The Alternating Knot 9₃₃

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Acknowledgement

KnotPlot

PD Presentation: X₄₂₅₁ X_12,8,13,7 X₈₃₉₄ X_2,9,3,10 X_18,13,1,14 X_14,5,15,6 X_6,17,7,18 X_16,12,17,11 X_10,16,11,15

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

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

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

Chiral 1 3 3 / 4--6 1

Alexander Polynomial: - t^-3 + 6t^-2 - 14t^-1 + 19 - 14t + 6t² - t³

Conway Polynomial: 1 + z² - z⁶

Other knots with the same Alexander/Conway Polynomial: {K11n55, ...}

Determinant and Signature: {61, 0}

Jones Polynomial: - q^-5 + 3q^-4 - 6q^-3 + 9q^-2 - 10q^-1 + 11 - 9q + 7q² - 4q³ + q⁴

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

A2 (sl(3)) Invariant: - q^-16 + q^-12 - 2q^-10 + 2q^-8 + 3q^-2 - 1 + 3q² - 2q⁴ + q⁸ - 2q¹⁰ + q¹²

HOMFLY-PT Polynomial: a^-2z² + a^-2z⁴ - 3z² - 3z⁴ - z⁶ + 2a² + 4a²z² + 2a²z⁴ - a⁴ - a⁴z²

Kauffman Polynomial: a^-4z⁴ - 3a^-3z³ + 4a^-3z⁵ + 3a^-2z² - 9a^-2z⁴ + 7a^-2z⁶ - a^-1z³ - 5a^-1z⁵ + 6a^-1z⁷ + 9z² - 20z⁴ + 9z⁶ + 2z⁸ + 5az³ - 16az⁵ + 10az⁷ - 2a² + 10a²z² - 16a²z⁴ + 5a²z⁶ + 2a²z⁸ + a³z + a³z³ - 6a³z⁵ + 4a³z⁷ - a⁴ + 4a⁴z² - 6a⁴z⁴ + 3a⁴z⁶ + a⁵z - 2a⁵z³ + a⁵z⁵

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

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 9₃₃. 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 4 1

j = 3 5 3

j = 1 6 4

j = -1 5 6

j = -3 4 5

j = -5 2 5

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 + 10q^-12 - 17q^-11 - 4q^-10 + 40q^-9 - 37q^-8 - 27q^-7 + 83q^-6 - 46q^-5 - 60q^-4 + 113q^-3 - 39q^-2 - 83q^-1 + 115 - 22q - 83q² + 86q³ - 3q⁴ - 59q⁵ + 43q⁶ + 6q⁷ - 25q⁸ + 11q⁹ + 3q¹⁰ - 4q¹¹ + q¹²

3 - q^-30 + 3q^-29 - q^-28 - 5q^-27 - 2q^-26 + 17q^-25 + 7q^-24 - 36q^-23 - 25q^-22 + 61q^-21 + 64q^-20 - 83q^-19 - 131q^-18 + 97q^-17 + 214q^-16 - 82q^-15 - 312q^-14 + 38q^-13 + 411q^-12 + 23q^-11 - 486q^-10 - 111q^-9 + 550q^-8 + 193q^-7 - 577q^-6 - 282q^-5 + 595q^-4 + 342q^-3 - 567q^-2 - 407q^-1 + 536 + 431q - 460q² - 452q³ + 382q⁴ + 430q⁵ - 276q⁶ - 392q⁷ + 179q⁸ + 326q⁹ - 94q¹⁰ - 247q¹¹ + 34q¹² + 166q¹³ - q¹⁴ - 96q¹⁵ - 14q¹⁶ + 53q¹⁷ + 7q¹⁸ - 20q¹⁹ - 5q²⁰ + 7q²¹ + 3q²² - 4q²³ + q²⁴

4 q^-50 - 3q^-49 + q^-48 + 5q^-47 - 3q^-46 + 2q^-45 - 20q^-44 + 5q^-43 + 38q^-42 + 4q^-41 + 2q^-40 - 111q^-39 - 33q^-38 + 137q^-37 + 119q^-36 + 97q^-35 - 342q^-34 - 298q^-33 + 161q^-32 + 423q^-31 + 584q^-30 - 513q^-29 - 904q^-28 - 268q^-27 + 641q^-26 + 1590q^-25 - 172q^-24 - 1506q^-23 - 1290q^-22 + 288q^-21 + 2717q^-20 + 806q^-19 - 1596q^-18 - 2498q^-17 - 701q^-16 + 3422q^-15 + 2012q^-14 - 1092q^-13 - 3396q^-12 - 1903q^-11 + 3566q^-10 + 2982q^-9 - 316q^-8 - 3814q^-7 - 2904q^-6 + 3288q^-5 + 3546q^-4 + 478q^-3 - 3782q^-2 - 3560q^-1 + 2665 + 3666q + 1225q² - 3254q³ - 3797q⁴ + 1682q⁵ + 3236q⁶ + 1822q⁷ - 2208q⁸ - 3454q⁹ + 557q¹⁰ + 2229q¹¹ + 1963q¹² - 948q¹³ - 2483q¹⁴ - 227q¹⁵ + 1012q¹⁶ + 1494q¹⁷ - 56q¹⁸ - 1285q¹⁹ - 384q²⁰ + 180q²¹ + 752q²² + 202q²³ - 438q²⁴ - 189q²⁵ - 73q²⁶ + 237q²⁷ + 113q²⁸ - 99q²⁹ - 34q³⁰ - 47q³¹ + 46q³² + 27q³³ - 19q³⁴ - 9q³⁶ + 7q³⁷ + 3q³⁸ - 4q³⁹ + q⁴⁰

