The Alternating Knot 9₃₀

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Acknowledgement

KnotPlot

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

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

DT (Dowker-Thistlethwaite) Code: 4 8 10 14 2 16 6 18 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 + 5t^-2 - 12t^-1 + 17 - 12t + 5t² - t³

Conway Polynomial: 1 - z² - z⁴ - z⁶

Other knots with the same Alexander/Conway Polynomial: {K11n130, ...}

Determinant and Signature: {53, 0}

Jones Polynomial: - q^-5 + 3q^-4 - 5q^-3 + 8q^-2 - 9q^-1 + 9 - 8q + 6q² - 3q³ + q⁴

Other knots (up to mirrors) with the same Jones Polynomial: {K11n114, ...}

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

HOMFLY-PT Polynomial: 2a^-2 + 2a^-2z² + a^-2z⁴ - 4 - 7z² - 4z⁴ - z⁶ + 4a² + 5a²z² + 2a²z⁴ - a⁴ - a⁴z²

Kauffman Polynomial: - a^-4z² + a^-4z⁴ + a^-3z - 3a^-3z³ + 3a^-3z⁵ - 2a^-2 + 5a^-2z² - 7a^-2z⁴ + 5a^-2z⁶ + a^-1z - 2a^-1z³ - 2a^-1z⁵ + 4a^-1z⁷ - 4 + 17z² - 23z⁴ + 10z⁶ + z⁸ + az - 9az⁵ + 7az⁷ - 4a² + 16a²z² - 22a²z⁴ + 8a²z⁶ + a²z⁸ + 2a³z - 3a³z³ - 3a³z⁵ + 3a³z⁷ - a⁴ + 5a⁴z² - 7a⁴z⁴ + 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 9₃₀. 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 2

j = 5 4 1

j = 3 4 2

j = 1 5 4

j = -1 5 5

j = -3 3 4

j = -5 2 5

j = -7 1 3

j = -9 2

j = -11 1

n Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q^-15 - 3q^-14 + 10q^-12 - 13q^-11 - 6q^-10 + 33q^-9 - 26q^-8 - 25q^-7 + 64q^-6 - 31q^-5 - 50q^-4 + 85q^-3 - 25q^-2 - 66q^-1 + 85 - 13q - 63q² + 63q³ - q⁴ - 43q⁵ + 31q⁶ + 4q⁷ - 18q⁸ + 8q⁹ + 2q¹⁰ - 3q¹¹ + q¹²

3 - q^-30 + 3q^-29 - 5q^-27 - 5q^-26 + 13q^-25 + 13q^-24 - 23q^-23 - 29q^-22 + 32q^-21 + 59q^-20 - 39q^-19 - 99q^-18 + 34q^-17 + 152q^-16 - 19q^-15 - 203q^-14 - 21q^-13 + 262q^-12 + 60q^-11 - 297q^-10 - 121q^-9 + 332q^-8 + 173q^-7 - 344q^-6 - 226q^-5 + 347q^-4 + 264q^-3 - 330q^-2 - 293q^-1 + 302 + 306q - 260q² - 303q³ + 209q⁴ + 282q⁵ - 151q⁶ - 248q⁷ + 97q⁸ + 201q⁹ - 49q¹⁰ - 153q¹¹ + 21q¹² + 99q¹³ + q¹⁴ - 61q¹⁵ - 6q¹⁶ + 32q¹⁷ + 5q¹⁸ - 14q¹⁹ - 2q²⁰ + 4q²¹ + 2q²² - 3q²³ + q²⁴

4 q^-50 - 3q^-49 + 5q^-47 + 5q^-45 - 20q^-44 - 6q^-43 + 23q^-42 + 12q^-41 + 32q^-40 - 73q^-39 - 53q^-38 + 45q^-37 + 65q^-36 + 145q^-35 - 153q^-34 - 200q^-33 - 14q^-32 + 143q^-31 + 445q^-30 - 147q^-29 - 439q^-28 - 288q^-27 + 105q^-26 + 928q^-25 + 106q^-24 - 599q^-23 - 783q^-22 - 220q^-21 + 1401q^-20 + 611q^-19 - 496q^-18 - 1308q^-17 - 810q^-16 + 1652q^-15 + 1180q^-14 - 138q^-13 - 1671q^-12 - 1454q^-11 + 1647q^-10 + 1616q^-9 + 308q^-8 - 1806q^-7 - 1949q^-6 + 1448q^-5 + 1834q^-4 + 718q^-3 - 1721q^-2 - 2213q^-1 + 1104 + 1806q + 1031q² - 1404q³ - 2198q⁴ + 637q⁵ + 1502q⁶ + 1191q⁷ - 886q⁸ - 1870q⁹ + 164q¹⁰ + 970q¹¹ + 1105q¹² - 332q¹³ - 1284q¹⁴ - 133q¹⁵ + 411q¹⁶ + 779q¹⁷ + 24q¹⁸ - 661q¹⁹ - 172q²⁰ + 55q²¹ + 387q²² + 112q²³ - 238q²⁴ - 82q²⁵ - 46q²⁶ + 129q²⁷ + 61q²⁸ - 61q²⁹ - 13q³⁰ - 28q³¹ + 27q³² + 16q³³ - 13q³⁴ + 2q³⁵ - 6q³⁶ + 4q³⁷ + 2q³⁸ - 3q³⁹ + q⁴⁰

