The Alternating Knot 9₄₀

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Acknowledgement

KnotPlot

PD Presentation: X₁₆₂₇ X_7,12,8,13 X_5,15,6,14 X_11,3,12,2 X_15,10,16,11 X_3,16,4,17 X_9,4,10,5 X_17,9,18,8 X_13,18,14,1

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

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

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 2

Alexander Polynomial: t^-3 - 7t^-2 + 18t^-1 - 23 + 18t - 7t² + t³

Conway Polynomial: 1 - z² - z⁴ + z⁶

Other knots with the same Alexander/Conway Polynomial: {10₅₉, K11n66, ...}

Determinant and Signature: {75, -2}

Jones Polynomial: q^-7 - 4q^-6 + 8q^-5 - 11q^-4 + 13q^-3 - 13q^-2 + 11q^-1 - 8 + 5q - q²

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

A2 (sl(3)) Invariant: q^-22 - q^-20 - 2q^-18 + 3q^-16 - q^-14 + 2q^-12 + q^-10 - 3q^-8 + q^-6 - 4q^-4 + 3q^-2 + 1 + 3q⁴ - q⁶

HOMFLY-PT Polynomial: 2 - z⁴ - 2a² + 2a²z⁴ + a²z⁶ + a⁴ - 2a⁴z² - 2a⁴z⁴ + a⁶z²

Kauffman Polynomial: a^-1z⁵ + 2 - 7z⁴ + 5z⁶ + 6az³ - 15az⁵ + 8az⁷ + 2a² + 3a²z² - 17a²z⁴ + 4a²z⁶ + 4a²z⁸ - a³z + 14a³z³ - 32a³z⁵ + 17a³z⁷ + a⁴ + 7a⁴z² - 20a⁴z⁴ + 7a⁴z⁶ + 4a⁴z⁸ - a⁵z + 6a⁵z³ - 12a⁵z⁵ + 9a⁵z⁷ + 4a⁶z² - 9a⁶z⁴ + 8a⁶z⁶ - 2a⁷z³ + 4a⁷z⁵ + a⁸z⁴

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=-2 is the signature of 9₄₀. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

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

j = 5 1

j = 3 4

j = 1 4 1

j = -1 7 4

j = -3 7 5

j = -5 6 6

j = -7 5 7

j = -9 3 6

j = -11 1 5

j = -13 3

j = -15 1

n Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q^-20 - 4q^-19 + 4q^-18 + 9q^-17 - 29q^-16 + 17q^-15 + 45q^-14 - 84q^-13 + 17q^-12 + 107q^-11 - 132q^-10 - 5q^-9 + 157q^-8 - 140q^-7 - 34q^-6 + 166q^-5 - 109q^-4 - 55q^-3 + 133q^-2 - 56q^-1 - 55 + 72q - 11q² - 31q³ + 19q⁴ + 3q⁵ - 5q⁶ + q⁷

3 q^-39 - 4q^-38 + 4q^-37 + 5q^-36 - 9q^-35 - 15q^-34 + 25q^-33 + 46q^-32 - 63q^-31 - 104q^-30 + 98q^-29 + 219q^-28 - 115q^-27 - 390q^-26 + 96q^-25 + 584q^-24 - 12q^-23 - 781q^-22 - 124q^-21 + 942q^-20 + 293q^-19 - 1048q^-18 - 461q^-17 + 1083q^-16 + 620q^-15 - 1066q^-14 - 743q^-13 + 994q^-12 + 834q^-11 - 880q^-10 - 891q^-9 + 728q^-8 + 910q^-7 - 544q^-6 - 886q^-5 + 344q^-4 + 815q^-3 - 158q^-2 - 680q^-1 - 11 + 525q + 107q² - 342q³ - 151q⁴ + 194q⁵ + 126q⁶ - 73q⁷ - 94q⁸ + 25q⁹ + 40q¹⁰ + q¹¹ - 14q¹² - 3q¹³ + 5q¹⁴ - q¹⁵

4 q^-64 - 4q^-63 + 4q^-62 + 5q^-61 - 13q^-60 + 5q^-59 - 7q^-58 + 36q^-57 + 17q^-56 - 108q^-55 - 21q^-54 + 22q^-53 + 259q^-52 + 155q^-51 - 473q^-50 - 412q^-49 - 90q^-48 + 1029q^-47 + 1049q^-46 - 889q^-45 - 1701q^-44 - 1284q^-43 + 1991q^-42 + 3403q^-41 - 139q^-40 - 3362q^-39 - 4297q^-38 + 1712q^-37 + 6394q^-36 + 2497q^-35 - 3786q^-34 - 8024q^-33 - 421q^-32 + 8187q^-31 + 5814q^-30 - 2414q^-29 - 10564q^-28 - 3234q^-27 + 8099q^-26 + 8166q^-25 - 219q^-24 - 11264q^-23 - 5423q^-22 + 6778q^-21 + 9105q^-20 + 1879q^-19 - 10527q^-18 - 6752q^-17 + 4753q^-16 + 8945q^-15 + 3786q^-14 - 8642q^-13 - 7392q^-12 + 2089q^-11 + 7709q^-10 + 5407q^-9 - 5605q^-8 - 6995q^-7 - 819q^-6 + 5176q^-5 + 5964q^-4 - 2006q^-3 - 5063q^-2 - 2721q^-1 + 1940 + 4687q + 613q² - 2214q³ - 2607q⁴ - 369q⁵ + 2280q⁶ + 1186q⁷ - 159q⁸ - 1224q⁹ - 830q¹⁰ + 505q¹¹ + 537q¹² + 329q¹³ - 217q¹⁴ - 354q¹⁵ - 8q¹⁶ + 63q¹⁷ + 123q¹⁸ + 12q¹⁹ - 54q²⁰ - 10q²¹ - 6q²² + 14q²³ + 3q²⁴ - 5q²⁵ + q²⁶

