The Alternating Knot 9₃₄

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Acknowledgement

KnotPlot

PD Presentation: X₆₂₇₁ X_16,8,17,7 X₈₃₉₄ X_2,15,3,16 X_14,9,15,10 X_10,6,11,5 X_4,14,5,13 X_18,11,1,12 X_12,17,13,18

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

DT (Dowker-Thistlethwaite) Code: 6 8 10 16 14 18 4 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 1 3 3 / 4--6 1

Alexander Polynomial: - t^-3 + 6t^-2 - 16t^-1 + 23 - 16t + 6t² - t³

Conway Polynomial: 1 - z² - z⁶

Other knots with the same Alexander/Conway Polynomial: {K11n32, K11n119, ...}

Determinant and Signature: {69, 0}

Jones Polynomial: - q^-5 + 4q^-4 - 7q^-3 + 10q^-2 - 12q^-1 + 12 - 10q + 8q² - 4q³ + q⁴

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

A2 (sl(3)) Invariant: - q^-16 + q^-14 + 2q^-12 - 2q^-10 + 2q^-8 - q^-6 - q^-4 + 2q^-2 - 2 + 3q² - 2q⁴ + q⁶ + 2q⁸ - 2q¹⁰ + q¹²

HOMFLY-PT Polynomial: a^-2 + a^-2z² + a^-2z⁴ - 1 - 4z² - 3z⁴ - z⁶ + a² + 3a²z² + 2a²z⁴ - a⁴z²

Kauffman Polynomial: a^-4z⁴ - 2a^-3z³ + 4a^-3z⁵ - a^-2 + 4a^-2z² - 10a^-2z⁴ + 8a^-2z⁶ - a^-1z + 4a^-1z³ - 10a^-1z⁵ + 8a^-1z⁷ - 1 + 11z² - 23z⁴ + 9z⁶ + 3z⁸ - az + 12az³ - 26az⁵ + 14az⁷ - a² + 10a²z² - 19a²z⁴ + 5a²z⁶ + 3a²z⁸ + 5a³z³ - 11a³z⁵ + 6a³z⁷ + 3a⁴z² - 7a⁴z⁴ + 4a⁴z⁶ - a⁵z³ + 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 9₃₄. 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 = 9 1

j = 7 3

j = 5 5 1

j = 3 5 3

j = 1 7 5

j = -1 6 6

j = -3 4 6

j = -5 3 6

j = -7 1 4

j = -9 3

j = -11 1

n Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q^-15 - 4q^-14 + 2q^-13 + 14q^-12 - 24q^-11 - 6q^-10 + 55q^-9 - 48q^-8 - 39q^-7 + 109q^-6 - 54q^-5 - 83q^-4 + 143q^-3 - 41q^-2 - 112q^-1 + 140 - 17q - 109q² + 101q³ + 6q⁴ - 74q⁵ + 47q⁶ + 12q⁷ - 29q⁸ + 10q⁹ + 4q¹⁰ - 4q¹¹ + q¹²

3 - q^-30 + 4q^-29 - 2q^-28 - 9q^-27 + 27q^-25 + 12q^-24 - 63q^-23 - 40q^-22 + 97q^-21 + 113q^-20 - 130q^-19 - 218q^-18 + 129q^-17 + 354q^-16 - 87q^-15 - 492q^-14 - 7q^-13 + 623q^-12 + 126q^-11 - 714q^-10 - 269q^-9 + 771q^-8 + 410q^-7 - 793q^-6 - 532q^-5 + 778q^-4 + 632q^-3 - 728q^-2 - 707q^-1 + 652 + 741q - 536q² - 747q³ + 410q⁴ + 693q⁵ - 258q⁶ - 611q⁷ + 132q⁸ + 482q⁹ - 25q¹⁰ - 350q¹¹ - 26q¹² + 214q¹³ + 49q¹⁴ - 116q¹⁵ - 42q¹⁶ + 56q¹⁷ + 20q¹⁸ - 19q¹⁹ - 9q²⁰ + 6q²¹ + 4q²² - 4q²³ + q²⁴

4 q^-50 - 4q^-49 + 2q^-48 + 9q^-47 - 5q^-46 - 3q^-45 - 33q^-44 + 12q^-43 + 74q^-42 + 17q^-41 - 15q^-40 - 211q^-39 - 76q^-38 + 256q^-37 + 277q^-36 + 191q^-35 - 622q^-34 - 641q^-33 + 189q^-32 + 844q^-31 + 1195q^-30 - 721q^-29 - 1744q^-28 - 861q^-27 + 993q^-26 + 3002q^-25 + 330q^-24 - 2485q^-23 - 2852q^-22 - 136q^-21 + 4597q^-20 + 2412q^-19 - 2001q^-18 - 4761q^-17 - 2308q^-16 + 5119q^-15 + 4516q^-14 - 524q^-13 - 5807q^-12 - 4533q^-11 + 4680q^-10 + 5920q^-9 + 1158q^-8 - 5993q^-7 - 6164q^-6 + 3731q^-5 + 6547q^-4 + 2637q^-3 - 5517q^-2 - 7085q^-1 + 2405 + 6391q + 3852q² - 4285q³ - 7160q⁴ + 689q⁵ + 5225q⁶ + 4550q⁷ - 2301q⁸ - 6066q⁹ - 926q¹⁰ + 3112q¹¹ + 4168q¹² - 277q¹³ - 3886q¹⁴ - 1582q¹⁵ + 949q¹⁶ + 2693q¹⁷ + 743q¹⁸ - 1643q¹⁹ - 1115q²⁰ - 183q²¹ + 1098q²² + 633q²³ - 384q²⁴ - 385q²⁵ - 272q²⁶ + 257q²⁷ + 221q²⁸ - 48q²⁹ - 49q³⁰ - 90q³¹ + 37q³² + 39q³³ - 11q³⁴ + q³⁵ - 13q³⁶ + 6q³⁷ + 4q³⁸ - 4q³⁹ + q⁴⁰