5 - q^-75 + 3q^-74 - 5q^-72 + 2q^-69 + 13q^-68 + 6q^-67 - 23q^-66 - 21q^-65 - 6q^-64 + 18q^-63 + 58q^-62 + 45q^-61 - 42q^-60 - 116q^-59 - 95q^-58 + 26q^-57 + 190q^-56 + 236q^-55 + 34q^-54 - 294q^-53 - 446q^-52 - 193q^-51 + 347q^-50 + 746q^-49 + 540q^-48 - 301q^-47 - 1113q^-46 - 1072q^-45 + 36q^-44 + 1422q^-43 + 1830q^-42 + 543q^-41 - 1593q^-40 - 2692q^-39 - 1485q^-38 + 1429q^-37 + 3600q^-36 + 2735q^-35 - 884q^-34 - 4284q^-33 - 4248q^-32 - 146q^-31 + 4767q^-30 + 5784q^-29 + 1488q^-28 - 4739q^-27 - 7283q^-26 - 3165q^-25 + 4453q^-24 + 8528q^-23 + 4840q^-22 - 3729q^-21 - 9486q^-20 - 6571q^-19 + 2900q^-18 + 10132q^-17 + 8046q^-16 - 1882q^-15 - 10500q^-14 - 9355q^-13 + 901q^-12 + 10624q^-11 + 10376q^-10 + 93q^-9 - 10548q^-8 - 11181q^-7 - 1006q^-6 + 10252q^-5 + 11735q^-4 + 1923q^-3 - 9757q^-2 - 12056q^-1 - 2802 + 9009q + 12089q² + 3677q³ - 7986q⁴ - 11822q⁵ - 4480q⁶ + 6701q⁷ + 11178q⁸ + 5155q⁹ - 5200q¹⁰ - 10153q¹¹ - 5597q¹² + 3590q¹³ + 8776q¹⁴ + 5724q¹⁵ - 2033q¹⁶ - 7149q¹⁷ - 5450q¹⁸ + 663q¹⁹ + 5394q²⁰ + 4883q²¹ + 325q²² - 3742q²³ - 3985q²⁴ - 936q²⁵ + 2281q²⁶ + 3039q²⁷ + 1154q²⁸ - 1232q²⁹ - 2062q³⁰ - 1082q³¹ + 500q³² + 1284q³³ + 856q³⁴ - 120q³⁵ - 711q³⁶ - 575q³⁷ - 46q³⁸ + 352q³⁹ + 334q⁴⁰ + 82q⁴¹ - 145q⁴² - 184q⁴³ - 61q⁴⁴ + 63q⁴⁵ + 78q⁴⁶ + 26q⁴⁷ - 6q⁴⁸ - 37q⁴⁹ - 22q⁵⁰ + 15q⁵¹ + 11q⁵² - 2q⁵³ + 3q⁵⁴ - 2q⁵⁵ - 6q⁵⁶ + 4q⁵⁷ + 2q⁵⁸ - 3q⁵⁹ + q⁶⁰