5 - q^-75 + 3q^-74 - q^-73 - 5q^-72 + 3q^-71 + 3q^-70 + q^-69 + 8q^-68 - 7q^-67 - 33q^-66 - 7q^-65 + 33q^-64 + 52q^-63 + 56q^-62 - 31q^-61 - 161q^-60 - 170q^-59 + 19q^-58 + 280q^-57 + 411q^-56 + 176q^-55 - 406q^-54 - 844q^-53 - 621q^-52 + 352q^-51 + 1388q^-50 + 1473q^-49 + 121q^-48 - 1861q^-47 - 2742q^-46 - 1262q^-45 + 1950q^-44 + 4243q^-43 + 3155q^-42 - 1207q^-41 - 5601q^-40 - 5803q^-39 - 573q^-38 + 6375q^-37 + 8777q^-36 + 3483q^-35 - 6070q^-34 - 11671q^-33 - 7333q^-32 + 4553q^-31 + 13979q^-30 + 11622q^-29 - 1851q^-28 - 15251q^-27 - 15968q^-26 - 1769q^-25 + 15547q^-24 + 19810q^-23 + 5780q^-22 - 14763q^-21 - 22943q^-20 - 9926q^-19 + 13403q^-18 + 25228q^-17 + 13655q^-16 - 11484q^-15 - 26787q^-14 - 16995q^-13 + 9544q^-12 + 27652q^-11 + 19688q^-10 - 7391q^-9 - 28057q^-8 - 22014q^-7 + 5437q^-6 + 27888q^-5 + 23779q^-4 - 3159q^-3 - 27295q^-2 - 25297q^-1 + 943 + 25992q + 26187q² + 1744q³ - 23971q⁴ - 26648q⁵ - 4385q⁶ + 21050q⁷ + 26127q⁸ + 7175q⁹ - 17328q¹⁰ - 24715q¹¹ - 9472q¹² + 12986q¹³ + 22129q¹⁴ + 11117q¹⁵ - 8472q¹⁶ - 18631q¹⁷ - 11655q¹⁸ + 4271q¹⁹ + 14476q²⁰ + 11121q²¹ - 936q²² - 10233q²³ - 9570q²⁴ - 1275q²⁵ + 6404q²⁶ + 7426q²⁷ + 2350q²⁸ - 3440q²⁹ - 5175q³⁰ - 2440q³¹ + 1461q³² + 3150q³³ + 2021q³⁴ - 338q³⁵ - 1752q³⁶ - 1353q³⁷ - 80q³⁸ + 791q³⁹ + 781q⁴⁰ + 213q⁴¹ - 335q⁴² - 403q⁴³ - 130q⁴⁴ + 111q⁴⁵ + 165q⁴⁶ + 71q⁴⁷ - 21q⁴⁸ - 72q⁴⁹ - 37q⁵⁰ + 23q⁵¹ + 20q⁵² + q⁵³ + q⁵⁴ - 4q⁵⁵ - 9q⁵⁶ + 7q⁵⁷ + 3q⁵⁸ - 4q⁵⁹ + q⁶⁰

6 q^-105 - 3q^-104 + q^-103 + 5q^-102 - 3q^-101 - 3q^-100 - 6q^-99 + 11q^-98 - 6q^-97 + 2q^-96 + 36q^-95 - 12q^-94 - 34q^-93 - 63q^-92 + 12q^-91 + 7q^-90 + 65q^-89 + 215q^-88 + 60q^-87 - 127q^-86 - 391q^-85 - 266q^-84 - 212q^-83 + 220q^-82 + 1000q^-81 + 916q^-80 + 333q^-79 - 1015q^-78 - 1606q^-77 - 2121q^-76 - 915q^-75 + 2079q^-74 + 3883q^-73 + 4009q^-72 + 764q^-71 - 2882q^-70 - 7524q^-69 - 7641q^-68 - 1467q^-67 + 6416q^-66 + 12832q^-65 + 11205q^-64 + 3842q^-63 - 11473q^-62 - 21421q^-61 - 17972q^-60 - 2915q^-59 + 18440q^-58 + 30486q^-57 + 28461q^-56 + 1254q^-55 - 29796q^-54 - 45436q^-53 - 34698q^-52 + 2172q^-51 + 41954q^-50 + 65843q^-49 + 40900q^-48 - 11723q^-47 - 63304q^-46 - 80241q^-45 - 44665q^-44 + 23687q^-43 + 92289q^-42 + 95210q^-41 + 38832q^-40 - 50107q^-39 - 114093q^-38 - 106483q^-37 - 26967q^-36 + 87871q^-35 + 138130q^-34 + 103658q^-33 - 6329q^-32 - 118833q^-31 - 157590q^-30 - 90163q^-29 + 55581q^-28 + 154703q^-27 + 158360q^-26 + 48156q^-25 - 99105q^-24 - 185196q^-23 - 143716q^-22 + 13884q^-21 + 149869q^-20 + 192292q^-19 + 94486q^-18 - 70883q^-17 - 193704q^-16 - 179367q^-15 - 22434q^-14 + 136239q^-13 + 209410q^-12 + 127353q^-11 - 44618q^-10 - 192445q^-9 - 201224q^-8 - 51346q^-7 + 120052q^-6 + 216694q^-5 + 151725q^-4 - 19189q^-3 - 184164q^-2 - 214818q^-1 - 78957 + 97330q + 214259q² + 172178q³ + 12510q⁴ - 162501q⁵ - 218005q⁶ - 108722q⁷ + 60262q⁸ + 193635q⁹ + 184020q¹⁰ + 52288q¹¹ - 119348q¹² - 200071q¹³ - 132607q¹⁴ + 9390q¹⁵ + 146686q¹⁶ + 173726q¹⁷ + 88047q¹⁸ - 58309q¹⁹ - 152968q²⁰ - 133685q²¹ - 38506q²² + 80365q²³ + 133075q²⁴ + 99561q²⁵ - 817q²⁶ - 86828q²⁷ - 103623q²⁸ - 60527q²⁹ + 19753q³⁰ + 74624q³¹ + 79230q³² + 29035q³³ - 28710q³⁴ - 56897q³⁵ - 50779q³⁶ - 12018q³⁷ + 25450q³⁸ + 43332q³⁹ + 28074q⁴⁰ + 1159q⁴¹ - 19157q⁴² - 26770q⁴³ - 15319q⁴⁴ + 1604q⁴⁵ + 15272q⁴⁶ + 14254q⁴⁷ + 6490q⁴⁸ - 2120q⁴⁹ - 8728q⁵⁰ - 7713q⁵¹ - 2969q⁵² + 3071q⁵³ + 4151q⁵⁴ + 3197q⁵⁵ + 1143q⁵⁶ - 1601q⁵⁷ - 2193q⁵⁸ - 1502q⁵⁹ + 286q⁶⁰ + 634q⁶¹ + 755q⁶² + 585q⁶³ - 122q⁶⁴ - 385q⁶⁵ - 372q⁶⁶ + 45q⁶⁷ + 26q⁶⁸ + 86q⁶⁹ + 130q⁷⁰ - 3q⁷¹ - 46q⁷² - 65q⁷³ + 33q⁷⁴ - 3q⁷⁵ - 6q⁷⁶ + 21q⁷⁷ - 3q⁷⁸ - 4q⁷⁹ - 9q⁸⁰ + 7q⁸¹ + 3q⁸² - 4q⁸³ + q⁸⁴