5 - q^-75 + 4q^-74 - 2q^-73 - 9q^-72 + 5q^-71 + 8q^-70 + 9q^-69 + 9q^-68 - 23q^-67 - 67q^-66 - 22q^-65 + 79q^-64 + 147q^-63 + 126q^-62 - 86q^-61 - 373q^-60 - 438q^-59 - 7q^-58 + 678q^-57 + 1028q^-56 + 531q^-55 - 810q^-54 - 2043q^-53 - 1768q^-52 + 451q^-51 + 3097q^-50 + 3834q^-49 + 1183q^-48 - 3662q^-47 - 6681q^-46 - 4309q^-45 + 2841q^-44 + 9415q^-43 + 9102q^-42 + 191q^-41 - 11161q^-40 - 14834q^-39 - 5684q^-38 + 10678q^-37 + 20490q^-36 + 13309q^-35 - 7381q^-34 - 24730q^-33 - 22142q^-32 + 1204q^-31 + 26727q^-30 + 30858q^-29 + 7098q^-28 - 25897q^-27 - 38434q^-26 - 16564q^-25 + 22696q^-24 + 44084q^-23 + 25954q^-22 - 17694q^-21 - 47595q^-20 - 34528q^-19 + 11825q^-18 + 49272q^-17 + 41709q^-16 - 5832q^-15 - 49515q^-14 - 47403q^-13 + 119q^-12 + 48808q^-11 + 51828q^-10 + 5096q^-9 - 47442q^-8 - 55165q^-7 - 9932q^-6 + 45366q^-5 + 57635q^-4 + 14722q^-3 - 42519q^-2 - 59226q^-1 - 19500 + 38437q + 59579q² + 24528q³ - 32910q⁴ - 58452q⁵ - 29202q⁶ + 25783q⁷ + 55091q⁸ + 33241q⁹ - 17413q¹⁰ - 49461q¹¹ - 35518q¹² + 8486q¹³ + 41435q¹⁴ + 35624q¹⁵ - 198q¹⁶ - 31910q¹⁷ - 32886q¹⁸ - 6404q¹⁹ + 21796q²⁰ + 27973q²¹ + 10324q²² - 12686q²³ - 21368q²⁴ - 11494q²⁵ + 5444q²⁶ + 14640q²⁷ + 10381q²⁸ - 897q²⁹ - 8689q³⁰ - 7900q³¹ - 1364q³² + 4347q³³ + 5191q³⁴ + 1892q³⁵ - 1772q³⁶ - 2883q³⁷ - 1505q³⁸ + 462q³⁹ + 1364q⁴⁰ + 957q⁴¹ - 30q⁴² - 571q⁴³ - 449q⁴⁴ - 60q⁴⁵ + 190q⁴⁶ + 181q⁴⁷ + 59q⁴⁸ - 72q⁴⁹ - 74q⁵⁰ - q⁵¹ + 20q⁵² + 8q⁵³ + 9q⁵⁴ - 3q⁵⁵ - 13q⁵⁶ + 6q⁵⁷ + 4q⁵⁸ - 4q⁵⁹ + q⁶⁰