6 q^-105 - 3q^-104 + 5q^-102 - 7q^-99 + 5q^-98 - 13q^-97 - 6q^-96 + 32q^-95 + 12q^-94 + 6q^-93 - 35q^-92 - 3q^-91 - 61q^-90 - 37q^-89 + 106q^-88 + 97q^-87 + 83q^-86 - 78q^-85 - 50q^-84 - 281q^-83 - 234q^-82 + 189q^-81 + 368q^-80 + 487q^-79 + 95q^-78 - 27q^-77 - 880q^-76 - 1062q^-75 - 200q^-74 + 669q^-73 + 1561q^-72 + 1252q^-71 + 917q^-70 - 1544q^-69 - 3022q^-68 - 2336q^-67 - 342q^-66 + 2696q^-65 + 4072q^-64 + 4665q^-63 - 208q^-62 - 5060q^-61 - 7072q^-60 - 5206q^-59 + 903q^-58 + 6920q^-57 + 12097q^-56 + 6384q^-55 - 3182q^-54 - 12040q^-53 - 14685q^-52 - 7707q^-51 + 4875q^-50 + 19971q^-49 + 18726q^-48 + 6871q^-47 - 11495q^-46 - 24707q^-45 - 23166q^-44 - 6334q^-43 + 21942q^-42 + 32015q^-41 + 24455q^-40 - 1411q^-39 - 28734q^-38 - 39997q^-37 - 25239q^-36 + 14429q^-35 + 39740q^-34 + 43584q^-33 + 16060q^-32 - 23870q^-31 - 51820q^-30 - 45519q^-29 + 65q^-28 + 39502q^-27 + 58150q^-26 + 34702q^-25 - 13004q^-24 - 56500q^-23 - 61480q^-22 - 15434q^-21 + 33915q^-20 + 66234q^-19 + 49597q^-18 - 1056q^-17 - 56063q^-16 - 71459q^-15 - 28185q^-14 + 26675q^-13 + 69304q^-12 + 59637q^-11 + 9292q^-10 - 52952q^-9 - 76599q^-8 - 37814q^-7 + 19152q^-6 + 68839q^-5 + 65948q^-4 + 18458q^-3 - 47408q^-2 - 77764q^-1 - 45553 + 10197q + 64272q² + 68954q³ + 27760q⁴ - 37744q⁵ - 73801q⁶ - 51360q⁷ - 1437q⁸ + 53604q⁹ + 66871q¹⁰ + 36529q¹¹ - 23006q¹² - 62465q¹³ - 52623q¹⁴ - 14213q¹⁵ + 36348q¹⁶ + 57063q¹⁷ + 41144q¹⁸ - 5863q¹⁹ - 43764q²⁰ - 46107q²¹ - 23280q²² + 16353q²³ + 39809q²⁴ + 37724q²⁵ + 7565q²⁶ - 22522q²⁷ - 32162q²⁸ - 24126q²⁹ + 655q³⁰ + 20456q³¹ + 26764q³² + 12392q³³ - 6154q³⁴ - 16308q³⁵ - 17419q³⁶ - 6085q³⁷ + 6142q³⁸ + 13916q³⁹ + 9631q⁴⁰ + 1365q⁴¹ - 5038q⁴² - 8702q⁴³ - 5504q⁴⁴ - 202q⁴⁵ + 4970q⁴⁶ + 4613q⁴⁷ + 2254q⁴⁸ - 298q⁴⁹ - 2873q⁵⁰ - 2662q⁵¹ - 1175q⁵² + 1143q⁵³ + 1356q⁵⁴ + 1060q⁵⁵ + 505q⁵⁶ - 583q⁵⁷ - 805q⁵⁸ - 583q⁵⁹ + 183q⁶⁰ + 222q⁶¹ + 255q⁶² + 255q⁶³ - 60q⁶⁴ - 166q⁶⁵ - 165q⁶⁶ + 47q⁶⁷ + 10q⁶⁸ + 26q⁶⁹ + 68q⁷⁰ - 3q⁷¹ - 24q⁷² - 36q⁷³ + 21q⁷⁴ - q⁷⁵ - 7q⁷⁶ + 14q⁷⁷ - q⁷⁸ - 2q⁷⁹ - 6q⁸⁰ + 4q⁸¹ + 2q⁸² - 3q⁸³ + q⁸⁴

7 - q^-140 + 3q^-139 - 5q^-137 + 7q^-134 - 5q^-132 + 13q^-131 - 3q^-130 - 23q^-129 - 12q^-128 - 6q^-127 + 35q^-126 + 31q^-125 - 5q^-124 + 42q^-123 - 16q^-122 - 87q^-121 - 87q^-120 - 86q^-119 + 90q^-118 + 170q^-117 + 121q^-116 + 202q^-115 + 9q^-114 - 265q^-113 - 402q^-112 - 538q^-111 - 79q^-110 + 429q^-109 + 671q^-108 + 1047q^-107 + 598q^-106 - 277q^-105 - 1162q^-104 - 2149q^-103 - 1623q^-102 - 168q^-101 + 1452q^-100 + 3494q^-99 + 3592q^-98 + 1882q^-97 - 1033q^-96 - 5221q^-95 - 6764q^-94 - 5177q^-93 - 907q^-92 + 6184q^-91 + 10739q^-90 + 10887q^-89 + 5906q^-88 - 5135q^-87 - 14824q^-86 - 18971q^-85 - 14834q^-84 + 106q^-83 + 16730q^-82 + 28303q^-81 + 28515q^-80 + 11200q^-79 - 13917q^-78 - 36794q^-77 - 46006q^-76 - 29803q^-75 + 3154q^-74 + 40523q^-73 + 64964q^-72 + 55884q^-71 + 17916q^-70 - 35776q^-69 - 81246q^-68 - 87023q^-67 - 50124q^-66 + 18811q^-65 + 90197q^-64 + 119293q^-63 + 91670q^-62 + 12099q^-61 - 87038q^-60 - 147372q^-59 - 139368q^-58 - 56314q^-57 + 69551q^-56 + 166245q^-55 + 187209q^-54 + 110653q^-53 - 36434q^-52 - 172069q^-51 - 230685q^-50 - 170584q^-49 - 9038q^-48 + 163756q^-47 + 264303q^-46 + 230046q^-45 + 63812q^-44 - 141581q^-43 - 286634q^-42 - 284915q^-41 - 121869q^-40 + 109569q^-39 + 296342q^-38 + 330748q^-37 + 179153q^-36 - 70743q^-35 - 295693q^-34 - 367093q^-33 - 231134q^-32 + 30306q^-31 + 286912q^-30 + 393037q^-29 + 276036q^-28 + 9110q^-27 - 273437q^-26 - 410688q^-25 - 312837q^-24 - 44562q^-23 + 257631q^-22 + 421529q^-21 + 342270q^-20 + 75535q^-19 - 241544q^-18 - 427749q^-17 - 365409q^-16 - 102197q^-15 + 225747q^-14 + 430664q^-13 + 384105q^-12 + 125684q^-11 - 210137q^-10 - 430936q^-9 - 399497q^-8 - 147605q^-7 + 193300q^-6 + 428330q^-5 + 412620q^-4 + 169564q^-3 - 173788q^-2 - 421788q^-1 - 423183 - 192532q + 149583q² + 409333q³ + 430356q⁴ + 216898q⁵ - 119422q⁶ - 389088q⁷ - 432054q⁸ - 241485q⁹ + 82746q¹⁰ + 358976q¹¹ + 425772q¹² + 264251q¹³ - 40378q¹⁴ - 318196q¹⁵ - 409021q¹⁶ - 281941q¹⁷ - 5088q¹⁸ + 267301q¹⁹ + 379836q²⁰ + 290962q²¹ + 49860q²² - 208618q²³ - 338081q²⁴ - 288323q²⁵ - 89189q²⁶ + 146556q²⁷ + 285618q²⁸ + 272168q²⁹ + 118414q³⁰ - 86496q³¹ - 225975q³² - 243111q³³ - 134696q³⁴ + 34010q³⁵ + 165116q³⁶ + 204052q³⁷ + 136276q³⁸ + 6363q³⁹ - 108154q⁴⁰ - 159320q⁴¹ - 125498q⁴² - 32714q⁴³ + 60818q⁴⁴ + 114826q⁴⁵ + 105238q⁴⁶ + 44770q⁴⁷ - 25322q⁴⁸ - 74920q⁴⁹ - 80761q⁵⁰ - 45848q⁵¹ + 2667q⁵² + 43541q⁵³ + 56377q⁵⁴ + 39285q⁵⁵ + 8931q⁵⁶ - 21280q⁵⁷ - 35566q⁵⁸ - 29561q⁵⁹ - 12632q⁶⁰ + 7772q⁶¹ + 20162q⁶² + 19699q⁶³ + 11553q⁶⁴ - 939q⁶⁵ - 9984q⁶⁶ - 11668q⁶⁷ - 8598q⁶⁸ - 1697q⁶⁹ + 4188q⁷⁰ + 6200q⁷¹ + 5545q⁷² + 2017q⁷³ - 1462q⁷⁴ - 2855q⁷⁵ - 3073q⁷⁶ - 1533q⁷⁷ + 234q⁷⁸ + 1136q⁷⁹ + 1661q⁸⁰ + 938q⁸¹ - 11q⁸² - 416q⁸³ - 697q⁸⁴ - 439q⁸⁵ - 118q⁸⁶ + 57q⁸⁷ + 360q⁸⁸ + 256q⁸⁹ + 10q⁹⁰ - 41q⁹¹ - 120q⁹² - 48q⁹³ - 26q⁹⁴ - 43q⁹⁵ + 66q⁹⁶ + 54q⁹⁷ - 3q⁹⁸ - 11q⁹⁹ - 23q¹⁰⁰ + 7q¹⁰¹ + 5q¹⁰² - 19q¹⁰³ + 9q¹⁰⁴ + 10q¹⁰⁵ - q¹⁰⁶ - 2q¹⁰⁷ - 6q¹⁰⁸ + 4q¹⁰⁹ + 2q¹¹⁰ - 3q¹¹¹ + 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, 30]]