7 - q^-140 + 3q^-139 - q^-138 - 5q^-137 + 3q^-136 + 3q^-135 + 6q^-134 - 6q^-133 - 13q^-132 + 11q^-131 - 5q^-130 - 17q^-129 + 13q^-128 + 29q^-127 + 55q^-126 - 85q^-124 - 43q^-123 - 88q^-122 - 88q^-121 + 54q^-120 + 188q^-119 + 409q^-118 + 288q^-117 - 118q^-116 - 376q^-115 - 791q^-114 - 917q^-113 - 420q^-112 + 395q^-111 + 1790q^-110 + 2427q^-109 + 1678q^-108 + 206q^-107 - 2525q^-106 - 4780q^-105 - 5102q^-104 - 3152q^-103 + 2220q^-102 + 7989q^-101 + 10926q^-100 + 9813q^-99 + 1991q^-98 - 9240q^-97 - 18756q^-96 - 22439q^-95 - 13638q^-94 + 4737q^-93 + 25351q^-92 + 39979q^-91 + 35831q^-90 + 12089q^-89 - 23549q^-88 - 58261q^-87 - 69150q^-86 - 46924q^-85 + 4057q^-84 + 67702q^-83 + 107960q^-82 + 101582q^-81 + 42625q^-80 - 54247q^-79 - 140049q^-78 - 171176q^-77 - 121531q^-76 + 4595q^-75 + 148133q^-74 + 241102q^-73 + 228197q^-72 + 90343q^-71 - 112897q^-70 - 290825q^-69 - 348951q^-68 - 229324q^-67 + 22031q^-66 + 297765q^-65 + 460243q^-64 + 399023q^-63 + 127750q^-62 - 243628q^-61 - 537227q^-60 - 577412q^-59 - 326102q^-58 + 122500q^-57 + 558450q^-56 + 736826q^-55 + 551937q^-54 + 60416q^-53 - 512171q^-52 - 854456q^-51 - 779577q^-50 - 286916q^-49 + 400916q^-48 + 914623q^-47 + 982955q^-46 + 533062q^-45 - 236456q^-44 - 913617q^-43 - 1145254q^-42 - 774451q^-41 + 40157q^-40 + 859423q^-39 + 1256627q^-38 + 990462q^-37 + 167085q^-36 - 765645q^-35 - 1319360q^-34 - 1170292q^-33 - 364860q^-32 + 650820q^-31 + 1340844q^-30 + 1308914q^-29 + 541117q^-28 - 530059q^-27 - 1334094q^-26 - 1410389q^-25 - 688591q^-24 + 416318q^-23 + 1310013q^-22 + 1481362q^-21 + 808575q^-20 - 315323q^-19 - 1279794q^-18 - 1531416q^-17 - 904233q^-16 + 228507q^-15 + 1247728q^-14 + 1568187q^-13 + 984896q^-12 - 151677q^-11 - 1217127q^-10 - 1598273q^-9 - 1056484q^-8 + 78632q^-7 + 1182743q^-6 + 1623094q^-5 + 1128107q^-4 + 335q^-3 - 1140218q^-2 - 1641850q^-1 - 1201173 - 92091q + 1077731q² + 1646488q³ + 1277395q⁴ + 204023q⁵ - 987518q⁶ - 1628437q⁷ - 1348375q⁸ - 335270q⁹ + 859516q¹⁰ + 1573547q¹¹ + 1404365q¹² + 481739q¹³ - 692247q¹⁴ - 1472214q¹⁵ - 1428674q¹⁶ - 628897q¹⁷ + 489369q¹⁸ + 1316820q¹⁹ + 1406486q²⁰ + 759123q²¹ - 265534q²² - 1110482q²³ - 1325913q²⁴ - 850809q²⁵ + 41939q²⁶ + 865421q²⁷ + 1185063q²⁸ + 886693q²⁹ + 155533q³⁰ - 604378q³¹ - 992330q³² - 858018q³³ - 304353q³⁴ + 354914q³⁵ + 768535q³⁶ + 768599q³⁷ + 388993q³⁸ - 143541q³⁹ - 539777q⁴⁰ - 633728q⁴¹ - 408448q⁴² - 11592q⁴³ + 333623q⁴⁴ + 477633q⁴⁵ + 373033q⁴⁶ + 103593q⁴⁷ - 169637q⁴⁸ - 324942q⁴⁹ - 302390q⁵⁰ - 139663q⁵¹ + 56536q⁵² + 196159q⁵³ + 219360q⁵⁴ + 134706q⁵⁵ + 7287q⁵⁶ - 101281q⁵⁷ - 141435q⁵⁸ - 107299q⁵⁹ - 34268q⁶⁰ + 41025q⁶¹ + 81192q⁶² + 73834q⁶³ + 36910q⁶⁴ - 9504q⁶⁵ - 40107q⁶⁶ - 44109q⁶⁷ - 29195q⁶⁸ - 3684q⁶⁹ + 16832q⁷⁰ + 23424q⁷¹ + 18797q⁷² + 6189q⁷³ - 5481q⁷⁴ - 10542q⁷⁵ - 10410q⁷⁶ - 5092q⁷⁷ + 933q⁷⁸ + 4209q⁷⁹ + 5199q⁸⁰ + 2973q⁸¹ + 160q⁸² - 1371q⁸³ - 2136q⁸⁴ - 1462q⁸⁵ - 399q⁸⁶ + 297q⁸⁷ + 935q⁸⁸ + 684q⁸⁹ + 121q⁹⁰ - 86q⁹¹ - 291q⁹² - 180q⁹³ - 86q⁹⁴ - 62q⁹⁵ + 132q⁹⁶ + 111q⁹⁷ + 4q⁹⁸ - 18q⁹⁹ - 39q¹⁰⁰ + 5q¹⁰¹ + 7q¹⁰² - 29q¹⁰³ + 14q¹⁰⁴ + 17q¹⁰⁵ - 3q¹⁰⁶ - 4q¹⁰⁷ - 9q¹⁰⁸ + 7q¹⁰⁹ + 3q¹¹⁰ - 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, 33]]