5 q^-95 - 4q^-94 + 4q^-93 + 5q^-92 - 13q^-91 + q^-90 + 13q^-89 + 4q^-88 + 7q^-87 - 18q^-86 - 75q^-85 - 21q^-84 + 134q^-83 + 196q^-82 + 65q^-81 - 290q^-80 - 588q^-79 - 346q^-78 + 578q^-77 + 1491q^-76 + 1131q^-75 - 787q^-74 - 2986q^-73 - 3096q^-72 + 254q^-71 + 5164q^-70 + 6816q^-69 + 1841q^-68 - 7184q^-67 - 12469q^-66 - 6869q^-65 + 7818q^-64 + 19647q^-63 + 15448q^-62 - 5474q^-61 - 26773q^-60 - 27371q^-59 - 1343q^-58 + 31974q^-57 + 41431q^-56 + 12728q^-55 - 33418q^-54 - 55278q^-53 - 27849q^-52 + 30045q^-51 + 66904q^-50 + 44655q^-49 - 22256q^-48 - 74559q^-47 - 60878q^-46 + 11310q^-45 + 77742q^-44 + 74637q^-43 + 945q^-42 - 76887q^-41 - 84934q^-40 - 12823q^-39 + 73166q^-38 + 91462q^-37 + 23316q^-36 - 67633q^-35 - 94929q^-34 - 32026q^-33 + 61317q^-32 + 95958q^-31 + 39130q^-30 - 54407q^-29 - 95300q^-28 - 45264q^-27 + 46955q^-26 + 93288q^-25 + 50814q^-24 - 38508q^-23 - 89768q^-22 - 56107q^-21 + 28638q^-20 + 84414q^-19 + 60870q^-18 - 17251q^-17 - 76575q^-16 - 64378q^-15 + 4624q^-14 + 65809q^-13 + 65649q^-12 + 8290q^-11 - 52284q^-10 - 63518q^-9 - 19905q^-8 + 36598q^-7 + 57365q^-6 + 28725q^-5 - 20537q^-4 - 47451q^-3 - 32966q^-2 + 5850q^-1 + 34873 + 32487q + 5201q² - 21711q³ - 27556q⁴ - 11699q⁵ + 10114q⁶ + 20201q⁷ + 13297q⁸ - 1714q⁹ - 12207q¹⁰ - 11610q¹¹ - 2816q¹² + 5796q¹³ + 7952q¹⁴ + 4095q¹⁵ - 1434q¹⁶ - 4496q¹⁷ - 3473q¹⁸ - 388q¹⁹ + 1848q²⁰ + 2087q²¹ + 913q²² - 470q²³ - 1030q²⁴ - 625q²⁵ - 15q²⁶ + 339q²⁷ + 305q²⁸ + 108q²⁹ - 82q³⁰ - 128q³¹ - 41q³² + 17q³³ + 24q³⁴ + 15q³⁵ + 6q³⁶ - 14q³⁷ - 3q³⁸ + 5q³⁹ - q⁴⁰