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

In[3]:=
GaussCode[Knot[9, 30]]

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

In[4]:=
DTCode[Knot[9, 30]]

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

In[5]:=
br = BR[Knot[9, 30]]

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

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

Out[6]=
{4, 9}

In[7]:=
BraidIndex[Knot[9, 30]]

Out[7]=
4

In[8]:=
Show[DrawMorseLink[Knot[9, 30]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[9, 30]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

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

In[10]:=
alex = Alexander[Knot[9, 30]][t]

Out[10]=
-3 5 12 2 3 17 - t + -- - -- - 12 t + 5 t - t 2 t t

In[11]:=
Conway[Knot[9, 30]][z]

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

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

Out[12]=
{Knot[9, 30], Knot[11, NonAlternating, 130]}

In[13]:=
{KnotDet[Knot[9, 30]], KnotSignature[Knot[9, 30]]}

Out[13]=
{53, 0}

In[14]:=
Jones[Knot[9, 30]][q]

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

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

Out[15]=
{Knot[9, 30], Knot[11, NonAlternating, 114]}

In[16]:=
A2Invariant[Knot[9, 30]][q]

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

In[17]:=
HOMFLYPT[Knot[9, 30]][a, z]

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

In[18]:=
Kauffman[Knot[9, 30]][a, z]

Out[18]=
2 2 2 2 4 z z 3 5 2 z 5 z -4 - -- - 4 a - a + -- + - + a z + 2 a z + a z + 17 z - -- + ---- + 2 3 a 4 2 a a a a 3 3 4 4 2 2 4 2 3 z 2 z 3 3 5 3 4 z 7 z > 16 a z + 5 a z - ---- - ---- - 3 a z - 2 a z - 23 z + -- - ---- - 3 a 4 2 a a a 5 5 2 4 4 4 3 z 2 z 5 3 5 5 5 6 > 22 a z - 7 a z + ---- - ---- - 9 a z - 3 a z + a z + 10 z + 3 a a 6 7 5 z 2 6 4 6 4 z 7 3 7 8 2 8 > ---- + 8 a z + 3 a z + ---- + 7 a z + 3 a z + z + a z 2 a a

In[19]:=
{Vassiliev[2][Knot[9, 30]], Vassiliev[3][Knot[9, 30]]}

Out[19]=
{-1, -1}

In[20]:=
Kh[Knot[9, 30]][q, t]