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

In[3]:=
GaussCode[Knot[9, 33]]

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

In[4]:=
DTCode[Knot[9, 33]]

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

In[5]:=
br = BR[Knot[9, 33]]

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

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

Out[6]=
{4, 9}

In[7]:=
BraidIndex[Knot[9, 33]]

Out[7]=
4

In[8]:=
Show[DrawMorseLink[Knot[9, 33]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[9, 33]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

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

In[10]:=
alex = Alexander[Knot[9, 33]][t]

Out[10]=
-3 6 14 2 3 19 - t + -- - -- - 14 t + 6 t - t 2 t t

In[11]:=
Conway[Knot[9, 33]][z]

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

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

Out[12]=
{Knot[9, 33], Knot[11, NonAlternating, 55]}

In[13]:=
{KnotDet[Knot[9, 33]], KnotSignature[Knot[9, 33]]}

Out[13]=
{61, 0}

In[14]:=
Jones[Knot[9, 33]][q]

Out[14]=
-5 3 6 9 10 2 3 4 11 - q + -- - -- + -- - -- - 9 q + 7 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, 33]}

In[16]:=
A2Invariant[Knot[9, 33]][q]

Out[16]=
-16 -12 2 2 3 2 4 8 10 12 -1 - q + q - --- + -- + -- + 3 q - 2 q + q - 2 q + q 10 8 2 q q q

In[17]:=
HOMFLYPT[Knot[9, 33]][a, z]

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

In[18]:=
Kauffman[Knot[9, 33]][a, z]

Out[18]=
2 3 3 2 4 3 5 2 3 z 2 2 4 2 3 z z -2 a - a + a z + a z + 9 z + ---- + 10 a z + 4 a z - ---- - -- + 2 3 a a a 4 4 5 3 3 3 5 3 4 z 9 z 2 4 4 4 4 z > 5 a z + a z - 2 a z - 20 z + -- - ---- - 16 a z - 6 a z + ---- - 4 2 3 a a a 5 6 7 5 z 5 3 5 5 5 6 7 z 2 6 4 6 6 z > ---- - 16 a z - 6 a z + a z + 9 z + ---- + 5 a z + 3 a z + ---- + a 2 a a 7 3 7 8 2 8 > 10 a z + 4 a z + 2 z + 2 a z

In[19]:=
{Vassiliev[2][Knot[9, 33]], Vassiliev[3][Knot[9, 33]]}

Out[19]=
{1, -1}

In[20]:=
Kh[Knot[9, 33]][q, t]

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

In[21]:=
ColouredJones[Knot[9, 33], 2][q]