6 q^-105 - 4q^-104 + 2q^-103 + 9q^-102 - 5q^-101 - 8q^-100 - 14q^-99 + 15q^-98 + 2q^-97 + 16q^-96 + 72q^-95 - 26q^-94 - 92q^-93 - 163q^-92 - 27q^-91 + 59q^-90 + 242q^-89 + 552q^-88 + 212q^-87 - 320q^-86 - 1063q^-85 - 1006q^-84 - 680q^-83 + 623q^-82 + 2695q^-81 + 2913q^-80 + 1492q^-79 - 2203q^-78 - 4888q^-77 - 6640q^-76 - 3742q^-75 + 4175q^-74 + 10756q^-73 + 13020q^-72 + 5890q^-71 - 5535q^-70 - 20110q^-69 - 24328q^-68 - 11458q^-67 + 11198q^-66 + 33982q^-65 + 38216q^-64 + 22114q^-63 - 18946q^-62 - 55241q^-61 - 61282q^-60 - 29789q^-59 + 30049q^-58 + 80016q^-57 + 94087q^-56 + 38852q^-55 - 48992q^-54 - 118831q^-53 - 123612q^-52 - 45861q^-51 + 71966q^-50 + 170542q^-49 + 156660q^-48 + 42172q^-47 - 115811q^-46 - 217687q^-45 - 186479q^-44 - 29574q^-43 + 178557q^-42 + 271427q^-41 + 199049q^-40 - 16250q^-39 - 239573q^-38 - 321541q^-37 - 193995q^-36 + 92020q^-35 + 315023q^-34 + 348024q^-33 + 140530q^-32 - 173593q^-31 - 389585q^-30 - 348258q^-29 - 44606q^-28 + 280278q^-27 + 435588q^-26 + 286036q^-25 - 65372q^-24 - 389077q^-23 - 447370q^-22 - 170599q^-21 + 209578q^-20 + 463670q^-19 + 384479q^-18 + 34442q^-17 - 356319q^-16 - 495577q^-15 - 259898q^-14 + 141818q^-13 + 462028q^-12 + 442844q^-11 + 109774q^-10 - 318813q^-9 - 517342q^-8 - 323134q^-7 + 83170q^-6 + 448797q^-5 + 482484q^-4 + 175604q^-3 - 273997q^-2 - 523707q^-1 - 380135 + 14014q + 414062q² + 508808q³ + 250860q⁴ - 197997q⁵ - 498059q⁶ - 430541q⁷ - 83203q⁸ + 330085q⁹ + 497760q¹⁰ + 328123q¹¹ - 75882q¹² - 408894q¹³ - 441171q¹⁴ - 191750q¹⁵ + 186123q¹⁶ + 413536q¹⁷ + 363413q¹⁸ + 63988q¹⁹ - 251033q²⁰ - 371665q²¹ - 255646q²² + 23502q²³ + 257079q²⁴ + 312752q²⁵ + 154170q²⁶ - 78266q²⁷ - 229647q²⁸ - 229420q²⁹ - 82084q³⁰ + 92051q³¹ + 191466q³² + 151409q³³ + 30912q³⁴ - 85215q³⁵ - 136646q³⁶ - 94578q³⁷ - 6610q³⁸ + 72590q³⁹ + 87565q⁴⁰ + 51710q⁴¹ - 4566q⁴² - 48973q⁴³ - 53389q⁴⁴ - 27268q⁴⁵ + 10529q⁴⁶ + 29057q⁴⁷ + 28184q⁴⁸ + 12518q⁴⁹ - 7389q⁵⁰ - 16750q⁵¹ - 14376q⁵² - 3277q⁵³ + 4218q⁵⁴ + 7879q⁵⁵ + 6329q⁵⁶ + 1083q⁵⁷ - 2705q⁵⁸ - 3767q⁵⁹ - 1676q⁶⁰ - 260q⁶¹ + 1095q⁶² + 1477q⁶³ + 616q⁶⁴ - 201q⁶⁵ - 596q⁶⁶ - 226q⁶⁷ - 172q⁶⁸ + 60q⁶⁹ + 214q⁷⁰ + 93q⁷¹ - 16q⁷² - 87q⁷³ + 15q⁷⁴ - 18q⁷⁵ - 11q⁷⁶ + 28q⁷⁷ + 5q⁷⁸ - 3q⁷⁹ - 13q⁸⁰ + 6q⁸¹ + 4q⁸² - 4q⁸³ + q⁸⁴

7 - q^-140 + 4q^-139 - 2q^-138 - 9q^-137 + 5q^-136 + 8q^-135 + 14q^-134 - 10q^-133 - 26q^-132 + 5q^-131 - 21q^-130 - 24q^-129 + 39q^-128 + 92q^-127 + 141q^-126 + 20q^-125 - 197q^-124 - 232q^-123 - 344q^-122 - 253q^-121 + 178q^-120 + 675q^-119 + 1289q^-118 + 1072q^-117 - 33q^-116 - 1311q^-115 - 2854q^-114 - 3381q^-113 - 1979q^-112 + 985q^-111 + 5620q^-110 + 8655q^-109 + 7576q^-108 + 2517q^-107 - 7040q^-106 - 16215q^-105 - 19946q^-104 - 15056q^-103 + 1684q^-102 + 23395q^-101 + 39464q^-100 + 41277q^-99 + 19848q^-98 - 18707q^-97 - 59274q^-96 - 83861q^-95 - 69180q^-94 - 14057q^-93 + 63605q^-92 + 132781q^-91 + 149147q^-90 + 93644q^-89 - 23369q^-88 - 162516q^-87 - 249493q^-86 - 229710q^-85 - 89221q^-84 + 133949q^-83 + 333928q^-82 + 407982q^-81 + 292569q^-80 - 1773q^-79 - 349646q^-78 - 587886q^-77 - 574945q^-76 - 260227q^-75 + 237065q^-74 + 700205q^-73 + 889664q^-72 + 648950q^-71 + 46196q^-70 - 672613q^-69 - 1160225q^-68 - 1116612q^-67 - 504567q^-66 + 447271q^-65 + 1299853q^-64 + 1583164q^-63 + 1097392q^-62 - 6866q^-61 - 1238823q^-60 - 1954553q^-59 - 1746377q^-58 - 617661q^-57 + 946552q^-56 + 2150543q^-55 + 2354958q^-54 + 1355500q^-53 - 438709q^-52 - 2127516q^-51 - 2838187q^-50 - 2115194q^-49 - 224497q^-48 + 1886766q^-47 + 3139347q^-46 + 2808633q^-45 + 962632q^-44 - 1469069q^-43 - 3244729q^-42 - 3373205q^-41 - 1691930q^-40 + 942327q^-39 + 3175544q^-38 + 3778094q^-37 + 2346613q^-36 - 377124q^-35 - 2978499q^-34 - 4027779q^-33 - 2887912q^-32 - 164342q^-31 + 2710187q^-30 + 4148097q^-29 + 3304202q^-28 + 641548q^-27 - 2420560q^-26 - 4177817q^-25 - 3607912q^-24 - 1036621q^-23 + 2147420q^-22 + 4156866q^-21 + 3823412q^-20 + 1351280q^-19 - 1909743q^-18 - 4116003q^-17 - 3981713q^-16 - 1604177q^-15 + 1709872q^-14 + 4076227q^-13 + 4112104q^-12 + 1820136q^-11 - 1535296q^-10 - 4042274q^-9 - 4235883q^-8 - 2029292q^-7 + 1360966q^-6 + 4006145q^-5 + 4364984q^-4 + 2257688q^-3 - 1157376q^-2 - 3945409q^-1 - 4494135 - 2522244q + 890834q² + 3825894q³ + 4604709q⁴ + 2826220q⁵ - 537899q⁶ - 3610169q⁷ - 4658362q⁸ - 3149968q⁹ + 87493q¹⁰ + 3261430q¹¹ + 4609291q¹² + 3454987q¹³ + 444233q¹⁴ - 2762212q¹⁵ - 4408420q¹⁶ - 3681754q¹⁷ - 1015128q¹⁸ + 2119469q¹⁹ + 4023424q²⁰ + 3768099q²¹ + 1555703q²² - 1377286q²³ - 3449927q²⁴ - 3661296q²⁵ - 1986923q²⁶ + 609765q²⁷ + 2725638q²⁸ + 3341119q²⁹ + 2235800q³⁰ + 89307q³¹ - 1922465q³² - 2829249q³³ - 2263880q³⁴ - 633602q³⁵ + 1139954q³⁶ + 2192067q³⁷ + 2073346q³⁸ + 964613q³⁹ - 470669q⁴⁰ - 1520174q⁴¹ - 1718343q⁴² - 1073367q⁴³ - 15765q⁴⁴ + 910274q⁴⁵ + 1278907q⁴⁶ + 993820q⁴⁷ + 298839q⁴⁸ - 430400q⁴⁹ - 843686q⁵⁰ - 797484q⁵¹ - 400811q⁵² + 112560q⁵³ + 480840q⁵⁴ + 559280q⁵⁵ + 375169q⁵⁶ + 56051q⁵⁷ - 223202q⁵⁸ - 341928q⁵⁹ - 286642q⁶⁰ - 113375q⁶¹ + 71464q⁶² + 179387q⁶³ + 185323q⁶⁴ + 106234q⁶⁵ + 217q⁶⁶ - 77368q⁶⁷ - 103193q⁶⁸ - 75414q⁶⁹ - 21814q⁷⁰ + 25233q⁷¹ + 49176q⁷² + 43605q⁷³ + 20489q⁷⁴ - 3481q⁷⁵ - 19779q⁷⁶ - 21631q⁷⁷ - 13194q⁷⁸ - 2254q⁷⁹ + 6836q⁸⁰ + 9138q⁸¹ + 6470q⁸² + 2371q⁸³ - 1711q⁸⁴ - 3329q⁸⁵ - 2844q⁸⁶ - 1471q⁸⁷ + 482q⁸⁸ + 1208q⁸⁹ + 917q⁹⁰ + 534q⁹¹ - 66q⁹² - 260q⁹³ - 318q⁹⁴ - 291q⁹⁵ + 48q⁹⁶ + 158q⁹⁷ + 75q⁹⁸ + 28q⁹⁹ - 31q¹⁰⁰ + 2q¹⁰¹ - 2q¹⁰² - 49q¹⁰³ + 9q¹⁰⁴ + 24q¹⁰⁵ + 5q¹⁰⁶ - 3q¹⁰⁷ - 13q¹⁰⁸ + 6q¹⁰⁹ + 4q¹¹⁰ - 4q¹¹¹ + q¹¹²

