 4 6 10 -q + q - --- - q + --- + -- + q - 2 q + q - q + q 14 10 6 q q q

In[17]:=
HOMFLYPT[Knot[9, 29]][a, z]

Out[17]=
2 -2 2 4 2 z 2 2 4 2 4 2 4 -3 + a + 5 a - 2 a - 5 z + -- + 7 a z - 2 a z - 2 z + 4 a z - 2 a 4 4 2 6 > a z + a z

In[18]:=
Kauffman[Knot[9, 29]][a, z]

Out[18]=
2 -2 2 4 z 3 5 2 3 z 2 2 -3 - a - 5 a - 2 a - - - a z + 2 a z + 2 a z + 12 z + ---- + 17 a z + a 2 a 3 4 4 2 9 z 3 3 3 5 3 7 3 4 3 z > 8 a z + ---- + 14 a z - a z - 5 a z + a z - 11 z - ---- - a 2 a 5 2 4 4 4 6 4 10 z 5 3 5 5 5 6 > 24 a z - 13 a z + 3 a z - ----- - 24 a z - 8 a z + 6 a z - z + a 6 7 z 2 6 4 6 3 z 7 3 7 8 2 8 > -- + 6 a z + 8 a z + ---- + 9 a z + 6 a z + 2 z + 2 a z 2 a a

In[19]:=
{Vassiliev[2][Knot[9, 29]], Vassiliev[3][Knot[9, 29]]}

Out[19]=
{1, -2}

In[20]:=
Kh[Knot[9, 29]][q, t]

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

In[21]:=
ColouredJones[Knot[9, 29], 2][q]

Out[21]=
-17 3 3 4 16 15 12 42 29 27 65 32 43 4 + q - --- + --- + --- - --- + --- + --- - --- + -- + -- - -- + -- + -- - 16 15 14 13 12 11 10 9 8 7 6 5 q q q q q q q q q q q q 71 20 51 60 2 3 4 5 6 7 > -- + -- + -- - -- + 46 q - 37 q - 8 q + 30 q - 13 q - 9 q + 11 q - 4 3 2 q q q q 8 9 10 > q - 3 q + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 9₂₉