Out[20]=
5 1 2 1 3 2 5 3 4 5 - + 5 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 + 4 q t + 2 q t + 4 q t + q t + 2 q t + q t

In[21]:=
ColouredJones[Knot[9, 30], 2][q]

Out[21]=
-15 3 10 13 6 33 26 25 64 31 50 85 25 85 + q - --- + --- - --- - --- + -- - -- - -- + -- - -- - -- + -- - -- - 14 12 11 10 9 8 7 6 5 4 3 2 q q q q q q q q q q q q 66 2 3 4 5 6 7 8 9 > -- - 13 q - 63 q + 63 q - q - 43 q + 31 q + 4 q - 18 q + 8 q + q 10 11 12 > 2 q - 3 q + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 9₃₀

9₂₉
9₃₁

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-15 - 3q^-14 + 10q^-12 - 13q^-11 - 6q^-10 + 33q^-9 - 26q^-8 - 25q^-7 + 64q^-6 - 31q^-5 - 50q^-4 + 85q^-3 - 25q^-2 - 66q^-1 + 85 - 13q - 63q² + 63q³ - q⁴ - 43q⁵ + 31q⁶ + 4q⁷ - 18q⁸ + 8q⁹ + 2q¹⁰ - 3q¹¹ + q¹²
3	- q^-30 + 3q^-29 - 5q^-27 - 5q^-26 + 13q^-25 + 13q^-24 - 23q^-23 - 29q^-22 + 32q^-21 + 59q^-20 - 39q^-19 - 99q^-18 + 34q^-17 + 152q^-16 - 19q^-15 - 203q^-14 - 21q^-13 + 262q^-12 + 60q^-11 - 297q^-10 - 121q^-9 + 332q^-8 + 173q^-7 - 344q^-6 - 226q^-5 + 347q^-4 + 264q^-3 - 330q^-2 - 293q^-1 + 302 + 306q - 260q² - 303q³ + 209q⁴ + 282q⁵ - 151q⁶ - 248q⁷ + 97q⁸ + 201q⁹ - 49q¹⁰ - 153q¹¹ + 21q¹² + 99q¹³ + q¹⁴ - 61q¹⁵ - 6q¹⁶ + 32q¹⁷ + 5q¹⁸ - 14q¹⁹ - 2q²⁰ + 4q²¹ + 2q²² - 3q²³ + q²⁴
4	q^-50 - 3q^-49 + 5q^-47 + 5q^-45 - 20q^-44 - 6q^-43 + 23q^-42 + 12q^-41 + 32q^-40 - 73q^-39 - 53q^-38 + 45q^-37 + 65q^-36 + 145q^-35 - 153q^-34 - 200q^-33 - 14q^-32 + 143q^-31 + 445q^-30 - 147q^-29 - 439q^-28 - 288q^-27 + 105q^-26 + 928q^-25 + 106q^-24 - 599q^-23 - 783q^-22 - 220q^-21 + 1401q^-20 + 611q^-19 - 496q^-18 - 1308q^-17 - 810q^-16 + 1652q^-15 + 1180q^-14 - 138q^-13 - 1671q^-12 - 1454q^-11 + 1647q^-10 + 1616q^-9 + 308q^-8 - 1806q^-7 - 1949q^-6 + 1448q^-5 + 1834q^-4 + 718q^-3 - 1721q^-2 - 2213q^-1 + 1104 + 1806q + 1031q² - 1404q³ - 2198q⁴ + 637q⁵ + 1502q⁶ + 1191q⁷ - 886q⁸ - 1870q⁹ + 164q¹⁰ + 970q¹¹ + 1105q¹² - 332q¹³ - 1284q¹⁴ - 133q¹⁵ + 411q¹⁶ + 779q¹⁷ + 24q¹⁸ - 661q¹⁹ - 172q²⁰ + 55q²¹ + 387q²² + 112q²³ - 238q²⁴ - 82q²⁵ - 46q²⁶ + 129q²⁷ + 61q²⁸ - 61q²⁹ - 13q³⁰ - 28q³¹ + 27q³² + 16q³³ - 13q³⁴ + 2q³⁵ - 6q³⁶ + 4q³⁷ + 2q³⁸ - 3q³⁹ + q⁴⁰
5	- q^-75 + 3q^-74 - 5q^-72 + 2q^-69 + 13q^-68 + 6q^-67 - 23q^-66 - 21q^-65 - 6q^-64 + 18q^-63 + 58q^-62 + 45q^-61 - 42q^-60 - 116q^-59 - 95q^-58 + 26q^-57 + 190q^-56 + 236q^-55 + 34q^-54 - 294q^-53 - 446q^-52 - 193q^-51 + 347q^-50 + 746q^-49 + 540q^-48 - 301q^-47 - 1113q^-46 - 1072q^-45 + 36q^-44 + 1422q^-43 + 1830q^-42 + 543q^-41 - 1593q^-40 - 2692q^-39 - 1485q^-38 + 1429q^-37 + 3600q^-36 + 2735q^-35 - 884q^-34 - 4284q^-33 - 4248q^-32 - 146q^-31 + 4767q^-30 + 5784q^-29 + 1488q^-28 - 4739q^-27 - 7283q^-26 - 3165q^-25 + 4453q^-24 + 8528q^-23 + 4840q^-22 - 3729q^-21 - 9486q^-20 - 6571q^-19 + 2900q^-18 + 10132q^-17 + 8046q^-16 - 1882q^-15 - 10500q^-14 - 9355q^-13 + 901q^-12 + 10624q^-11 + 10376q^-10 + 93q^-9 - 10548q^-8 - 11181q^-7 - 1006q^-6 + 10252q^-5 + 11735q^-4 + 1923q^-3 - 9757q^-2 - 12056q^-1 - 2802 + 9009q + 12089q² + 3677q³ - 7986q⁴ - 11822q⁵ - 4480q⁶ + 6701q⁷ + 11178q⁸ + 5155q⁹ - 5200q¹⁰ - 10153q¹¹ - 5597q¹² + 3590q¹³ + 8776q¹⁴ + 5724q¹⁵ - 2033q¹⁶ - 7149q¹⁷ - 5450q¹⁸ + 663q¹⁹ + 5394q²⁰ + 4883q²¹ + 325q²² - 3742q²³ - 3985q²⁴ - 936q²⁵ + 2281q²⁶ + 3039q²⁷ + 1154q²⁸ - 1232q²⁹ - 2062q³⁰ - 1082q³¹ + 500q³² + 1284q³³ + 856q³⁴ - 120q³⁵ - 711q³⁶ - 575q³⁷ - 46q³⁸ + 352q³⁹ + 334q⁴⁰ + 82q⁴¹ - 145q⁴² - 184q⁴³ - 61q⁴⁴ + 63q⁴⁵ + 78q⁴⁶ + 26q⁴⁷ - 6q⁴⁸ - 37q⁴⁹ - 22q⁵⁰ + 15q⁵¹ + 11q⁵² - 2q⁵³ + 3q⁵⁴ - 2q⁵⁵ - 6q⁵⁶ + 4q⁵⁷ + 2q⁵⁸ - 3q⁵⁹ + q⁶⁰
6	q^-105 - 3q^-104 + 5q^-102 - 7q^-99 + 5q^-98 - 13q^-97 - 6q^-96 + 32q^-95 + 12q^-94 + 6q^-93 - 35q^-92 - 3q^-91 - 61q^-90 - 37q^-89 + 106q^-88 + 97q^-87 + 83q^-86 - 78q^-85 - 50q^-84 - 281q^-83 - 234q^-82 + 189q^-81 + 368q^-80 + 487q^-79 + 95q^-78 - 27q^-77 - 880q^-76 - 1062q^-75 - 200q^-74 + 669q^-73 + 1561q^-72 + 1252q^-71 + 917q^-70 - 1544q^-69 - 3022q^-68 - 2336q^-67 - 342q^-66 + 2696q^-65 + 4072q^-64 + 4665q^-63 - 208q^-62 - 5060q^-61 - 7072q^-60 - 5206q^-59 + 903q^-58 + 6920q^-57 + 12097q^-56 + 6384q^-55 - 3182q^-54 - 12040q^-53 - 14685q^-52 - 7707q^-51 + 4875q^-50 + 19971q^-49 + 18726q^-48 + 6871q^-47 - 11495q^-46 - 24707q^-45 - 23166q^-44 - 6334q^-43 + 21942q^-42 + 32015q^-41 + 24455q^-40 - 1411q^-39 - 28734q^-38 - 39997q^-37 - 25239q^-36 + 14429q^-35 + 39740q^-34 + 43584q^-33 + 16060q^-32 - 23870q^-31 - 51820q^-30 - 45519q^-29 + 65q^-28 + 39502q^-27 + 58150q^-26 + 34702q^-25 - 13004q^-24 - 56500q^-23 - 61480q^-22 - 15434q^-21 + 33915q^-20 + 66234q^-19 + 49597q^-18 - 1056q^-17 - 56063q^-16 - 71459q^-15 - 28185q^-14 + 26675q^-13 + 69304q^-12 + 59637q^-11 + 9292q^-10 - 52952q^-9 - 76599q^-8 - 37814q^-7 + 19152q^-6 + 68839q^-5 + 65948q^-4 + 18458q^-3 - 47408q^-2 - 77764q^-1 - 45553 + 10197q + 64272q² + 68954q³ + 27760q⁴ - 37744q⁵ - 73801q⁶ - 51360q⁷ - 1437q⁸ + 53604q⁹ + 66871q¹⁰ + 36529q¹¹ - 23006q¹² - 62465q¹³ - 52623q¹⁴ - 14213q¹⁵ + 36348q¹⁶ + 57063q¹⁷ + 41144q¹⁸ - 5863q¹⁹ - 43764q²⁰ - 46107q²¹ - 23280q²² + 16353q²³ + 39809q²⁴ + 37724q²⁵ + 7565q²⁶ - 22522q²⁷ - 32162q²⁸ - 24126q²⁹ + 655q³⁰ + 20456q³¹ + 26764q³² + 12392q³³ - 6154q³⁴ - 16308q³⁵ - 17419q³⁶ - 6085q³⁷ + 6142q³⁸ + 13916q³⁹ + 9631q⁴⁰ + 1365q⁴¹ - 5038q⁴² - 8702q⁴³ - 5504q⁴⁴ - 202q⁴⁵ + 4970q⁴⁶ + 4613q⁴⁷ + 2254q⁴⁸ - 298q⁴⁹ - 2873q⁵⁰ - 2662q⁵¹ - 1175q⁵² + 1143q⁵³ + 1356q⁵⁴ + 1060q⁵⁵ + 505q⁵⁶ - 583q⁵⁷ - 805q⁵⁸ - 583q⁵⁹ + 183q⁶⁰ + 222q⁶¹ + 255q⁶² + 255q⁶³ - 60q⁶⁴ - 166q⁶⁵ - 165q⁶⁶ + 47q⁶⁷ + 10q⁶⁸ + 26q⁶⁹ + 68q⁷⁰ - 3q⁷¹ - 24q⁷² - 36q⁷³ + 21q⁷⁴ - q⁷⁵ - 7q⁷⁶ + 14q⁷⁷ - q⁷⁸ - 2q⁷⁹ - 6q⁸⁰ + 4q⁸¹ + 2q⁸² - 3q⁸³ + q⁸⁴