6 q^-132 - 4q^-131 + 4q^-130 + 5q^-129 - 13q^-128 + q^-127 + 9q^-126 + 24q^-125 - 25q^-124 - 28q^-123 + 15q^-122 - 65q^-121 + 41q^-120 + 147q^-119 + 178q^-118 - 105q^-117 - 379q^-116 - 310q^-115 - 384q^-114 + 386q^-113 + 1312q^-112 + 1583q^-111 + 123q^-110 - 2142q^-109 - 3366q^-108 - 3713q^-107 + 262q^-106 + 6256q^-105 + 10279q^-104 + 6741q^-103 - 3802q^-102 - 14987q^-101 - 22497q^-100 - 12780q^-99 + 11186q^-98 + 36943q^-97 + 42301q^-96 + 17754q^-95 - 26641q^-94 - 72673q^-93 - 75149q^-92 - 22897q^-91 + 64452q^-90 + 126502q^-89 + 114861q^-88 + 20859q^-87 - 122835q^-86 - 206333q^-85 - 163059q^-84 + 11002q^-83 + 208105q^-82 + 300160q^-81 + 203709q^-80 - 69254q^-79 - 330890q^-78 - 406467q^-77 - 200735q^-76 + 165921q^-75 + 472200q^-74 + 496034q^-73 + 157242q^-72 - 317127q^-71 - 626795q^-70 - 518543q^-69 - 53388q^-68 + 496689q^-67 + 752315q^-66 + 479294q^-65 - 130939q^-64 - 694526q^-63 - 787792q^-62 - 353776q^-61 + 358388q^-60 + 855244q^-59 + 744150q^-58 + 123130q^-57 - 610257q^-56 - 909875q^-55 - 595293q^-54 + 160417q^-53 + 818376q^-52 + 875471q^-51 + 326403q^-50 - 468676q^-49 - 908816q^-48 - 725155q^-47 - 3220q^-46 + 724933q^-45 + 904821q^-44 + 448490q^-43 - 340050q^-42 - 855997q^-41 - 780219q^-40 - 120688q^-39 + 625002q^-38 + 892351q^-37 + 529518q^-36 - 222197q^-35 - 785117q^-34 - 810609q^-33 - 233228q^-32 + 505896q^-31 + 857823q^-30 + 610250q^-29 - 74077q^-28 - 674236q^-27 - 822791q^-26 - 371957q^-25 + 326559q^-24 + 768671q^-23 + 682241q^-22 + 125390q^-21 - 479914q^-20 - 770134q^-19 - 510911q^-18 + 75505q^-17 + 575644q^-16 + 680589q^-15 + 330483q^-14 - 198691q^-13 - 594619q^-12 - 564400q^-11 - 181245q^-10 + 282605q^-9 + 537846q^-8 + 434931q^-7 + 84205q^-6 - 308568q^-5 - 458656q^-4 - 320564q^-3 - 6881q^-2 + 277833q^-1 + 364754 + 235195q - 32602q² - 233296q³ - 276577q⁴ - 155693q⁵ + 35402q⁶ + 178683q⁷ + 204585q⁸ + 100451q⁹ - 32561q¹⁰ - 128030q¹¹ - 135402q¹² - 69538q¹³ + 23123q¹⁴ + 88090q¹⁵ + 85559q¹⁶ + 43740q¹⁷ - 14516q¹⁸ - 51067q¹⁹ - 54410q²⁰ - 26949q²¹ + 9228q²² + 27867q²³ + 30283q²⁴ + 15291q²⁵ - 2539q²⁶ - 15544q²⁷ - 15758q²⁸ - 7395q²⁹ + 501q³⁰ + 6891q³¹ + 7213q³² + 4314q³³ - 557q³⁴ - 2878q³⁵ - 2711q³⁶ - 1815q³⁷ + 22q³⁸ + 970q³⁹ + 1244q⁴⁰ + 449q⁴¹ - 34q⁴² - 236q⁴³ - 367q⁴⁴ - 158q⁴⁵ - 3q⁴⁶ + 147q⁴⁷ + 46q⁴⁸ + 12q⁴⁹ + 13q⁵⁰ - 29q⁵¹ - 15q⁵² - 6q⁵³ + 14q⁵⁴ + 3q⁵⁵ - 5q⁵⁶ + q⁵⁷

7 q^-175 - 4q^-174 + 4q^-173 + 5q^-172 - 13q^-171 + q^-170 + 9q^-169 + 20q^-168 - 5q^-167 - 60q^-166 + 5q^-165 + 25q^-164 - 3q^-163 + 64q^-162 + 79q^-161 + 84q^-160 - 122q^-159 - 454q^-158 - 281q^-157 + 52q^-156 + 463q^-155 + 1045q^-154 + 998q^-153 + 433q^-152 - 1222q^-151 - 3419q^-150 - 3579q^-149 - 1570q^-148 + 2835q^-147 + 8202q^-146 + 10225q^-145 + 7031q^-144 - 3253q^-143 - 17943q^-142 - 26826q^-141 - 23120q^-140 - 1907q^-139 + 31757q^-138 + 58792q^-137 + 62462q^-136 + 27835q^-135 - 40805q^-134 - 110794q^-133 - 142875q^-132 - 99576q^-131 + 22452q^-130 + 172482q^-129 + 276253q^-128 + 252753q^-127 + 70313q^-126 - 208683q^-125 - 458776q^-124 - 519690q^-123 - 297183q^-122 + 152790q^-121 + 644597q^-120 + 903805q^-119 + 718366q^-118 + 90916q^-117 - 744407q^-116 - 1360872q^-115 - 1356868q^-114 - 611740q^-113 + 632115q^-112 + 1779506q^-111 + 2170262q^-110 + 1460192q^-109 - 180061q^-108 - 2006206q^-107 - 3040392q^-106 - 2604575q^-105 - 686673q^-104 + 1882299q^-103 + 3786854q^-102 + 3922146q^-101 + 1957131q^-100 - 1299571q^-99 - 4225837q^-98 - 5223160q^-97 - 3513648q^-96 + 249033q^-95 + 4219104q^-94 + 6297475q^-93 + 5165363q^-92 + 1175115q^-91 - 3725086q^-90 - 6984368q^-89 - 6695555q^-88 - 2793574q^-87 + 2808350q^-86 + 7207331q^-85 + 7923098q^-84 + 4400454q^-83 - 1614595q^-82 - 6991824q^-81 - 8747457q^-80 - 5814345q^-79 + 325035q^-78 + 6443244q^-77 + 9157727q^-76 + 6919798q^-75 + 895583q^-74 - 5704977q^-73 - 9218072q^-72 - 7682771q^-71 - 1934342q^-70 + 4916830q^-69 + 9036407q^-68 + 8135949q^-67 + 2741764q^-66 - 4181694q^-65 - 8723609q^-64 - 8354428q^-63 - 3331008q^-62 + 3552214q^-61 + 8372101q^-60 + 8428340q^-59 + 3751721q^-58 - 3034014q^-57 - 8035187q^-56 - 8436095q^-55 - 4075215q^-54 + 2592903q^-53 + 7730287q^-52 + 8435186q^-51 + 4371827q^-50 - 2175554q^-49 - 7440451q^-48 - 8450748q^-47 - 4698297q^-46 + 1717549q^-45 + 7122092q^-44 + 8479451q^-43 + 5090863q^-42 - 1160176q^-41 - 6718006q^-40 - 8486939q^-39 - 5553743q^-38 + 459355q^-37 + 6161476q^-36 + 8413636q^-35 + 6060211q^-34 + 402193q^-33 - 5395536q^-32 - 8182962q^-31 - 6544746q^-30 - 1402181q^-29 + 4384567q^-28 + 7711927q^-27 + 6911668q^-26 + 2474147q^-25 - 3133803q^-24 - 6936254q^-23 - 7049080q^-22 - 3503822q^-21 + 1705045q^-20 + 5831383q^-19 + 6850240q^-18 + 4347929q^-17 - 216180q^-16 - 4436603q^-15 - 6252848q^-14 - 4862053q^-13 - 1165051q^-12 + 2864862q^-11 + 5261483q^-10 + 4937335q^-9 + 2263048q^-8 - 1287412q^-7 - 3970524q^-6 - 4544974q^-5 - 2932065q^-4 - 94561q^-3 + 2546754q^-2 + 3748955q^-1 + 3106578 + 1111249q - 1198124q² - 2705364q³ - 2822242q⁴ - 1665147q⁵ + 112556q⁶ + 1615593q⁷ + 2213668q⁸ + 1764879q⁹ + 590454q¹⁰ - 673337q¹¹ - 1465900q¹² - 1516157q¹³ - 897258q¹⁴ + 7574q¹⁵ + 766515q¹⁶ + 1082336q¹⁷ + 880257q¹⁸ + 346214q¹⁹ - 238988q²⁰ - 625966q²¹ - 676126q²² - 439265q²³ - 66388q²⁴ + 264128q²⁵ + 414939q²⁶ + 367971q²⁷ + 182613q²⁸ - 40511q²⁹ - 195483q³⁰ - 238141q³¹ - 175676q³² - 54766q³³ + 55653q³⁴ + 117895q³⁵ + 118760q³⁶ + 70522q³⁷ + 8542q³⁸ - 41189q³⁹ - 61295q⁴⁰ - 50025q⁴¹ - 23636q⁴² + 4388q⁴³ + 22951q⁴⁴ + 25994q⁴⁵ + 18756q⁴⁶ + 5972q⁴⁷ - 5395q⁴⁸ - 9986q⁴⁹ - 9547q⁵⁰ - 5407q⁵¹ - 522q⁵² + 2496q⁵³ + 3770q⁵⁴ + 2987q⁵⁵ + 997q⁵⁶ - 339q⁵⁷ - 1076q⁵⁸ - 1015q⁵⁹ - 526q⁶⁰ - 207q⁶¹ + 225q⁶² + 363q⁶³ + 205q⁶⁴ + 53q⁶⁵ - 62q⁶⁶ - 65q⁶⁷ - 17q⁶⁸ - 42q⁶⁹ - 8q⁷⁰ + 29q⁷¹ + 15q⁷² + 6q⁷³ - 14q⁷⁴ - 3q⁷⁵ + 5q⁷⁶ - q⁷⁷