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, 34]]

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

In[3]:=
GaussCode[Knot[9, 34]]

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

In[4]:=
DTCode[Knot[9, 34]]

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

In[5]:=
br = BR[Knot[9, 34]]

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

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

Out[6]=
{4, 9}

In[7]:=
BraidIndex[Knot[9, 34]]

Out[7]=
4

In[8]:=
Show[DrawMorseLink[Knot[9, 34]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[9, 34]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

Out[9]=
{Reversible, 1, 3, 3, {4, 6}, 1}

In[10]:=
alex = Alexander[Knot[9, 34]][t]

Out[10]=
-3 6 16 2 3 23 - t + -- - -- - 16 t + 6 t - t 2 t t

In[11]:=
Conway[Knot[9, 34]][z]

Out[11]=
2 6 1 - z - z

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

Out[12]=
{Knot[9, 34], Knot[11, NonAlternating, 32], Knot[11, NonAlternating, 119]}

In[13]:=
{KnotDet[Knot[9, 34]], KnotSignature[Knot[9, 34]]}

Out[13]=
{69, 0}

In[14]:=
Jones[Knot[9, 34]][q]

Out[14]=
-5 4 7 10 12 2 3 4 12 - q + -- - -- + -- - -- - 10 q + 8 q - 4 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[9, 34]}

In[16]:=
A2Invariant[Knot[9, 34]][q]

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

In[17]:=
HOMFLYPT[Knot[9, 34]][a, z]

Out[17]=
2 4 -2 2 2 z 2 2 4 2 4 z 2 4 6 -1 + a + a - 4 z + -- + 3 a z - a z - 3 z + -- + 2 a z - z 2 2 a a

In[18]:=
Kauffman[Knot[9, 34]][a, z]

Out[18]=
2 3 3 -2 2 z 2 4 z 2 2 4 2 2 z 4 z -1 - a - a - - - a z + 11 z + ---- + 10 a z + 3 a z - ---- + ---- + a 2 3 a a a 4 4 3 3 3 5 3 4 z 10 z 2 4 4 4 > 12 a z + 5 a z - a z - 23 z + -- - ----- - 19 a z - 7 a z + 4 2 a a 5 5 6 4 z 10 z 5 3 5 5 5 6 8 z 2 6 > ---- - ----- - 26 a z - 11 a z + a z + 9 z + ---- + 5 a z + 3 a 2 a a 7 4 6 8 z 7 3 7 8 2 8 > 4 a z + ---- + 14 a z + 6 a z + 3 z + 3 a z a

In[19]:=
{Vassiliev[2][Knot[9, 34]], Vassiliev[3][Knot[9, 34]]}

Out[19]=
{-1, 0}

In[20]:=
Kh[Knot[9, 34]][q, t]