7	- q^-140 + 3q^-139 - 5q^-137 + 7q^-134 - 5q^-132 + 13q^-131 - 3q^-130 - 23q^-129 - 12q^-128 - 6q^-127 + 35q^-126 + 31q^-125 - 5q^-124 + 42q^-123 - 16q^-122 - 87q^-121 - 87q^-120 - 86q^-119 + 90q^-118 + 170q^-117 + 121q^-116 + 202q^-115 + 9q^-114 - 265q^-113 - 402q^-112 - 538q^-111 - 79q^-110 + 429q^-109 + 671q^-108 + 1047q^-107 + 598q^-106 - 277q^-105 - 1162q^-104 - 2149q^-103 - 1623q^-102 - 168q^-101 + 1452q^-100 + 3494q^-99 + 3592q^-98 + 1882q^-97 - 1033q^-96 - 5221q^-95 - 6764q^-94 - 5177q^-93 - 907q^-92 + 6184q^-91 + 10739q^-90 + 10887q^-89 + 5906q^-88 - 5135q^-87 - 14824q^-86 - 18971q^-85 - 14834q^-84 + 106q^-83 + 16730q^-82 + 28303q^-81 + 28515q^-80 + 11200q^-79 - 13917q^-78 - 36794q^-77 - 46006q^-76 - 29803q^-75 + 3154q^-74 + 40523q^-73 + 64964q^-72 + 55884q^-71 + 17916q^-70 - 35776q^-69 - 81246q^-68 - 87023q^-67 - 50124q^-66 + 18811q^-65 + 90197q^-64 + 119293q^-63 + 91670q^-62 + 12099q^-61 - 87038q^-60 - 147372q^-59 - 139368q^-58 - 56314q^-57 + 69551q^-56 + 166245q^-55 + 187209q^-54 + 110653q^-53 - 36434q^-52 - 172069q^-51 - 230685q^-50 - 170584q^-49 - 9038q^-48 + 163756q^-47 + 264303q^-46 + 230046q^-45 + 63812q^-44 - 141581q^-43 - 286634q^-42 - 284915q^-41 - 121869q^-40 + 109569q^-39 + 296342q^-38 + 330748q^-37 + 179153q^-36 - 70743q^-35 - 295693q^-34 - 367093q^-33 - 231134q^-32 + 30306q^-31 + 286912q^-30 + 393037q^-29 + 276036q^-28 + 9110q^-27 - 273437q^-26 - 410688q^-25 - 312837q^-24 - 44562q^-23 + 257631q^-22 + 421529q^-21 + 342270q^-20 + 75535q^-19 - 241544q^-18 - 427749q^-17 - 365409q^-16 - 102197q^-15 + 225747q^-14 + 430664q^-13 + 384105q^-12 + 125684q^-11 - 210137q^-10 - 430936q^-9 - 399497q^-8 - 147605q^-7 + 193300q^-6 + 428330q^-5 + 412620q^-4 + 169564q^-3 - 173788q^-2 - 421788q^-1 - 423183 - 192532q + 149583q² + 409333q³ + 430356q⁴ + 216898q⁵ - 119422q⁶ - 389088q⁷ - 432054q⁸ - 241485q⁹ + 82746q¹⁰ + 358976q¹¹ + 425772q¹² + 264251q¹³ - 40378q¹⁴ - 318196q¹⁵ - 409021q¹⁶ - 281941q¹⁷ - 5088q¹⁸ + 267301q¹⁹ + 379836q²⁰ + 290962q²¹ + 49860q²² - 208618q²³ - 338081q²⁴ - 288323q²⁵ - 89189q²⁶ + 146556q²⁷ + 285618q²⁸ + 272168q²⁹ + 118414q³⁰ - 86496q³¹ - 225975q³² - 243111q³³ - 134696q³⁴ + 34010q³⁵ + 165116q³⁶ + 204052q³⁷ + 136276q³⁸ + 6363q³⁹ - 108154q⁴⁰ - 159320q⁴¹ - 125498q⁴² - 32714q⁴³ + 60818q⁴⁴ + 114826q⁴⁵ + 105238q⁴⁶ + 44770q⁴⁷ - 25322q⁴⁸ - 74920q⁴⁹ - 80761q⁵⁰ - 45848q⁵¹ + 2667q⁵² + 43541q⁵³ + 56377q⁵⁴ + 39285q⁵⁵ + 8931q⁵⁶ - 21280q⁵⁷ - 35566q⁵⁸ - 29561q⁵⁹ - 12632q⁶⁰ + 7772q⁶¹ + 20162q⁶² + 19699q⁶³ + 11553q⁶⁴ - 939q⁶⁵ - 9984q⁶⁶ - 11668q⁶⁷ - 8598q⁶⁸ - 1697q⁶⁹ + 4188q⁷⁰ + 6200q⁷¹ + 5545q⁷² + 2017q⁷³ - 1462q⁷⁴ - 2855q⁷⁵ - 3073q⁷⁶ - 1533q⁷⁷ + 234q⁷⁸ + 1136q⁷⁹ + 1661q⁸⁰ + 938q⁸¹ - 11q⁸² - 416q⁸³ - 697q⁸⁴ - 439q⁸⁵ - 118q⁸⁶ + 57q⁸⁷ + 360q⁸⁸ + 256q⁸⁹ + 10q⁹⁰ - 41q⁹¹ - 120q⁹² - 48q⁹³ - 26q⁹⁴ - 43q⁹⁵ + 66q⁹⁶ + 54q⁹⁷ - 3q⁹⁸ - 11q⁹⁹ - 23q¹⁰⁰ + 7q¹⁰¹ + 5q¹⁰² - 19q¹⁰³ + 9q¹⁰⁴ + 10q¹⁰⁵ - q¹⁰⁶ - 2q¹⁰⁷ - 6q¹⁰⁸ + 4q¹⁰⁹ + 2q¹¹⁰ - 3q¹¹¹ + q¹¹²