Out[21]=
-15 3 -13 10 17 4 40 37 27 83 46 60 113 115 + q - --- + q + --- - --- - --- + -- - -- - -- + -- - -- - -- + --- - 14 12 11 10 9 8 7 6 5 4 3 q q q q q q q q q q q 39 83 2 3 4 5 6 7 8 > -- - -- - 22 q - 83 q + 86 q - 3 q - 59 q + 43 q + 6 q - 25 q + 2 q q 9 10 11 12 > 11 q + 3 q - 4 q + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 9₃₃

9₃₂
9₃₄

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-15 - 3q^-14 + q^-13 + 10q^-12 - 17q^-11 - 4q^-10 + 40q^-9 - 37q^-8 - 27q^-7 + 83q^-6 - 46q^-5 - 60q^-4 + 113q^-3 - 39q^-2 - 83q^-1 + 115 - 22q - 83q² + 86q³ - 3q⁴ - 59q⁵ + 43q⁶ + 6q⁷ - 25q⁸ + 11q⁹ + 3q¹⁰ - 4q¹¹ + q¹²
3	- q^-30 + 3q^-29 - q^-28 - 5q^-27 - 2q^-26 + 17q^-25 + 7q^-24 - 36q^-23 - 25q^-22 + 61q^-21 + 64q^-20 - 83q^-19 - 131q^-18 + 97q^-17 + 214q^-16 - 82q^-15 - 312q^-14 + 38q^-13 + 411q^-12 + 23q^-11 - 486q^-10 - 111q^-9 + 550q^-8 + 193q^-7 - 577q^-6 - 282q^-5 + 595q^-4 + 342q^-3 - 567q^-2 - 407q^-1 + 536 + 431q - 460q² - 452q³ + 382q⁴ + 430q⁵ - 276q⁶ - 392q⁷ + 179q⁸ + 326q⁹ - 94q¹⁰ - 247q¹¹ + 34q¹² + 166q¹³ - q¹⁴ - 96q¹⁵ - 14q¹⁶ + 53q¹⁷ + 7q¹⁸ - 20q¹⁹ - 5q²⁰ + 7q²¹ + 3q²² - 4q²³ + q²⁴
4	q^-50 - 3q^-49 + q^-48 + 5q^-47 - 3q^-46 + 2q^-45 - 20q^-44 + 5q^-43 + 38q^-42 + 4q^-41 + 2q^-40 - 111q^-39 - 33q^-38 + 137q^-37 + 119q^-36 + 97q^-35 - 342q^-34 - 298q^-33 + 161q^-32 + 423q^-31 + 584q^-30 - 513q^-29 - 904q^-28 - 268q^-27 + 641q^-26 + 1590q^-25 - 172q^-24 - 1506q^-23 - 1290q^-22 + 288q^-21 + 2717q^-20 + 806q^-19 - 1596q^-18 - 2498q^-17 - 701q^-16 + 3422q^-15 + 2012q^-14 - 1092q^-13 - 3396q^-12 - 1903q^-11 + 3566q^-10 + 2982q^-9 - 316q^-8 - 3814q^-7 - 2904q^-6 + 3288q^-5 + 3546q^-4 + 478q^-3 - 3782q^-2 - 3560q^-1 + 2665 + 3666q + 1225q² - 3254q³ - 3797q⁴ + 1682q⁵ + 3236q⁶ + 1822q⁷ - 2208q⁸ - 3454q⁹ + 557q¹⁰ + 2229q¹¹ + 1963q¹² - 948q¹³ - 2483q¹⁴ - 227q¹⁵ + 1012q¹⁶ + 1494q¹⁷ - 56q¹⁸ - 1285q¹⁹ - 384q²⁰ + 180q²¹ + 752q²² + 202q²³ - 438q²⁴ - 189q²⁵ - 73q²⁶ + 237q²⁷ + 113q²⁸ - 99q²⁹ - 34q³⁰ - 47q³¹ + 46q³² + 27q³³ - 19q³⁴ - 9q³⁶ + 7q³⁷ + 3q³⁸ - 4q³⁹ + q⁴⁰
5	- q^-75 + 3q^-74 - q^-73 - 5q^-72 + 3q^-71 + 3q^-70 + q^-69 + 8q^-68 - 7q^-67 - 33q^-66 - 7q^-65 + 33q^-64 + 52q^-63 + 56q^-62 - 31q^-61 - 161q^-60 - 170q^-59 + 19q^-58 + 280q^-57 + 411q^-56 + 176q^-55 - 406q^-54 - 844q^-53 - 621q^-52 + 352q^-51 + 1388q^-50 + 1473q^-49 + 121q^-48 - 1861q^-47 - 2742q^-46 - 1262q^-45 + 1950q^-44 + 4243q^-43 + 3155q^-42 - 1207q^-41 - 5601q^-40 - 5803q^-39 - 573q^-38 + 6375q^-37 + 8777q^-36 + 3483q^-35 - 6070q^-34 - 11671q^-33 - 7333q^-32 + 4553q^-31 + 13979q^-30 + 11622q^-29 - 1851q^-28 - 15251q^-27 - 15968q^-26 - 1769q^-25 + 15547q^-24 + 19810q^-23 + 5780q^-22 - 14763q^-21 - 22943q^-20 - 9926q^-19 + 13403q^-18 + 25228q^-17 + 13655q^-16 - 11484q^-15 - 26787q^-14 - 16995q^-13 + 9544q^-12 + 27652q^-11 + 19688q^-10 - 7391q^-9 - 28057q^-8 - 22014q^-7 + 5437q^-6 + 27888q^-5 + 23779q^-4 - 3159q^-3 - 27295q^-2 - 25297q^-1 + 