Out[20]=
6 1 3 1 4 3 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 9 4 > 5 q t + 5 q t + 3 q t + 5 q t + q t + 3 q t + q t

In[21]:=
ColouredJones[Knot[9, 34], 2][q]

Out[21]=
-15 4 2 14 24 6 55 48 39 109 54 83 143 140 + q - --- + --- + --- - --- - --- + -- - -- - -- + --- - -- - -- + --- - 14 13 12 11 10 9 8 7 6 5 4 3 q q q q q q q q q q q q 41 112 2 3 4 5 6 7 8 > -- - --- - 17 q - 109 q + 101 q + 6 q - 74 q + 47 q + 12 q - 29 q + 2 q q 9 10 11 12 > 10 q + 4 q - 4 q + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 9₃₄

9₃₃
9₃₅

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-15 - 4q^-14 + 2q^-13 + 14q^-12 - 24q^-11 - 6q^-10 + 55q^-9 - 48q^-8 - 39q^-7 + 109q^-6 - 54q^-5 - 83q^-4 + 143q^-3 - 41q^-2 - 112q^-1 + 140 - 17q - 109q² + 101q³ + 6q⁴ - 74q⁵ + 47q⁶ + 12q⁷ - 29q⁸ + 10q⁹ + 4q¹⁰ - 4q¹¹ + q¹²
3	- q^-30 + 4q^-29 - 2q^-28 - 9q^-27 + 27q^-25 + 12q^-24 - 63q^-23 - 40q^-22 + 97q^-21 + 113q^-20 - 130q^-19 - 218q^-18 + 129q^-17 + 354q^-16 - 87q^-15 - 492q^-14 - 7q^-13 + 623q^-12 + 126q^-11 - 714q^-10 - 269q^-9 + 771q^-8 + 410q^-7 - 793q^-6 - 532q^-5 + 778q^-4 + 632q^-3 - 728q^-2 - 707q^-1 + 652 + 741q - 536q² - 747q³ + 410q⁴ + 693q⁵ - 258q⁶ - 611q⁷ + 132q⁸ + 482q⁹ - 25q¹⁰ - 350q¹¹ - 26q¹² + 214q¹³ + 49q¹⁴ - 116q¹⁵ - 42q¹⁶ + 56q¹⁷ + 20q¹⁸ - 19q¹⁹ - 9q²⁰ + 6q²¹ + 4q²² - 4q²³ + q²⁴
4	q^-50 - 4q^-49 + 2q^-48 + 9q^-47 - 5q^-46 - 3q^-45 - 33q^-44 + 12q^-43 + 74q^-42 + 17q^-41 - 15q^-40 - 211q^-39 - 76q^-38 + 256q^-37 + 277q^-36 + 191q^-35 - 622q^-34 - 641q^-33 + 189q^-32 + 844q^-31 + 1195q^-30 - 721q^-29 - 1744q^-28 - 861q^-27 + 993q^-26 + 3002q^-25 + 330q^-24 - 2485q^-23 - 2852q^-22 - 136q^-21 + 4597q^-20 + 2412q^-19 - 2001q^-18 - 4761q^-17 - 2308q^-16 + 5119q^-15 + 4516q^-14 - 524q^-13 - 5807q^-12 - 4533q^-11 + 4680q^-10 + 5920q^-9 + 1158q^-8 - 5993q^-7 - 6164q^-6 + 3731q^-5 + 6547q^-4 + 2637q^-3 - 5517q^-2 - 7085q^-1 + 2405 + 6391q + 3852q² - 4285q³ - 7160q⁴ + 689q⁵ + 5225q⁶ + 4550q⁷ - 2301q⁸ - 6066q⁹ - 926q¹⁰ + 3112q¹¹ + 4168q¹² - 277q¹³ - 3886q¹⁴ - 1582q¹⁵ + 949q¹⁶ + 2693q¹⁷ + 743q¹⁸ - 1643q¹⁹ - 1115q²⁰ - 183q²¹ + 1098q²² + 633q²³ - 384q²⁴ - 385q²⁵ - 272q²⁶ + 257q²⁷ + 221q²⁸ - 48q²⁹ - 49q³⁰ - 90q³¹ + 37q³² + 39q³³ - 11q³⁴ + q³⁵ - 13q³⁶ + 6q³⁷ + 4q³⁸ - 4q³⁹ + q⁴⁰
5	- q^-75 + 4q^-74 - 2q^-73 - 9q^-72 + 5q^-71 + 8q^-70 + 9q^-69 + 9q^-68 - 23q^-67 - 67q^-66 - 22q^-65 + 79q^-64 + 147q^-63 + 126q^-62 - 86q^-61 - 373q^-60 - 438q^-59 - 7q^-58 + 678q^-57 + 1028q^-56 + 531q^-55 - 810q^-54 - 2043q^-53 - 1768q^-52 + 451q^-51 + 3097q^-50 + 3834q^-49 + 1183q^-48 - 3662q^-47 - 6681q^-46 - 4309q^-45 + 2841q^-44 + 9415q^-43 + 9102q^-42 + 191q^-41 - 11161q^-40 - 14834q^-39 - 5684q^-38 + 10678q^-37 + 20490q^-36 + 13309q^-35 - 7381q^-34 - 24730q^-33 - 22142q^-32 + 1204q^-31 + 26727q^-30 + 30858q^-29 + 7098q^-28 - 25897q^-27 - 38434q^-26 - 16564q^-25 + 22696q^-24 + 44084q^-23 + 25954q^-22 - 17694q^-21 - 47595q^-20 - 34528q^-19 + 11825q^-18 + 49272q^-17 + 41709q^-16 - 5832q^-15 - 49515q^-14 - 47403q^-13 + 119q^-12 + 48808q^-11 + 51828q^-10 + 5096q^-9 - 47442q^-8 - 55165q^-7 - 9932q^-6 + 45366q^-5 + 57635q^-4 + 14722q^-3 - 42519q^-2 - 59226q^-1 - 19500 + 38437q + 59579q² + 24528q³ - 32910q⁴ - 58452q⁵ - 29202q⁶ + 25783q⁷ + 55091q⁸ + 33241q⁹ - 17413q¹⁰ - 49461q¹¹ - 35518q¹² + 8486q¹³ + 41435q¹⁴ + 35624q¹⁵ - 198q¹⁶ - 31910q¹⁷ - 32886q¹⁸ - 6404q¹⁹ + 21796q²⁰ + 27973q²¹ + 10324q²² - 12686q²³ - 21368q²⁴ - 11494q²⁵ + 5444q²⁶ + 14640q²⁷ + 10381q²⁸ - 897q²⁹ - 8689q³⁰ - 7900q³¹ - 1364q³² + 4347q³³ + 5191q³⁴ + 1892q³⁵ - 1772q³⁶ - 2883q³⁷ - 1505q³⁸ + 462q³⁹ + 1364q⁴⁰ + 957q⁴¹ - 30q⁴² - 571q⁴³ - 449q⁴⁴ - 60q⁴⁵ + 190q⁴⁶ + 181q⁴⁷ + 59q⁴⁸ - 72q⁴⁹ - 74q⁵⁰ - q⁵¹ + 20q⁵² + 8q⁵³ + 9q⁵⁴ - 3q⁵⁵ - 13q⁵⁶ + 6q⁵⁷ + 4q⁵⁸ - 4q⁵⁹ + q⁶⁰
6	q^-105 - 4q^-104 + 2q^-103 + 9q^-102 - 5q^-101 - 8q^-100 - 14q^-99 + 15q^-98 + 2q^-97 + 16q^-96 + 72q^-95 - 26q^-94 - 92q^-93 - 163q^-92 - 27q^-91 + 59q^-90 + 242q^-89 + 552q^-88 + 212q^-87 - 320q^-86 - 1063q^-85 - 1006q^-84 - 680q^-83 + 623q^-82 + 2695q^-81 + 2913q^-80 + 1492q^-79 - 2203q^-78 - 4888q^-77 - 6640q^-76 - 3742q^-75 + 4175q^-74 + 10756q^-73 + 13020q^-72 + 5890q^-71 - 5535q^-70 - 20110q^-69 - 24328q^-68 - 11458q^-67 + 11198q^-66 + 33982q^-65 + 38216q^-64 + 22114q^-63 - 18946q^-62 - 55241q^-61 - 61282q^-60 - 29789q^-59 + 30049q^-58 + 80016q^-57 + 94087q^-56 + 38852q^-55 - 48992q^-54 - 118831q^-53 - 123612q^-52 - 45861q^-51 + 71966q^-50 + 170542q^-49 + 156660q^-48 + 42172q^-47 - 115811q^-46 - 217687q^-45 - 186479q^-44 - 29574q^-43 + 178557q^-42 + 271427q^-41 + 199049q^-40 - 16250q^-39 - 239573q^-38 - 321541q^-37 - 193995q^-36 + 92020q^-35 + 315023q^-34 + 348024q^-33 + 140530q^-32 - 173593q^-31 - 389585q^-30 - 348258q^-29 - 44606q^-28 + 280278q^-27 + 435588q^-26 + 286036q^-25 - 65372q^-24 - 389077q^-23 - 447370q^-22 - 170599q^-21 + 209578q^-20 + 463670q^-19 + 384479q^-18 + 34442q^-17 - 356319q^-16 - 495577q^-15 - 259898q^-14 + 141818q^-13 + 462028q^-12 + 442844q^-11 + 109774q^-10 - 318813q^-9 - 517342q^-8 - 323134q^-7 + 83170q^-6 + 448797q^-5 + 482484q^-4 + 175604q^-3 - 273997q^-2 - 523707q^-1 - 380135 + 14014q + 414062q² + 508808q³ + 250860q⁴ - 197997q⁵ - 498059q⁶ - 430541q⁷ - 83203q⁸ + 330085q⁹ + 497760q¹⁰ + 328123q¹¹ - 75882q¹² - 408894q¹³ - 441171q¹⁴ - 191750q¹⁵ + 186123q¹⁶ + 413536q¹⁷ + 363413q¹⁸ + 63988q¹⁹ - 251033q²⁰ - 371665q²¹ - 255646q²² + 23502q²³ + 257079q²⁴ + 312752q²⁵ + 154170q²⁶ - 78266q²⁷ - 229647q²⁸ - 229420q²⁹ - 82084q³⁰ + 92051q³¹ + 191466q³² + 151409q³³ + 30912q³⁴ - 85215q³⁵ - 136646q³⁶ - 94578q³⁷ - 6610q³⁸ + 72590q³⁹ + 87565q⁴⁰ + 51710q⁴¹ - 4566q⁴² - 48973q⁴³ - 53389q⁴⁴ - 27268q⁴⁵ + 10529q⁴⁶ + 29057q⁴⁷ + 28184q⁴⁸ + 12518q⁴⁹ - 7389q⁵⁰ - 16750q⁵¹ - 14376q⁵² - 3277q⁵³ + 4218q⁵⁴ + 7879q⁵⁵ + 6329q⁵⁶ + 1083q⁵⁷ - 2705q⁵⁸ - 3767q⁵⁹ - 1676q⁶⁰ - 260q⁶¹ + 1095q⁶² + 1477q⁶³ + 616q⁶⁴ - 201q⁶⁵ - 596q⁶⁶ - 226q⁶⁷ - 172q⁶⁸ + 60q⁶⁹ + 214q⁷⁰ + 93q⁷¹ - 16q⁷² - 87q⁷³ + 15q⁷⁴ - 18q⁷⁵ - 11q⁷⁶ + 28q⁷⁷ + 5q⁷⁸ - 3q⁷⁹ - 13q⁸⁰ + 6q⁸¹ + 4q⁸² - 4q⁸³ + q⁸⁴