9₂₈
9₃₀

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-17 - 3q^-16 + 3q^-15 + 4q^-14 - 16q^-13 + 15q^-12 + 12q^-11 - 42q^-10 + 29q^-9 + 27q^-8 - 65q^-7 + 32q^-6 + 43q^-5 - 71q^-4 + 20q^-3 + 51q^-2 - 60q^-1 + 4 + 46q - 37q² - 8q³ + 30q⁴ - 13q⁵ - 9q⁶ + 11q⁷ - q⁸ - 3q⁹ + q¹⁰
3	- q^-33 + 3q^-32 - 3q^-31 - q^-30 + 4q^-29 + 3q^-28 - 12q^-27 - 3q^-26 + 29q^-25 + 2q^-24 - 58q^-23 - 2q^-22 + 94q^-21 + 16q^-20 - 144q^-19 - 37q^-18 + 190q^-17 + 69q^-16 - 222q^-15 - 113q^-14 + 243q^-13 + 147q^-12 - 232q^-11 - 190q^-10 + 222q^-9 + 206q^-8 - 180q^-7 - 231q^-6 + 149q^-5 + 230q^-4 - 96q^-3 - 237q^-2 + 57q^-1 + 220 - 8q - 199q² - 32q³ + 165q⁴ + 62q⁵ - 121q⁶ - 78q⁷ + 76q⁸ + 78q⁹ - 36q¹⁰ - 64q¹¹ + 8q¹² + 42q¹³ + 7q¹⁴ - 23q¹⁵ - 9q¹⁶ + 9q¹⁷ + 5q¹⁸ - q¹⁹ - 3q²⁰ + q²¹
4	q^-54 - 3q^-53 + 3q^-52 + q^-51 - 7q^-50 + 9q^-49 - 6q^-48 + 9q^-47 - 8q^-46 - 30q^-45 + 48q^-44 + 8q^-43 + 13q^-42 - 67q^-41 - 110q^-40 + 156q^-39 + 124q^-38 + 49q^-37 - 257q^-36 - 364q^-35 + 307q^-34 + 444q^-33 + 260q^-32 - 522q^-31 - 889q^-30 + 298q^-29 + 862q^-28 + 743q^-27 - 608q^-26 - 1495q^-25 + 10q^-24 + 1060q^-23 + 1284q^-22 - 383q^-21 - 1830q^-20 - 375q^-19 + 909q^-18 + 1583q^-17 - 18q^-16 - 1782q^-15 - 631q^-14 + 552q^-13 + 1593q^-12 + 314q^-11 - 1498q^-10 - 747q^-9 + 153q^-8 + 1429q^-7 + 588q^-6 - 1091q^-5 - 785q^-4 - 258q^-3 + 1139q^-2 + 809q^-1 - 585 - 713q - 625q² + 697q³ + 861q⁴ - 53q⁵ - 440q⁶ - 792q⁷ + 177q⁸ + 632q⁹ + 294q¹⁰ - 40q¹¹ - 630q¹² - 179q¹³ + 231q¹⁴ + 300q¹⁵ + 221q¹⁶ - 275q¹⁷ - 210q¹⁸ - 46q¹⁹ + 104q²⁰ + 203q²¹ - 27q²² - 72q²³ - 78q²⁴ - 18q²⁵ + 73q²⁶ + 19q²⁷ + 4q²⁸ - 21q²⁹ - 19q³⁰ + 9q³¹ + 3q³² + 5q³³ - q³⁴ - 3q³⁵ + q³⁶
5	- q^-80 + 3q^-79 - 3q^-78 - q^-77 + 7q^-76 - 6q^-75 - 6q^-74 + 9q^-73 + 2q^-72 + 3q^-71 + 4q^-70 - 28q^-69 - 28q^-68 + 37q^-67 + 71q^-66 + 33q^-65 - 71q^-64 - 172q^-63 - 98q^-62 + 170q^-61 + 392q^-60 + 220q^-59 - 323q^-58 - 764q^-57 - 511q^-56 + 490q^-55 + 1371q^-54 + 1066q^-53 - 579q^-52 - 2219q^-51 - 1978q^-50 + 473q^-49 + 3159q^-48 + 3291q^-47 + 42q^-46 - 4076q^-45 - 4922q^-44 - 964q^-43 + 4672q^-42 + 6628q^-41 + 2348q^-40 - 4816q^-39 - 8171q^-38 - 3979q^-37 + 4462q^-36 + 9284q^-35 + 5569q^-34 - 3624q^-33 - 9832q^-32 - 6985q^-31 + 2584q^-30 + 9875q^-29 + 7895q^-28 - 1445q^-27 - 9411q^-26 - 8492q^-25 + 451q^-24 + 8763q^-23 + 8574q^-22 + 434q^-21 - 7887q^-20 - 8550q^-19 - 1117q^-18 + 7044q^-17 + 8239q^-16 + 1787q^-15 - 6080q^-14 - 8012q^-13 - 2392q^-12 + 5147q^-11 + 7581q^-10 + 3081q^-9 - 4013q^-8 - 7193q^-7 - 3748q^-6 + 2836q^-5 + 6521q^-4 + 4371q^-3 - 1432q^-2 - 5707q^-1 - 4825 + 62q + 4547q² + 4966q³ + 1288q⁴ - 3172q⁵ - 4724q⁶ - 2353q⁷ + 1672q⁸ + 4016q⁹ + 3013q¹⁰ - 221q¹¹ - 2961q¹² - 3154q¹³ - 939q¹⁴ + 1718q¹⁵ + 2778q¹⁶ + 1651q¹⁷ - 532q¹⁸ - 2026q¹⁹ - 1852q²⁰ - 372q²¹ + 1140q²² + 1603q²³ + 854q²⁴ - 342q²⁵ - 1087q²⁶ - 946q²⁷ - 187q²⁸ + 547q²⁹ + 745q³⁰ + 397q³¹ - 119q³² - 436q³³ - 393q³⁴ - 97q³⁵ + 183q³⁶ + 254q³⁷ + 141q³⁸ - 14q³⁹ - 121q⁴⁰ - 116q⁴¹ - 30q⁴² + 40q⁴³ + 49q⁴⁴ + 31q⁴⁵ + 6q⁴⁶ - 24q⁴⁷ - 17q⁴⁸ - q⁴⁹ + 3q⁵⁰ + 3q⁵¹ + 5q⁵² - q⁵³ - 3q⁵⁴ + q⁵⁵
6	q^-111 - 3q^-110 + 3q^-109 + q^-108 - 7q^-107 + 6q^-106 + 3q^-105 + 3q^-104 - 20q^-103 + 3q^-102 + 23q^-101 - 18q^-100 + 24q^-99 + 8q^-98 - 32q^-97 - 81q^-96 + 12q^-95 + 137q^-94 + 43q^-93 + 68q^-92 - 77q^-91 - 294q^-90 - 337q^-89 + 117q^-88 + 695q^-87 + 558q^-86 + 282q^-85 - 611q^-84 - 1551q^-83 - 1472q^-82 + 272q^-81 + 2617q^-80 + 2954q^-79 + 1708q^-78 - 1818q^-77 - 5525q^-76 - 5801q^-75 - 987q^-74 + 6473q^-73 + 9869q^-72 + 7705q^-71 - 1642q^-70 - 12912q^-69 - 16791q^-68 - 8177q^-67 + 9222q^-66 + 21648q^-65 + 22233q^-64 + 5635q^-63 - 19131q^-62 - 33676q^-61 - 25265q^-60 + 3796q^-59 + 31684q^-58 + 42812q^-57 + 23543q^-56 - 15894q^-55 - 47621q^-54 - 47747q^-53 - 12630q^-52 + 30939q^-51 + 58777q^-50 + 45690q^-49 - 1443q^-48 - 49510q^-47 - 63985q^-46 - 32354q^-45 + 19047q^-44 + 61882q^-43 + 60548q^-42 + 15693q^-41 - 40453q^-40 - 67352q^-39 - 44936q^-38 + 4746q^-37 + 54687q^-36 + 63623q^-35 + 26676q^-34 - 28849q^-33 - 61616q^-32 - 48015q^-31 - 5020q^-30 + 44895q^-29 + 59547q^-28 + 30914q^-27 - 