943 + 25992q + 26187q² + 1744q³ - 23971q⁴ - 26648q⁵ - 4385q⁶ + 21050q⁷ + 26127q⁸ + 7175q⁹ - 17328q¹⁰ - 24715q¹¹ - 9472q¹² + 12986q¹³ + 22129q¹⁴ + 11117q¹⁵ - 8472q¹⁶ - 18631q¹⁷ - 11655q¹⁸ + 4271q¹⁹ + 14476q²⁰ + 11121q²¹ - 936q²² - 10233q²³ - 9570q²⁴ - 1275q²⁵ + 6404q²⁶ + 7426q²⁷ + 2350q²⁸ - 3440q²⁹ - 5175q³⁰ - 2440q³¹ + 1461q³² + 3150q³³ + 2021q³⁴ - 338q³⁵ - 1752q³⁶ - 1353q³⁷ - 80q³⁸ + 791q³⁹ + 781q⁴⁰ + 213q⁴¹ - 335q⁴² - 403q⁴³ - 130q⁴⁴ + 111q⁴⁵ + 165q⁴⁶ + 71q⁴⁷ - 21q⁴⁸ - 72q⁴⁹ - 37q⁵⁰ + 23q⁵¹ + 20q⁵² + q⁵³ + q⁵⁴ - 4q⁵⁵ - 9q⁵⁶ + 7q⁵⁷ + 3q⁵⁸ - 4q⁵⁹ + q⁶⁰
6	q^-105 - 3q^-104 + q^-103 + 5q^-102 - 3q^-101 - 3q^-100 - 6q^-99 + 11q^-98 - 6q^-97 + 2q^-96 + 36q^-95 - 12q^-94 - 34q^-93 - 63q^-92 + 12q^-91 + 7q^-90 + 65q^-89 + 215q^-88 + 60q^-87 - 127q^-86 - 391q^-85 - 266q^-84 - 212q^-83 + 220q^-82 + 1000q^-81 + 916q^-80 + 333q^-79 - 1015q^-78 - 1606q^-77 - 2121q^-76 - 915q^-75 + 2079q^-74 + 3883q^-73 + 4009q^-72 + 764q^-71 - 2882q^-70 - 7524q^-69 - 7641q^-68 - 1467q^-67 + 6416q^-66 + 12832q^-65 + 11205q^-64 + 3842q^-63 - 11473q^-62 - 21421q^-61 - 17972q^-60 - 2915q^-59 + 18440q^-58 + 30486q^-57 + 28461q^-56 + 1254q^-55 - 29796q^-54 - 45436q^-53 - 34698q^-52 + 2172q^-51 + 41954q^-50 + 65843q^-49 + 40900q^-48 - 11723q^-47 - 63304q^-46 - 80241q^-45 - 44665q^-44 + 23687q^-43 + 92289q^-42 + 95210q^-41 + 38832q^-40 - 50107q^-39 - 114093q^-38 - 106483q^-37 - 26967q^-36 + 87871q^-35 + 138130q^-34 + 103658q^-33 - 6329q^-32 - 118833q^-31 - 157590q^-30 - 90163q^-29 + 55581q^-28 + 154703q^-27 + 158360q^-26 + 48156q^-25 - 99105q^-24 - 185196q^-23 - 143716q^-22 + 13884q^-21 + 149869q^-20 + 192292q^-19 + 94486q^-18 - 70883q^-17 - 193704q^-16 - 179367q^-15 - 22434q^-14 + 136239q^-13 + 209410q^-12 + 127353q^-11 - 44618q^-10 - 192445q^-9 - 201224q^-8 - 51346q^-7 + 120052q^-6 + 216694q^-5 + 151725q^-4 - 19189q^-3 - 184164q^-2 - 214818q^-1 - 78957 + 97330q + 214259q² + 172178q³ + 12510q⁴ - 162501q⁵ - 218005q⁶ - 108722q⁷ + 60262q⁸ + 193635q⁹ + 184020q¹⁰ + 52288q¹¹ - 119348q¹² - 200071q¹³ - 132607q¹⁴ + 9390q¹⁵ + 146686q¹⁶ + 173726q¹⁷ + 88047q¹⁸ - 58309q¹⁹ - 152968q²⁰ - 133685q²¹ - 38506q²² + 80365q²³ + 133075q²⁴ + 99561q²⁵ - 817q²⁶ - 86828q²⁷ - 103623q²⁸ - 60527q²⁹ + 19753q³⁰ + 74624q³¹ + 79230q³² + 29035q³³ - 28710q³⁴ - 56897q³⁵ - 50779q³⁶ - 12018q³⁷ + 25450q³⁸ + 43332q³⁹ + 28074q⁴⁰ + 1159q⁴¹ - 19157q⁴² - 26770q⁴³ - 15319q⁴⁴ + 1604q⁴⁵ + 15272q⁴⁶ + 14254q⁴⁷ + 6490q⁴⁸ - 2120q⁴⁹ - 8728q⁵⁰ - 7713q⁵¹ - 2969q⁵² + 3071q⁵³ + 4151q⁵⁴ + 3197q⁵⁵ + 1143q⁵⁶ - 1601q⁵⁷ - 2193q⁵⁸ - 1502q⁵⁹ + 286q⁶⁰ + 634q⁶¹ + 755q⁶² + 585q⁶³ - 122q⁶⁴ - 385q⁶⁵ - 372q⁶⁶ + 45q⁶⁷ + 26q⁶⁸ + 86q⁶⁹ + 130q⁷⁰ - 3q⁷¹ - 46q⁷² - 65q⁷³ + 33q⁷⁴ - 3q⁷⁵ - 6q⁷⁶ + 21q⁷⁷ - 3q⁷⁸ - 4q⁷⁹ - 9q⁸⁰ + 7q⁸¹ + 3q⁸² - 4q⁸³ + q⁸⁴