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, 40]]

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

In[3]:=
GaussCode[Knot[9, 40]]

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

In[4]:=
DTCode[Knot[9, 40]]

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

In[5]:=
br = BR[Knot[9, 40]]

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

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

Out[6]=
{4, 9}

In[7]:=
BraidIndex[Knot[9, 40]]

Out[7]=
4

In[8]:=
Show[DrawMorseLink[Knot[9, 40]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[9, 40]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

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

In[10]:=
alex = Alexander[Knot[9, 40]][t]

Out[10]=
-3 7 18 2 3 -23 + t - -- + -- + 18 t - 7 t + t 2 t t

In[11]:=
Conway[Knot[9, 40]][z]

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

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

Out[12]=
{Knot[9, 40], Knot[10, 59], Knot[11, NonAlternating, 66]}

In[13]:=
{KnotDet[Knot[9, 40]], KnotSignature[Knot[9, 40]]}

Out[13]=
{75, -2}

In[14]:=
Jones[Knot[9, 40]][q]

Out[14]=
-7 4 8 11 13 13 11 2 -8 + q - -- + -- - -- + -- - -- + -- + 5 q - q 6 5 4 3 2 q q q q q q

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

Out[15]=
{Knot[9, 40]}

In[16]:=
A2Invariant[Knot[9, 40]][q]

Out[16]=
-22 -20 2 3 -14 2 -10 3 -6 4 3 4 6 1 + q - q - --- + --- - q + --- + q - -- + q - -- + -- + 3 q - q 18 16 12 8 4 2 q q q q q q

In[17]:=
HOMFLYPT[Knot[9, 40]][a, z]

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

In[18]:=
Kauffman[Knot[9, 40]][a, z]

Out[18]=
2 4 3 5 2 2 4 2 6 2 3 3 3 2 + 2 a + a - a z - a z + 3 a z + 7 a z + 4 a z + 6 a z + 14 a z + 5 5 3 7 3 4 2 4 4 4 6 4 8 4 z > 6 a z - 2 a z - 7 z - 17 a z - 20 a z - 9 a z + a z + -- - a 5 3 5 5 5 7 5 6 2 6 4 6 > 15 a z - 32 a z - 12 a z + 4 a z + 5 z + 4 a z + 7 a z + 6 6 7 3 7 5 7 2 8 4 8 > 8 a z + 8 a z + 17 a z + 9 a z + 4 a z + 4 a z

In[19]:=
{Vassiliev[2][Knot[9, 40]], Vassiliev[3][Knot[9, 40]]}

Out[19]=
{-1, 1}

In[20]:=
Kh[Knot[9, 40]][q, t]

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

In[21]:=
ColouredJones[Knot[9, 40], 2][q]