19825q^-26 - 53611q^-25 - 46555q^-24 - 10914q^-23 + 36146q^-22 + 54119q^-21 + 33152q^-20 - 12014q^-19 - 45769q^-18 - 45092q^-17 - 17057q^-16 + 26735q^-15 + 48571q^-14 + 36528q^-13 - 2035q^-12 - 36056q^-11 - 43540q^-10 - 25216q^-9 + 13842q^-8 + 40200q^-7 + 39573q^-6 + 10767q^-5 - 21762q^-4 - 38225q^-3 - 32653q^-2 - 2422q^-1 + 26018 + 37520q + 22509q² - 3472q³ - 25683q⁴ - 33580q⁵ - 17140q⁶ + 7049q⁷ + 26530q⁸ + 26434q⁹ + 12872q¹⁰ - 7497q¹¹ - 24071q¹² - 22746q¹³ - 9630q¹⁴ + 9107q¹⁵ + 18793q¹⁶ + 19072q¹⁷ + 8110q¹⁸ - 7864q¹⁹ - 16099q²⁰ - 15491q²¹ - 5335q²² + 4715q²³ + 12976q²⁴ + 12713q²⁵ + 4702q²⁶ - 3787q²⁷ - 9751q²⁸ - 8939q²⁹ - 5071q³⁰ + 2474q³¹ + 7131q³² + 6942q³³ + 3704q³⁴ - 1195q³⁵ - 4131q³⁶ - 5620q³⁷ - 2883q³⁸ + 412q³⁹ + 2741q⁴⁰ + 3431q⁴¹ + 2230q⁴² + 492q⁴³ - 1870q⁴⁴ - 2121q⁴⁵ - 1558q⁴⁶ - 376q⁴⁷ + 702q⁴⁸ + 1242q⁴⁹ + 1217q⁵⁰ + 207q⁵¹ - 271q⁵² - 647q⁵³ - 581q⁵⁴ - 324q⁵⁵ + 74q⁵⁶ + 393q⁵⁷ + 241q⁵⁸ + 180q⁵⁹ - 6q⁶⁰ - 101q⁶¹ - 164q⁶² - 90q⁶³ + 24q⁶⁴ + 18q⁶⁵ + 54q⁶⁶ + 32q⁶⁷ + 18q⁶⁸ - 22q⁶⁹ - 20q⁷⁰ + q⁷¹ - 7q⁷² + 3q⁷³ + 3q⁷⁴ + 5q⁷⁵ - q⁷⁶ - 3q⁷⁷ + q⁷⁸
7	- q^-147 + 3q^-146 - 3q^-145 - q^-144 + 7q^-143 - 6q^-142 - 3q^-141 + 8q^-139 + 15q^-138 - 29q^-137 - 9q^-136 + 22q^-135 - 10q^-134 + 11q^-133 + 12q^-132 + 27q^-131 + 9q^-130 - 128q^-129 - 71q^-128 + 64q^-127 + 102q^-126 + 201q^-125 + 97q^-124 - 74q^-123 - 283q^-122 - 623q^-121 - 325q^-120 + 359q^-119 + 983q^-118 + 1386q^-117 + 627q^-116 - 809q^-115 - 2289q^-114 - 3180q^-113 - 1701q^-112 + 1609q^-111 + 5121q^-110 + 6942q^-109 + 4013q^-108 - 2628q^-107 - 9910q^-106 - 13916q^-105 - 9348q^-104 + 2775q^-103 + 17297q^-102 + 26152q^-101 + 20036q^-100 - 477q^-99 - 26780q^-98 - 44848q^-97 - 39001q^-96 - 7869q^-95 + 36131q^-94 + 70437q^-93 + 69152q^-92 + 26537q^-91 - 41494q^-90 - 100865q^-89 - 111331q^-88 - 59298q^-87 + 36969q^-86 + 130712q^-85 + 163644q^-84 + 108723q^-83 - 16964q^-82 - 153536q^-81 - 220234q^-80 - 172351q^-79 - 21770q^-78 + 161037q^-77 + 272303q^-76 + 244798q^-75 + 78719q^-74 - 148664q^-73 - 311118q^-72 - 316152q^-71 - 147897q^-70 + 115480q^-69 + 329393q^-68 + 376410q^-67 + 220557q^-66 - 65768q^-65 - 325264q^-64 - 418086q^-63 - 286250q^-62 + 8157q^-61 + 301531q^-60 + 437136q^-59 + 336921q^-58 + 48726q^-57 - 264871q^-56 - 435963q^-55 - 368829q^-54 - 96262q^-53 + 223868q^-52 + 419089q^-51 + 381977q^-50 + 131056q^-49 - 184955q^-48 - 394208q^-47 - 381130q^-46 - 152125q^-45 + 153100q^-44 + 366896q^-43 + 371166q^-42 + 163029q^-41 - 128398q^-40 - 341928q^-39 - 358376q^-38 - 167583q^-37 + 109928q^-36 + 320128q^-35 + 345916q^-34 + 171077q^-33 - 93702q^-32 - 301601q^-31 - 336443q^-30 - 176163q^-29 + 76834q^-28 + 283380q^-27 + 329272q^-26 + 185744q^-25 - 55747q^-24 - 263598q^-23 - 323576q^-22 - 199228q^-21 + 29174q^-20 + 238621q^-19 + 316237q^-18 + 216215q^-17 + 4237q^-16 - 206976q^-15 - 304972q^-14 - 233495q^-13 - 42882q^-12 + 166696q^-11 + 285755q^-10 + 248069q^-9 + 85277q^-8 - 118052q^-7 - 256846q^-6 - 255411q^-5 - 126646q^-4 + 62457q^-3 + 215796q^-2 + 251327q^-1 + 162847 - 3501q - 163734q² - 232727q³ - 187532q⁴ - 53039q⁵ + 102902q⁶ + 198060q⁷ + 196107q⁸ + 100893q⁹ - 39161q¹⁰ - 149479q¹¹ - 185483q¹² - 132996q¹³ - 20006q¹⁴ + 91657q¹⁵ + 155949q¹⁶ + 144984q¹⁷ + 66891q¹⁸ - 32813q¹⁹ - 112112q²⁰ - 135477q²¹ - 94844q²² - 18257q²³ + 61475q²⁴ + 107869q²⁵ + 101325q²⁶ + 53920q²⁷ - 13652q²⁸ - 69424q²⁹ - 88369q³⁰ - 69988q³¹ - 22796q³² + 29443q³³ + 62149q³⁴ + 67283q³⁵ + 43079q³⁶ + 3323q³⁷ - 31706q³⁸ - 51133q³⁹ - 46602q⁴⁰ - 23262q⁴¹ + 5120q⁴² + 29101q⁴³ + 37723q⁴⁴ + 29687q⁴⁵ + 12056q⁴⁶ - 8982q⁴⁷ - 22994q⁴⁸ - 25401q⁴⁹ - 18514q⁵⁰ - 4633q⁵¹ + 8580q⁵² + 15988q⁵³ + 16875q⁵⁴ + 10330q⁵⁵ + 1127q⁵⁶ - 6485q⁵⁷ - 10883q⁵⁸ - 9858q⁵⁹ - 5479q⁶⁰ - 155q⁶¹ + 4805q⁶² + 6556q⁶³ + 5492q⁶⁴ + 2878q⁶⁵ - 617q⁶⁶ - 2798q⁶⁷ - 3567q⁶⁸ - 3091q⁶⁹ - 1225q⁷⁰ + 493q⁷¹ + 1583q⁷² + 1944q⁷³ + 1308q⁷⁴ + 540q⁷⁵ - 221q⁷⁶ - 884q⁷⁷ - 875q⁷⁸ - 602q⁷⁹ - 187q⁸⁰ + 226q⁸¹ + 314q⁸² + 342q⁸³ + 268q⁸⁴ + 42q⁸⁵ - 89q⁸⁶ - 152q⁸⁷ - 133q⁸⁸ - 35q⁸⁹ - 20q⁹⁰ + 19q⁹¹ + 57q⁹² + 37q⁹³ + 19q⁹⁴ - 10q⁹⁵ - 18q⁹⁶ - 2q⁹⁷ - 5q⁹⁸ - 7q⁹⁹ + 3q¹⁰⁰ + 3q¹⁰¹ + 5q¹⁰² - q¹⁰³ - 3q¹⁰⁴ + q¹⁰⁵