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

Out[8]=	-Graphics-
In[9]:=	#[Knot[9, 30]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 1, 3, 3, {4, 6}, 1}
In[10]:=	alex = Alexander[Knot[9, 30]][t]
Out[10]=	-3 5 12 2 3 17 - t + -- - -- - 12 t + 5 t - t 2 t t
In[11]:=	Conway[Knot[9, 30]][z]
Out[11]=	2 4 6 1 - z - z - z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[9, 30], Knot[11, NonAlternating, 130]}
In[13]:=	{KnotDet[Knot[9, 30]], KnotSignature[Knot[9, 30]]}
Out[13]=	{53, 0}
In[14]:=	Jones[Knot[9, 30]][q]
Out[14]=	-5 3 5 8 9 2 3 4 9 - q + -- - -- + -- - - - 8 q + 6 q - 3 q + q 4 3 2 q q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[9, 30], Knot[11, NonAlternating, 114]}
In[16]:=	A2Invariant[Knot[9, 30]][q]
Out[16]=	-16 -12 -10 3 -6 -2 2 4 6 8 10 12 -3 - q + q - q + -- + q + q + q - 2 q + q + 2 q - q + q 8 q
In[17]:=	HOMFLYPT[Knot[9, 30]][a, z]
Out[17]=	2 4 2 2 4 2 2 z 2 2 4 2 4 z 2 4 6 -4 + -- + 4 a - a - 7 z + ---- + 5 a z - a z - 4 z + -- + 2 a z - z 2 2 2 a a a
In[18]:=	Kauffman[Knot[9, 30]][a, z]
Out[18]=	2 2 2 2 4 z z 3 5 2 z 5 z -4 - -- - 4 a - a + -- + - + a z + 2 a z + a z + 17 z - -- + ---- + 2 3 a 4 2 a a a a 3 3 4 4 2 2 4 2 3 z 2 z 3 3 5 3 4 z 7 z > 16 a z + 5 a z - ---- - ---- - 3 a z - 2 a z - 23 z + -- - ---- - 3 a 4 2 a a a 5 5 2 4 4 4 3 z 2 z 5 3 5 5 5 6 > 22 a z - 7 a z + ---- - ---- - 9 a z - 3 a z + a z + 10 z + 3 a a 6 7 5 z 2 6 4 6 4 z 7 3 7 8 2 8 > ---- + 8 a z + 3 a z + ---- + 7 a z + 3 a z + z + a z 2 a a
In[19]:=	{Vassiliev[2][Knot[9, 30]], Vassiliev[3][Knot[9, 30]]}
Out[19]=	{-1, -1}
In[20]:=	Kh[Knot[9, 30]][q, t]
Out[20]=	5 1 2 1 3 2 5 3 4 5 - + 5 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 + 4 q t + 2 q t + 4 q t + q t + 2 q t + q t
In[21]:=	ColouredJones[Knot[9, 30], 2][q]
Out[21]=	-15 3 10 13 6 33 26 25 64 31 50 85 25 85 + q - --- + --- - --- - --- + -- - -- - -- + -- - -- - -- + -- - -- - 14 12 11 10 9 8 7 6 5 4 3 2 q q q q q q q q q q q q 66 2 3 4 5 6 7 8 9 > -- - 13 q - 63 q + 63 q - q - 43 q + 31 q + 4 q - 18 q + 8 q + q 10 11 12 > 2 q - 3 q + q

The Alternating Knot 930

The Alternating Knot 9₃₀