Out[21]=
-20 4 4 9 29 17 45 84 17 107 132 5 -55 + q - --- + --- + --- - --- + --- + --- - --- + --- + --- - --- - -- + 19 18 17 16 15 14 13 12 11 10 9 q q q q q q q q q q q 157 140 34 166 109 55 133 56 2 3 4 > --- - --- - -- + --- - --- - -- + --- - -- + 72 q - 11 q - 31 q + 19 q + 8 7 6 5 4 3 2 q q q q q q q q 5 6 7 > 3 q - 5 q + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 9₄₀

9₃₉
9₄₁

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-20 - 4q^-19 + 4q^-18 + 9q^-17 - 29q^-16 + 17q^-15 + 45q^-14 - 84q^-13 + 17q^-12 + 107q^-11 - 132q^-10 - 5q^-9 + 157q^-8 - 140q^-7 - 34q^-6 + 166q^-5 - 109q^-4 - 55q^-3 + 133q^-2 - 56q^-1 - 55 + 72q - 11q² - 31q³ + 19q⁴ + 3q⁵ - 5q⁶ + q⁷
3	q^-39 - 4q^-38 + 4q^-37 + 5q^-36 - 9q^-35 - 15q^-34 + 25q^-33 + 46q^-32 - 63q^-31 - 104q^-30 + 98q^-29 + 219q^-28 - 115q^-27 - 390q^-26 + 96q^-25 + 584q^-24 - 12q^-23 - 781q^-22 - 124q^-21 + 942q^-20 + 293q^-19 - 1048q^-18 - 461q^-17 + 1083q^-16 + 620q^-15 - 1066q^-14 - 743q^-13 + 994q^-12 + 834q^-11 - 880q^-10 - 891q^-9 + 728q^-8 + 910q^-7 - 544q^-6 - 886q^-5 + 344q^-4 + 815q^-3 - 158q^-2 - 680q^-1 - 11 + 525q + 107q² - 342q³ - 151q⁴ + 194q⁵ + 126q⁶ - 73q⁷ - 94q⁸ + 25q⁹ + 40q¹⁰ + q¹¹ - 14q¹² - 3q¹³ + 5q¹⁴ - q¹⁵
4	q^-64 - 4q^-63 + 4q^-62 + 5q^-61 - 13q^-60 + 5q^-59 - 7q^-58 + 36q^-57 + 17q^-56 - 108q^-55 - 21q^-54 + 22q^-53 + 259q^-52 + 155q^-51 - 473q^-50 - 412q^-49 - 90q^-48 + 1029q^-47 + 1049q^-46 - 889q^-45 - 1701q^-44 - 1284q^-43 + 1991q^-42 + 3403q^-41 - 139q^-40 - 3362q^-39 - 4297q^-38 + 1712q^-37 + 6394q^-36 + 2497q^-35 - 3786q^-34 - 8024q^-33 - 421q^-32 + 8187q^-31 + 5814q^-30 - 2414q^-29 - 10564q^-28 - 3234q^-27 + 8099q^-26 + 8166q^-25 - 219q^-24 - 11264q^-23 - 5423q^-22 + 6778q^-21 + 9105q^-20 + 1879q^-19 - 10527q^-18 - 6752q^-17 + 4753q^-16 + 8945q^-15 + 3786q^-14 - 8642q^-13 - 7392q^-12 + 2089q^-11 + 7709q^-10 + 5407q^-9 - 5605q^-8 - 6995q^-7 - 819q^-6 + 5176q^-5 + 5964q^-4 - 2006q^-3 - 5063q^-2 - 2721q^-1 + 1940 + 4687q + 613q² - 2214q³ - 2607q⁴ - 369q⁵ + 2280q⁶ + 1186q⁷ - 159q⁸ - 1224q⁹ - 830q¹⁰ + 505q¹¹ + 537q¹² + 329q¹³ - 217q¹⁴ - 354q¹⁵ - 8q¹⁶ + 63q¹⁷ + 123q¹⁸ + 12q¹⁹ - 54q²⁰ - 10q²¹ - 6q²² + 14q²³ + 3q²⁴ - 5q²⁵ + q²⁶
5	q^-95 - 4q^-94 + 4q^-93 + 5q^-92 - 13q^-91 + q^-90 + 13q^-89 + 4q^-88 + 7q^-87 - 18q^-86 - 75q^-85 - 21q^-84 + 134q^-83 + 196q^-82 + 65q^-81 - 290q^-80 - 588q^-79 - 346q^-78 + 578q^-77 + 1491q^-76 + 1131q^-75 - 787q^-74 - 2986q^-73 - 3096q^-72 + 254q^-71 + 5164q^-70 + 6816q^-69 + 1841q^-68 - 7184q^-67 - 12469q^-66 - 6869q^-65 + 7818q^-64 + 19647q^-63 + 15448q^-62 - 5474q^-61 - 26773q^-60 - 27371q^-59 - 1343q^-58 + 31974q^-57 + 41431q^-56 + 12728q^-55 - 33418q^-54 - 55278q^-53 - 27849q^-52 + 30045q^-51 + 66904q^-50 + 44655q^-49 - 22256q^-48 - 74559q^-47 - 60878q^-46 + 11310q^-45 + 77742q^-44 + 74637q^-43 + 945q^-42 - 76887q^-41 - 84934q^-40 - 12823q^-39 + 73166q^-38 + 91462q^-37 + 23316q^-36 - 67633q^-35 - 94929q^-34 - 32026q^-33 + 61317q^-32 + 95958q^-31 + 39130q^-30 - 54407q^-29 - 95300q^-28 - 45264q^-27 + 46955q^-26 + 93288q^-25 + 50814q^-24 - 38508q^-23 - 89768q^-22 - 56107q^-21 + 28638q^-20 + 84414q^-19 + 60870q^-18 - 