7	- q^-140 + 3q^-139 - q^-138 - 5q^-137 + 3q^-136 + 3q^-135 + 6q^-134 - 6q^-133 - 13q^-132 + 11q^-131 - 5q^-130 - 17q^-129 + 13q^-128 + 29q^-127 + 55q^-126 - 85q^-124 - 43q^-123 - 88q^-122 - 88q^-121 + 54q^-120 + 188q^-119 + 409q^-118 + 288q^-117 - 118q^-116 - 376q^-115 - 791q^-114 - 917q^-113 - 420q^-112 + 395q^-111 + 1790q^-110 + 2427q^-109 + 1678q^-108 + 206q^-107 - 2525q^-106 - 4780q^-105 - 5102q^-104 - 3152q^-103 + 2220q^-102 + 7989q^-101 + 10926q^-100 + 9813q^-99 + 1991q^-98 - 9240q^-97 - 18756q^-96 - 22439q^-95 - 13638q^-94 + 4737q^-93 + 25351q^-92 + 39979q^-91 + 35831q^-90 + 12089q^-89 - 23549q^-88 - 58261q^-87 - 69150q^-86 - 46924q^-85 + 4057q^-84 + 67702q^-83 + 107960q^-82 + 101582q^-81 + 42625q^-80 - 54247q^-79 - 140049q^-78 - 171176q^-77 - 121531q^-76 + 4595q^-75 + 148133q^-74 + 241102q^-73 + 228197q^-72 + 90343q^-71 - 112897q^-70 - 290825q^-69 - 348951q^-68 - 229324q^-67 + 22031q^-66 + 297765q^-65 + 460243q^-64 + 399023q^-63 + 127750q^-62 - 243628q^-61 - 537227q^-60 - 577412q^-59 - 326102q^-58 + 122500q^-57 + 558450q^-56 + 736826q^-55 + 551937q^-54 + 60416q^-53 - 512171q^-52 - 854456q^-51 - 779577q^-50 - 286916q^-49 + 400916q^-48 + 914623q^-47 + 982955q^-46 + 533062q^-45 - 236456q^-44 - 913617q^-43 - 1145254q^-42 - 774451q^-41 + 40157q^-40 + 859423q^-39 + 1256627q^-38 + 990462q^-37 + 167085q^-36 - 765645q^-35 - 1319360q^-34 - 1170292q^-33 - 364860q^-32 + 650820q^-31 + 1340844q^-30 + 1308914q^-29 + 541117q^-28 - 530059q^-27 - 1334094q^-26 - 1410389q^-25 - 688591q^-24 + 416318q^-23 + 1310013q^-22 + 1481362q^-21 + 808575q^-20 - 315323q^-19 - 1279794q^-18 - 1531416q^-17 - 904233q^-16 + 228507q^-15 + 1247728q^-14 + 1568187q^-13 + 984896q^-12 - 151677q^-11 - 1217127q^-10 - 1598273q^-9 - 1056484q^-8 + 78632q^-7 + 1182743q^-6 + 1623094q^-5 + 1128107q^-4 + 335q^-3 - 1140218q^-2 - 1641850q^-1 - 1201173 - 92091q + 1077731q² + 1646488q³ + 1277395q⁴ + 204023q⁵ - 987518q⁶ - 1628437q⁷ - 1348375q⁸ - 335270q⁹ + 859516q¹⁰ + 1573547q¹¹ + 1404365q¹² + 481739q¹³ - 692247q¹⁴ - 1472214q¹⁵ - 1428674q¹⁶ - 628897q¹⁷ + 489369q¹⁸ + 1316820q¹⁹ + 1406486q²⁰ + 759123q²¹ - 265534q²² - 1110482q²³ - 1325913q²⁴ - 850809q²⁵ + 41939q²⁶ + 865421q²⁷ + 1185063q²⁸ + 886693q²⁹ + 155533q³⁰ - 604378q³¹ - 992330q³² - 858018q³³ - 304353q³⁴ + 354914q³⁵ + 768535q³⁶ + 768599q³⁷ + 388993q³⁸ - 143541q³⁹ - 539777q⁴⁰ - 633728q⁴¹ - 408448q⁴² - 11592q⁴³ + 333623q⁴⁴ + 477633q⁴⁵ + 373033q⁴⁶ + 103593q⁴⁷ - 169637q⁴⁸ - 324942q⁴⁹ - 302390q⁵⁰ - 139663q⁵¹ + 56536q⁵² + 196159q⁵³ + 219360q⁵⁴ + 134706q⁵⁵ + 7287q⁵⁶ - 101281q⁵⁷ - 141435q⁵⁸ - 107299q⁵⁹ - 34268q⁶⁰ + 41025q⁶¹ + 81192q⁶² + 73834q⁶³ + 36910q⁶⁴ - 9504q⁶⁵ - 40107q⁶⁶ - 44109q⁶⁷ - 29195q⁶⁸ - 3684q⁶⁹ + 16832q⁷⁰ + 23424q⁷¹ + 18797q⁷² + 6189q⁷³ - 5481q⁷⁴ - 10542q⁷⁵ - 10410q⁷⁶ - 5092q⁷⁷ + 933q⁷⁸ + 4209q⁷⁹ + 5199q⁸⁰ + 2973q⁸¹ + 160q⁸² - 1371q⁸³ - 2136q⁸⁴ - 1462q⁸⁵ - 399q⁸⁶ + 297q⁸⁷ + 935q⁸⁸ + 684q⁸⁹ + 121q⁹⁰ - 86q⁹¹ - 291q⁹² - 180q⁹³ - 86q⁹⁴ - 62q⁹⁵ + 132q⁹⁶ + 111q⁹⁷ + 4q⁹⁸ - 18q⁹⁹ - 39q¹⁰⁰ + 5q¹⁰¹ + 7q¹⁰² - 29q¹⁰³ + 14q¹⁰⁴ + 17q¹⁰⁵ - 3q¹⁰⁶ - 4q¹⁰⁷ - 9q¹⁰⁸ + 7q¹⁰⁹ + 3q¹¹⁰ - 4q¹¹¹ + q¹¹²