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	PD[Knot[9, 29]]
Out[2]=	PD[X[6, 2, 7, 1], X[16, 11, 17, 12], X[10, 4, 11, 3], X[2, 15, 3, 16], > X[14, 5, 15, 6], X[18, 8, 1, 7], X[4, 10, 5, 9], X[12, 17, 13, 18], > X[8, 13, 9, 14]]
In[3]:=	GaussCode[Knot[9, 29]]
Out[3]=	GaussCode[1, -4, 3, -7, 5, -1, 6, -9, 7, -3, 2, -8, 9, -5, 4, -2, 8, -6]
In[4]:=	DTCode[Knot[9, 29]]
Out[4]=	DTCode[6, 10, 14, 18, 4, 16, 8, 2, 12]
In[5]:=	br = BR[Knot[9, 29]]
Out[5]=	BR[4, {1, -2, -2, 3, -2, 1, -2, 3, -2}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 9}
In[7]:=	BraidIndex[Knot[9, 29]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[9, 29]]]

Out[8]=	-Graphics-
In[9]:=	#[Knot[9, 29]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 3, 3, {4, 7}, 1}
In[10]:=	alex = Alexander[Knot[9, 29]][t]
Out[10]=	-3 5 12 2 3 -15 + t - -- + -- + 12 t - 5 t + t 2 t t
In[11]:=	Conway[Knot[9, 29]][z]
Out[11]=	2 4 6 1 + z + z + z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[9, 28], Knot[9, 29], Knot[10, 163], Knot[11, NonAlternating, 87]}
In[13]:=	{KnotDet[Knot[9, 29]], KnotSignature[Knot[9, 29]]}
Out[13]=	{51, -2}
In[14]:=	Jones[Knot[9, 29]][q]
Out[14]=	-6 3 6 8 8 9 2 3 -7 - q + -- - -- + -- - -- + - + 5 q - 3 q + q 5 4 3 2 q q q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[9, 29]}
In[16]:=	A2Invariant[Knot[9, 29]][q]
Out[16]=	-18 -16 2 -12 2 4 -2 2 4 6 10 -q + q - --- - q + --- + -- + q - 2 q + q - q + q 14 10 6 q q q
In[17]:=	HOMFLYPT[Knot[9, 29]][a, z]
Out[17]=	2 -2 2 4 2 z 2 2 4 2 4 2 4 -3 + a + 5 a - 2 a - 5 z + -- + 7 a z - 2 a z - 2 z + 4 a z - 2 a 4 4 2 6 > a z + a z
In[18]:=	Kauffman[Knot[9, 29]][a, z]
Out[18]=	2 -2 2 4 z 3 5 2 3 z 2 2 -3 - a - 5 a - 2 a - - - a z + 2 a z + 2 a z + 12 z + ---- + 17 a z + a 2 a 3 4 4 2 9 z 3 3 3 5 3 7 3 4 3 z > 8 a z + ---- + 14 a z - a z - 5 a z + a z - 11 z - ---- - a 2 a 5 2 4 4 4 6 4 10 z 5 3 5 5 5 6 > 24 a z - 13 a z + 3 a z - ----- - 24 a z - 8 a z + 6 a z - z + a 6 7 z 2 6 4 6 3 z 7 3 7 8 2 8 > -- + 6 a z + 8 a z + ---- + 9 a z + 6 a z + 2 z + 2 a z 2 a a
In[19]:=	{Vassiliev[2][Knot[9, 29]], Vassiliev[3][Knot[9, 29]]}
Out[19]=	{1, -2}
In[20]:=	Kh[Knot[9, 29]][q, t]
Out[20]=	5 5 1 2 1 4 2 4 4 4 -- + - + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ---- + 3 q 13 5 11 4 9 4 9 3 7 3 7 2 5 2 5 q q t q t q t q t q t q t q t q t 4 3 t 2 3 2 3 3 5 3 7 4 > ---- + --- + 4 q t + 2 q t + 3 q t + q t + 2 q t + q t 3 q q t
In[21]:=	ColouredJones[Knot[9, 29], 2][q]
Out[21]=	-17 3 3 4 16 15 12 42 29 27 65 32 43 4 + q - --- + --- + --- - --- + --- + --- - --- + -- + -- - -- + -- + -- - 16 15 14 13 12 11 10 9 8 7 6 5 q q q q q q q q q q q q 71 20 51 60 2 3 4 5 6 7 > -- + -- + -- - -- + 46 q - 37 q - 8 q + 30 q - 13 q - 9 q + 11 q - 4 3 2 q q q q 8 9 10 > q - 3 q + q

The Alternating Knot 929

The Alternating Knot 9₂₉