17251q^-17 - 76575q^-16 - 64378q^-15 + 4624q^-14 + 65809q^-13 + 65649q^-12 + 8290q^-11 - 52284q^-10 - 63518q^-9 - 19905q^-8 + 36598q^-7 + 57365q^-6 + 28725q^-5 - 20537q^-4 - 47451q^-3 - 32966q^-2 + 5850q^-1 + 34873 + 32487q + 5201q² - 21711q³ - 27556q⁴ - 11699q⁵ + 10114q⁶ + 20201q⁷ + 13297q⁸ - 1714q⁹ - 12207q¹⁰ - 11610q¹¹ - 2816q¹² + 5796q¹³ + 7952q¹⁴ + 4095q¹⁵ - 1434q¹⁶ - 4496q¹⁷ - 3473q¹⁸ - 388q¹⁹ + 1848q²⁰ + 2087q²¹ + 913q²² - 470q²³ - 1030q²⁴ - 625q²⁵ - 15q²⁶ + 339q²⁷ + 305q²⁸ + 108q²⁹ - 82q³⁰ - 128q³¹ - 41q³² + 17q³³ + 24q³⁴ + 15q³⁵ + 6q³⁶ - 14q³⁷ - 3q³⁸ + 5q³⁹ - q⁴⁰
6	q^-132 - 4q^-131 + 4q^-130 + 5q^-129 - 13q^-128 + q^-127 + 9q^-126 + 24q^-125 - 25q^-124 - 28q^-123 + 15q^-122 - 65q^-121 + 41q^-120 + 147q^-119 + 178q^-118 - 105q^-117 - 379q^-116 - 310q^-115 - 384q^-114 + 386q^-113 + 1312q^-112 + 1583q^-111 + 123q^-110 - 2142q^-109 - 3366q^-108 - 3713q^-107 + 262q^-106 + 6256q^-105 + 10279q^-104 + 6741q^-103 - 3802q^-102 - 14987q^-101 - 22497q^-100 - 12780q^-99 + 11186q^-98 + 36943q^-97 + 42301q^-96 + 17754q^-95 - 26641q^-94 - 72673q^-93 - 75149q^-92 - 22897q^-91 + 64452q^-90 + 126502q^-89 + 114861q^-88 + 20859q^-87 - 122835q^-86 - 206333q^-85 - 163059q^-84 + 11002q^-83 + 208105q^-82 + 300160q^-81 + 203709q^-80 - 69254q^-79 - 330890q^-78 - 406467q^-77 - 200735q^-76 + 165921q^-75 + 472200q^-74 + 496034q^-73 + 157242q^-72 - 317127q^-71 - 626795q^-70 - 518543q^-69 - 53388q^-68 + 496689q^-67 + 752315q^-66 + 479294q^-65 - 130939q^-64 - 694526q^-63 - 787792q^-62 - 353776q^-61 + 358388q^-60 + 855244q^-59 + 744150q^-58 + 123130q^-57 - 610257q^-56 - 909875q^-55 - 595293q^-54 + 160417q^-53 + 818376q^-52 + 875471q^-51 + 326403q^-50 - 468676q^-49 - 908816q^-48 - 725155q^-47 - 3220q^-46 + 724933q^-45 + 904821q^-44 + 448490q^-43 - 340050q^-42 - 855997q^-41 - 780219q^-40 - 120688q^-39 + 625002q^-38 + 892351q^-37 + 529518q^-36 - 222197q^-35 - 785117q^-34 - 810609q^-33 - 233228q^-32 + 505896q^-31 + 857823q^-30 + 610250q^-29 - 74077q^-28 - 674236q^-27 - 822791q^-26 - 371957q^-25 + 326559q^-24 + 768671q^-23 + 682241q^-22 + 125390q^-21 - 479914q^-20 - 770134q^-19 - 510911q^-18 + 75505q^-17 + 575644q^-16 + 680589q^-15 + 330483q^-14 - 198691q^-13 - 594619q^-12 - 564400q^-11 - 181245q^-10 + 282605q^-9 + 537846q^-8 + 434931q^-7 + 84205q^-6 - 308568q^-5 - 458656q^-4 - 320564q^-3 - 6881q^-2 + 277833q^-1 + 364754 + 235195q - 32602q² - 233296q³ - 276577q⁴ - 155693q⁵ + 35402q⁶ + 178683q⁷ + 204585q⁸ + 100451q⁹ - 32561q¹⁰ - 128030q¹¹ - 135402q¹² - 69538q¹³ + 23123q¹⁴ + 88090q¹⁵ + 85559q¹⁶ + 43740q¹⁷ - 14516q¹⁸ - 51067q¹⁹ - 54410q²⁰ - 26949q²¹ + 9228q²² + 27867q²³ + 30283q²⁴ + 15291q²⁵ - 2539q²⁶ - 15544q²⁷ - 15758q²⁸ - 7395q²⁹ + 501q³⁰ + 6891q³¹ + 7213q³² + 4314q³³ - 557q³⁴ - 2878q³⁵ - 2711q³⁶ - 1815q³⁷ + 22q³⁸ + 970q³⁹ + 1244q⁴⁰ + 449q⁴¹ - 34q⁴² - 236q⁴³ - 367q⁴⁴ - 158q⁴⁵ - 3q⁴⁶ + 147q⁴⁷ + 46q⁴⁸ + 12q⁴⁹ + 13q⁵⁰ - 29q⁵¹ - 15q⁵² - 6q⁵³ + 14q⁵⁴ + 3q⁵⁵ - 5q⁵⁶ + q⁵⁷