7	- q^-140 + 4q^-139 - 2q^-138 - 9q^-137 + 5q^-136 + 8q^-135 + 14q^-134 - 10q^-133 - 26q^-132 + 5q^-131 - 21q^-130 - 24q^-129 + 39q^-128 + 92q^-127 + 141q^-126 + 20q^-125 - 197q^-124 - 232q^-123 - 344q^-122 - 253q^-121 + 178q^-120 + 675q^-119 + 1289q^-118 + 1072q^-117 - 33q^-116 - 1311q^-115 - 2854q^-114 - 3381q^-113 - 1979q^-112 + 985q^-111 + 5620q^-110 + 8655q^-109 + 7576q^-108 + 2517q^-107 - 7040q^-106 - 16215q^-105 - 19946q^-104 - 15056q^-103 + 1684q^-102 + 23395q^-101 + 39464q^-100 + 41277q^-99 + 19848q^-98 - 18707q^-97 - 59274q^-96 - 83861q^-95 - 69180q^-94 - 14057q^-93 + 63605q^-92 + 132781q^-91 + 149147q^-90 + 93644q^-89 - 23369q^-88 - 162516q^-87 - 249493q^-86 - 229710q^-85 - 89221q^-84 + 133949q^-83 + 333928q^-82 + 407982q^-81 + 292569q^-80 - 1773q^-79 - 349646q^-78 - 587886q^-77 - 574945q^-76 - 260227q^-75 + 237065q^-74 + 700205q^-73 + 889664q^-72 + 648950q^-71 + 46196q^-70 - 672613q^-69 - 1160225q^-68 - 1116612q^-67 - 504567q^-66 + 447271q^-65 + 1299853q^-64 + 1583164q^-63 + 1097392q^-62 - 6866q^-61 - 1238823q^-60 - 1954553q^-59 - 1746377q^-58 - 617661q^-57 + 946552q^-56 + 2150543q^-55 + 2354958q^-54 + 1355500q^-53 - 438709q^-52 - 2127516q^-51 - 2838187q^-50 - 2115194q^-49 - 224497q^-48 + 1886766q^-47 + 3139347q^-46 + 2808633q^-45 + 962632q^-44 - 1469069q^-43 - 3244729q^-42 - 3373205q^-41 - 1691930q^-40 + 942327q^-39 + 3175544q^-38 + 3778094q^-37 + 2346613q^-36 - 377124q^-35 - 2978499q^-34 - 4027779q^-33 - 2887912q^-32 - 164342q^-31 + 2710187q^-30 + 4148097q^-29 + 3304202q^-28 + 641548q^-27 - 2420560q^-26 - 4177817q^-25 - 3607912q^-24 - 1036621q^-23 + 2147420q^-22 + 4156866q^-21 + 3823412q^-20 + 1351280q^-19 - 1909743q^-18 - 4116003q^-17 - 3981713q^-16 - 1604177q^-15 + 1709872q^-14 + 4076227q^-13 + 4112104q^-12 + 1820136q^-11 - 1535296q^-10 - 4042274q^-9 - 4235883q^-8 - 2029292q^-7 + 1360966q^-6 + 4006145q^-5 + 4364984q^-4 + 2257688q^-3 - 1157376q^-2 - 3945409q^-1 - 4494135 - 2522244q + 890834q² + 3825894q³ + 4604709q⁴ + 2826220q⁵ - 537899q⁶ - 3610169q⁷ - 4658362q⁸ - 3149968q⁹ + 87493q¹⁰ + 3261430q¹¹ + 4609291q¹² + 3454987q¹³ + 444233q¹⁴ - 2762212q¹⁵ - 4408420q¹⁶ - 3681754q¹⁷ - 1015128q¹⁸ + 2119469q¹⁹ + 4023424q²⁰ + 3768099q²¹ + 1555703q²² - 1377286q²³ - 3449927q²⁴ - 3661296q²⁵ - 1986923q²⁶ + 609765q²⁷ + 2725638q²⁸ + 3341119q²⁹ + 2235800q³⁰ + 89307q³¹ - 1922465q³² - 2829249q³³ - 2263880q³⁴ - 633602q³⁵ + 1139954q³⁶ + 2192067q³⁷ + 2073346q³⁸ + 964613q³⁹ - 470669q⁴⁰ - 1520174q⁴¹ - 1718343q⁴² - 1073367q⁴³ - 15765q⁴⁴ + 910274q⁴⁵ + 1278907q⁴⁶ + 993820q⁴⁷ + 298839q⁴⁸ - 430400q⁴⁹ - 843686q⁵⁰ - 797484q⁵¹ - 400811q⁵² + 112560q⁵³ + 480840q⁵⁴ + 559280q⁵⁵ + 375169q⁵⁶ + 56051q⁵⁷ - 223202q⁵⁸ - 341928q⁵⁹ - 286642q⁶⁰ - 113375q⁶¹ + 71464q⁶² + 179387q⁶³ + 185323q⁶⁴ + 106234q⁶⁵ + 217q⁶⁶ - 77368q⁶⁷ - 103193q⁶⁸ - 75414q⁶⁹ - 21814q⁷⁰ + 25233q⁷¹ + 49176q⁷² + 43605q⁷³ + 20489q⁷⁴ - 3481q⁷⁵ - 19779q⁷⁶ - 21631q⁷⁷ - 13194q⁷⁸ - 2254q⁷⁹ + 6836q⁸⁰ + 9138q⁸¹ + 6470q⁸² + 2371q⁸³ - 1711q⁸⁴ - 3329q⁸⁵ - 2844q⁸⁶ - 1471q⁸⁷ + 482q⁸⁸ + 1208q⁸⁹ + 917q⁹⁰ + 534q⁹¹ - 66q⁹² - 260q⁹³ - 318q⁹⁴ - 291q⁹⁵ + 48q⁹⁶ + 158q⁹⁷ + 75q⁹⁸ + 28q⁹⁹ - 31q¹⁰⁰ + 2q¹⁰¹ - 2q¹⁰² - 49q¹⁰³ + 9q¹⁰⁴ + 24q¹⁰⁵ + 5q¹⁰⁶ - 3q¹⁰⁷ - 13q¹⁰⁸ + 6q¹⁰⁹ + 4q¹¹⁰ - 4q¹¹¹ + q¹¹²