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	PD[Knot[9, 33]]
Out[2]=	PD[X[4, 2, 5, 1], X[12, 8, 13, 7], X[8, 3, 9, 4], X[2, 9, 3, 10], > X[18, 13, 1, 14], X[14, 5, 15, 6], X[6, 17, 7, 18], X[16, 12, 17, 11], > X[10, 16, 11, 15]]
In[3]:=	GaussCode[Knot[9, 33]]
Out[3]=	GaussCode[1, -4, 3, -1, 6, -7, 2, -3, 4, -9, 8, -2, 5, -6, 9, -8, 7, -5]
In[4]:=	DTCode[Knot[9, 33]]
Out[4]=	DTCode[4, 8, 14, 12, 2, 16, 18, 10, 6]
In[5]:=	br = BR[Knot[9, 33]]
Out[5]=	BR[4, {-1, 2, -1, 2, 2, -1, -3, 2, -3}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 9}
In[7]:=	BraidIndex[Knot[9, 33]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[9, 33]]]

Out[8]=	-Graphics-
In[9]:=	#[Knot[9, 33]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Chiral, 1, 3, 3, {4, 6}, 1}
In[10]:=	alex = Alexander[Knot[9, 33]][t]
Out[10]=	-3 6 14 2 3 19 - t + -- - -- - 14 t + 6 t - t 2 t t
In[11]:=	Conway[Knot[9, 33]][z]
Out[11]=	2 6 1 + z - z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[9, 33], Knot[11, NonAlternating, 55]}
In[13]:=	{KnotDet[Knot[9, 33]], KnotSignature[Knot[9, 33]]}
Out[13]=	{61, 0}
In[14]:=	Jones[Knot[9, 33]][q]
Out[14]=	-5 3 6 9 10 2 3 4 11 - q + -- - -- + -- - -- - 9 q + 7 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, 33]}
In[16]:=	A2Invariant[Knot[9, 33]][q]
Out[16]=	-16 -12 2 2 3 2 4 8 10 12 -1 - q + q - --- + -- + -- + 3 q - 2 q + q - 2 q + q 10 8 2 q q q
In[17]:=	HOMFLYPT[Knot[9, 33]][a, z]
Out[17]=	2 4 2 4 2 z 2 2 4 2 4 z 2 4 6 2 a - a - 3 z + -- + 4 a z - a z - 3 z + -- + 2 a z - z 2 2 a a
In[18]:=	Kauffman[Knot[9, 33]][a, z]
Out[18]=	2 3 3 2 4 3 5 2 3 z 2 2 4 2 3 z z -2 a - a + a z + a z + 9 z + ---- + 10 a z + 4 a z - ---- - -- + 2 3 a a a 4 4 5 3 3 3 5 3 4 z 9 z 2 4 4 4 4 z > 5 a z + a z - 2 a z - 20 z + -- - ---- - 16 a z - 6 a z + ---- - 4 2 3 a a a 5 6 7 5 z 5 3 5 5 5 6 7 z 2 6 4 6 6 z > ---- - 16 a z - 6 a z + a z + 9 z + ---- + 5 a z + 3 a z + ---- + a 2 a a 7 3 7 8 2 8 > 10 a z + 4 a z + 2 z + 2 a z
In[19]:=	{Vassiliev[2][Knot[9, 33]], Vassiliev[3][Knot[9, 33]]}
Out[19]=	{1, -1}
In[20]:=	Kh[Knot[9, 33]][q, t]
Out[20]=	6 1 2 1 4 2 5 4 5 5 - + 6 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ---- + --- + q 11 5 9 4 7 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t q t q t 3 3 2 5 2 5 3 7 3 9 4 > 4 q t + 5 q t + 3 q t + 4 q t + q t + 3 q t + q t
In[21]:=	ColouredJones[Knot[9, 33], 2][q]
Out[21]=	-15 3 -13 10 17 4 40 37 27 83 46 60 113 115 + q - --- + q + --- - --- - --- + -- - -- - -- + -- - -- - -- + --- - 14 12 11 10 9 8 7 6 5 4 3 q q q q q q q q q q q 39 83 2 3 4 5 6 7 8 > -- - -- - 22 q - 83 q + 86 q - 3 q - 59 q + 43 q + 6 q - 25 q + 2 q q 9 10 11 12 > 11 q + 3 q - 4 q + q

The Alternating Knot 933

The Alternating Knot 9₃₃