7	q^-175 - 4q^-174 + 4q^-173 + 5q^-172 - 13q^-171 + q^-170 + 9q^-169 + 20q^-168 - 5q^-167 - 60q^-166 + 5q^-165 + 25q^-164 - 3q^-163 + 64q^-162 + 79q^-161 + 84q^-160 - 122q^-159 - 454q^-158 - 281q^-157 + 52q^-156 + 463q^-155 + 1045q^-154 + 998q^-153 + 433q^-152 - 1222q^-151 - 3419q^-150 - 3579q^-149 - 1570q^-148 + 2835q^-147 + 8202q^-146 + 10225q^-145 + 7031q^-144 - 3253q^-143 - 17943q^-142 - 26826q^-141 - 23120q^-140 - 1907q^-139 + 31757q^-138 + 58792q^-137 + 62462q^-136 + 27835q^-135 - 40805q^-134 - 110794q^-133 - 142875q^-132 - 99576q^-131 + 22452q^-130 + 172482q^-129 + 276253q^-128 + 252753q^-127 + 70313q^-126 - 208683q^-125 - 458776q^-124 - 519690q^-123 - 297183q^-122 + 152790q^-121 + 644597q^-120 + 903805q^-119 + 718366q^-118 + 90916q^-117 - 744407q^-116 - 1360872q^-115 - 1356868q^-114 - 611740q^-113 + 632115q^-112 + 1779506q^-111 + 2170262q^-110 + 1460192q^-109 - 180061q^-108 - 2006206q^-107 - 3040392q^-106 - 2604575q^-105 - 686673q^-104 + 1882299q^-103 + 3786854q^-102 + 3922146q^-101 + 1957131q^-100 - 1299571q^-99 - 4225837q^-98 - 5223160q^-97 - 3513648q^-96 + 249033q^-95 + 4219104q^-94 + 6297475q^-93 + 5165363q^-92 + 1175115q^-91 - 3725086q^-90 - 6984368q^-89 - 6695555q^-88 - 2793574q^-87 + 2808350q^-86 + 7207331q^-85 + 7923098q^-84 + 4400454q^-83 - 1614595q^-82 - 6991824q^-81 - 8747457q^-80 - 5814345q^-79 + 325035q^-78 + 6443244q^-77 + 9157727q^-76 + 6919798q^-75 + 895583q^-74 - 5704977q^-73 - 9218072q^-72 - 7682771q^-71 - 1934342q^-70 + 4916830q^-69 + 9036407q^-68 + 8135949q^-67 + 2741764q^-66 - 4181694q^-65 - 8723609q^-64 - 8354428q^-63 - 3331008q^-62 + 3552214q^-61 + 8372101q^-60 + 8428340q^-59 + 3751721q^-58 - 3034014q^-57 - 8035187q^-56 - 8436095q^-55 - 4075215q^-54 + 2592903q^-53 + 7730287q^-52 + 8435186q^-51 + 4371827q^-50 - 2175554q^-49 - 7440451q^-48 - 8450748q^-47 - 4698297q^-46 + 1717549q^-45 + 7122092q^-44 + 8479451q^-43 + 5090863q^-42 - 1160176q^-41 - 6718006q^-40 - 8486939q^-39 - 5553743q^-38 + 459355q^-37 + 6161476q^-36 + 8413636q^-35 + 6060211q^-34 + 402193q^-33 - 5395536q^-32 - 8182962q^-31 - 6544746q^-30 - 1402181q^-29 + 4384567q^-28 + 7711927q^-27 + 6911668q^-26 + 2474147q^-25 - 3133803q^-24 - 6936254q^-23 - 7049080q^-22 - 3503822q^-21 + 1705045q^-20 + 5831383q^-19 + 6850240q^-18 + 4347929q^-17 - 216180q^-16 - 4436603q^-15 - 6252848q^-14 - 4862053q^-13 - 1165051q^-12 + 2864862q^-11 + 5261483q^-10 + 4937335q^-9 + 2263048q^-8 - 1287412q^-7 - 3970524q^-6 - 4544974q^-5 - 2932065q^-4 - 94561q^-3 + 2546754q^-2 + 3748955q^-1 + 3106578 + 1111249q - 1198124q² - 2705364q³ - 2822242q⁴ - 1665147q⁵ + 112556q⁶ + 1615593q⁷ + 2213668q⁸ + 1764879q⁹ + 590454q¹⁰ - 673337q¹¹ - 1465900q¹² - 1516157q¹³ - 897258q¹⁴ + 7574q¹⁵ + 766515q¹⁶ + 1082336q¹⁷ + 880257q¹⁸ + 346214q¹⁹ - 238988q²⁰ - 625966q²¹ - 676126q²² - 439265q²³ - 66388q²⁴ + 264128q²⁵ + 414939q²⁶ + 367971q²⁷ + 182613q²⁸ - 40511q²⁹ - 195483q³⁰ - 238141q³¹ - 175676q³² - 54766q³³ + 55653q³⁴ + 117895q³⁵ + 118760q³⁶ + 70522q³⁷ + 8542q³⁸ - 41189q³⁹ - 61295q⁴⁰ - 50025q⁴¹ - 23636q⁴² + 4388q⁴³ + 22951q⁴⁴ + 25994q⁴⁵ + 18756q⁴⁶ + 5972q⁴⁷ - 5395q⁴⁸ - 9986q⁴⁹ - 9547q⁵⁰ - 5407q⁵¹ - 522q⁵² + 2496q⁵³ + 3770q⁵⁴ + 2987q⁵⁵ + 997q⁵⁶ - 339q⁵⁷ - 1076q⁵⁸ - 1015q⁵⁹ - 526q⁶⁰ - 207q⁶¹ + 225q⁶² + 363q⁶³ + 205q⁶⁴ + 53q⁶⁵ - 62q⁶⁶ - 65q⁶⁷ - 17q⁶⁸ - 42q⁶⁹ - 8q⁷⁰ + 29q⁷¹ + 15q⁷² + 6q⁷³ - 14q⁷⁴ - 3q⁷⁵ + 5q⁷⁶ - q⁷⁷