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	PD[Knot[9, 34]]
Out[2]=	PD[X[6, 2, 7, 1], X[16, 8, 17, 7], X[8, 3, 9, 4], X[2, 15, 3, 16], > X[14, 9, 15, 10], X[10, 6, 11, 5], X[4, 14, 5, 13], X[18, 11, 1, 12], > X[12, 17, 13, 18]]
In[3]:=	GaussCode[Knot[9, 34]]
Out[3]=	GaussCode[1, -4, 3, -7, 6, -1, 2, -3, 5, -6, 8, -9, 7, -5, 4, -2, 9, -8]
In[4]:=	DTCode[Knot[9, 34]]
Out[4]=	DTCode[6, 8, 10, 16, 14, 18, 4, 2, 12]
In[5]:=	br = BR[Knot[9, 34]]
Out[5]=	BR[4, {-1, 2, -1, 2, -3, 2, -1, 2, -3}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 9}
In[7]:=	BraidIndex[Knot[9, 34]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[9, 34]]]

Out[8]=	-Graphics-
In[9]:=	#[Knot[9, 34]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 1, 3, 3, {4, 6}, 1}
In[10]:=	alex = Alexander[Knot[9, 34]][t]
Out[10]=	-3 6 16 2 3 23 - t + -- - -- - 16 t + 6 t - t 2 t t
In[11]:=	Conway[Knot[9, 34]][z]
Out[11]=	2 6 1 - z - z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[9, 34], Knot[11, NonAlternating, 32], Knot[11, NonAlternating, 119]}
In[13]:=	{KnotDet[Knot[9, 34]], KnotSignature[Knot[9, 34]]}
Out[13]=	{69, 0}
In[14]:=	Jones[Knot[9, 34]][q]
Out[14]=	-5 4 7 10 12 2 3 4 12 - q + -- - -- + -- - -- - 10 q + 8 q - 4 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[9, 34]}
In[16]:=	A2Invariant[Knot[9, 34]][q]
Out[16]=	-16 -14 2 2 2 -6 -4 2 2 4 6 8 -2 - q + q + --- - --- + -- - q - q + -- + 3 q - 2 q + q + 2 q - 12 10 8 2 q q q q 10 12 > 2 q + q
In[17]:=	HOMFLYPT[Knot[9, 34]][a, z]
Out[17]=	2 4 -2 2 2 z 2 2 4 2 4 z 2 4 6 -1 + a + a - 4 z + -- + 3 a z - a z - 3 z + -- + 2 a z - z 2 2 a a
In[18]:=	Kauffman[Knot[9, 34]][a, z]
Out[18]=	2 3 3 -2 2 z 2 4 z 2 2 4 2 2 z 4 z -1 - a - a - - - a z + 11 z + ---- + 10 a z + 3 a z - ---- + ---- + a 2 3 a a a 4 4 3 3 3 5 3 4 z 10 z 2 4 4 4 > 12 a z + 5 a z - a z - 23 z + -- - ----- - 19 a z - 7 a z + 4 2 a a 5 5 6 4 z 10 z 5 3 5 5 5 6 8 z 2 6 > ---- - ----- - 26 a z - 11 a z + a z + 9 z + ---- + 5 a z + 3 a 2 a a 7 4 6 8 z 7 3 7 8 2 8 > 4 a z + ---- + 14 a z + 6 a z + 3 z + 3 a z a
In[19]:=	{Vassiliev[2][Knot[9, 34]], Vassiliev[3][Knot[9, 34]]}
Out[19]=	{-1, 0}
In[20]:=	Kh[Knot[9, 34]][q, t]
Out[20]=	6 1 3 1 4 3 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 9 4 > 5 q t + 5 q t + 3 q t + 5 q t + q t + 3 q t + q t
In[21]:=	ColouredJones[Knot[9, 34], 2][q]
Out[21]=	-15 4 2 14 24 6 55 48 39 109 54 83 143 140 + q - --- + --- + --- - --- - --- + -- - -- - -- + --- - -- - -- + --- - 14 13 12 11 10 9 8 7 6 5 4 3 q q q q q q q q q q q q 41 112 2 3 4 5 6 7 8 > -- - --- - 17 q - 109 q + 101 q + 6 q - 74 q + 47 q + 12 q - 29 q + 2 q q 9 10 11 12 > 10 q + 4 q - 4 q + q

The Alternating Knot 934

The Alternating Knot 9₃₄