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	PD[Knot[9, 40]]
Out[2]=	PD[X[1, 6, 2, 7], X[7, 12, 8, 13], X[5, 15, 6, 14], X[11, 3, 12, 2], > X[15, 10, 16, 11], X[3, 16, 4, 17], X[9, 4, 10, 5], X[17, 9, 18, 8], > X[13, 18, 14, 1]]
In[3]:=	GaussCode[Knot[9, 40]]
Out[3]=	GaussCode[-1, 4, -6, 7, -3, 1, -2, 8, -7, 5, -4, 2, -9, 3, -5, 6, -8, 9]
In[4]:=	DTCode[Knot[9, 40]]
Out[4]=	DTCode[6, 16, 14, 12, 4, 2, 18, 10, 8]
In[5]:=	br = BR[Knot[9, 40]]
Out[5]=	BR[4, {-1, 2, -1, -3, 2, -1, -3, 2, -3}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 9}
In[7]:=	BraidIndex[Knot[9, 40]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[9, 40]]]

Out[8]=	-Graphics-
In[9]:=	#[Knot[9, 40]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 3, 3, 4, 2}
In[10]:=	alex = Alexander[Knot[9, 40]][t]
Out[10]=	-3 7 18 2 3 -23 + t - -- + -- + 18 t - 7 t + t 2 t t
In[11]:=	Conway[Knot[9, 40]][z]
Out[11]=	2 4 6 1 - z - z + z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[9, 40], Knot[10, 59], Knot[11, NonAlternating, 66]}
In[13]:=	{KnotDet[Knot[9, 40]], KnotSignature[Knot[9, 40]]}
Out[13]=	{75, -2}
In[14]:=	Jones[Knot[9, 40]][q]
Out[14]=	-7 4 8 11 13 13 11 2 -8 + q - -- + -- - -- + -- - -- + -- + 5 q - q 6 5 4 3 2 q q q q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[9, 40]}
In[16]:=	A2Invariant[Knot[9, 40]][q]
Out[16]=	-22 -20 2 3 -14 2 -10 3 -6 4 3 4 6 1 + q - q - --- + --- - q + --- + q - -- + q - -- + -- + 3 q - q 18 16 12 8 4 2 q q q q q q
In[17]:=	HOMFLYPT[Knot[9, 40]][a, z]
Out[17]=	2 4 4 2 6 2 4 2 4 4 4 2 6 2 - 2 a + a - 2 a z + a z - z + 2 a z - 2 a z + a z
In[18]:=	Kauffman[Knot[9, 40]][a, z]
Out[18]=	2 4 3 5 2 2 4 2 6 2 3 3 3 2 + 2 a + a - a z - a z + 3 a z + 7 a z + 4 a z + 6 a z + 14 a z + 5 5 3 7 3 4 2 4 4 4 6 4 8 4 z > 6 a z - 2 a z - 7 z - 17 a z - 20 a z - 9 a z + a z + -- - a 5 3 5 5 5 7 5 6 2 6 4 6 > 15 a z - 32 a z - 12 a z + 4 a z + 5 z + 4 a z + 7 a z + 6 6 7 3 7 5 7 2 8 4 8 > 8 a z + 8 a z + 17 a z + 9 a z + 4 a z + 4 a z
In[19]:=	{Vassiliev[2][Knot[9, 40]], Vassiliev[3][Knot[9, 40]]}
Out[19]=	{-1, 1}
In[20]:=	Kh[Knot[9, 40]][q, t]
Out[20]=	5 7 1 3 1 5 3 6 5 7 -- + - + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 3 q 15 6 13 5 11 5 11 4 9 4 9 3 7 3 7 2 q q t q t q t q t q t q t q t q t 6 6 7 4 t 2 3 2 5 3 > ----- + ---- + ---- + --- + 4 q t + q t + 4 q t + q t 5 2 5 3 q q t q t q t
In[21]:=	ColouredJones[Knot[9, 40], 2][q]
Out[21]=	-20 4 4 9 29 17 45 84 17 107 132 5 -55 + q - --- + --- + --- - --- + --- + --- - --- + --- + --- - --- - -- + 19 18 17 16 15 14 13 12 11 10 9 q q q q q q q q q q q 157 140 34 166 109 55 133 56 2 3 4 > --- - --- - -- + --- - --- - -- + --- - -- + 72 q - 11 q - 31 q + 19 q + 8 7 6 5 4 3 2 q q q q q q q q 5 6 7 > 3 q - 5 q + q

The Alternating Knot 940

The Alternating Knot 9₄₀