The Alternating Knot 10₁₁₆

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Acknowledgement

KnotPlot

PD Presentation: X₆₂₇₁ X_16,3,17,4 X_14,7,15,8 X_8,15,9,16 X_10,18,11,17 X_18,6,19,5 X_20,13,1,14 X_12,19,13,20 X_2,10,3,9 X_4,11,5,12

Gauss Code: {1, -9, 2, -10, 6, -1, 3, -4, 9, -5, 10, -8, 7, -3, 4, -2, 5, -6, 8, -7}

DT (Dowker-Thistlethwaite) Code: 6 16 18 14 2 4 20 8 10 12

Minimum Braid Representative:

Length is 10, width is 3
Braid index is 3

A Morse Link Presentation:

3D Invariants:

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

Reversible 2 4 3 / NotAvailable 1

Alexander Polynomial: - t^-4 + 5t^-3 - 12t^-2 + 19t^-1 - 21 + 19t - 12t² + 5t³ - t⁴

Conway Polynomial: 1 - 2z⁴ - 3z⁶ - z⁸

Other knots with the same Alexander/Conway Polynomial: {K11a7, K11a33, K11a82, ...}

Determinant and Signature: {95, -2}

Jones Polynomial: q^-7 - 4q^-6 + 8q^-5 - 12q^-4 + 15q^-3 - 16q^-2 + 15q^-1 - 11 + 8q - 4q² + q³

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

A2 (sl(3)) Invariant: q^-20 - 2q^-18 + 2q^-16 - 2q^-14 + 2q^-10 - 3q^-8 + 4q^-6 - 3q^-4 + 3q^-2 + 1 - q² + 2q⁴ - 2q⁶ + q⁸

HOMFLY-PT Polynomial: 1 + 2z² + 3z⁴ + z⁶ - 4a²z² - 8a²z⁴ - 5a²z⁶ - a²z⁸ + 2a⁴z² + 3a⁴z⁴ + a⁴z⁶

Kauffman Polynomial: a^-2z² - 2a^-2z⁴ + a^-2z⁶ - a^-1z + 6a^-1z³ - 10a^-1z⁵ + 4a^-1z⁷ + 1 - z² + 9z⁴ - 15z⁶ + 6z⁸ - 3az + 17az³ - 22az⁵ + 3az⁷ + 3az⁹ - 3a²z² + 19a²z⁴ - 32a²z⁶ + 14a²z⁸ - 3a³z + 19a³z³ - 29a³z⁵ + 9a³z⁷ + 3a³z⁹ + a⁴z² - a⁴z⁴ - 8a⁴z⁶ + 8a⁴z⁸ - a⁵z + 6a⁵z³ - 13a⁵z⁵ + 10a⁵z⁷ + 2a⁶z² - 8a⁶z⁴ + 8a⁶z⁶ - 2a⁷z³ + 4a⁷z⁵ + a⁸z⁴

V₂ and V₃, the type 2 and 3 Vassiliev invariants: {0, 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=-2 is the signature of 10₁₁₆. 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 r = 4

j = 7 1

j = 5 3

j = 3 5 1

j = 1 6 3

j = -1 9 5

j = -3 8 7

j = -5 7 8

j = -7 5 8

j = -9 3 7

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 + 8q^-17 - 26q^-16 + 20q^-15 + 30q^-14 - 81q^-13 + 46q^-12 + 80q^-11 - 155q^-10 + 52q^-9 + 143q^-8 - 199q^-7 + 27q^-6 + 182q^-5 - 186q^-4 - 12q^-3 + 176q^-2 - 129q^-1 - 42 + 129q - 58q² - 47q³ + 63q⁴ - 10q⁵ - 25q⁶ + 15q⁷ + 2q⁸ - 4q⁹ + q¹⁰

3 q^-39 - 4q^-38 + 4q^-37 + 4q^-36 - 6q^-35 - 11q^-34 + 15q^-33 + 24q^-32 - 43q^-31 - 34q^-30 + 86q^-29 + 68q^-28 - 168q^-27 - 135q^-26 + 280q^-25 + 258q^-24 - 403q^-23 - 446q^-22 + 504q^-21 + 687q^-20 - 553q^-19 - 945q^-18 + 532q^-17 + 1178q^-16 - 439q^-15 - 1360q^-14 + 305q^-13 + 1457q^-12 - 134q^-11 - 1485q^-10 - 41q^-9 + 1436q^-8 + 223q^-7 - 1341q^-6 - 376q^-5 + 1172q^-4 + 532q^-3 - 984q^-2 - 620q^-1 + 732 + 681q - 492q² - 647q³ + 248q⁴ + 562q⁵ - 63q⁶ - 421q⁷ - 59q⁸ + 274q⁹ + 100q¹⁰ - 143q¹¹ - 92q¹² + 55q¹³ + 63q¹⁴ - 16q¹⁵ - 28q¹⁶ + q¹⁷ + 9q¹⁸ + 2q¹⁹ - 4q²⁰ + q²¹

4 q^-64 - 4q^-63 + 4q^-62 + 4q^-61 - 10q^-60 + 9q^-59 - 16q^-58 + 19q^-57 + 11q^-56 - 52q^-55 + 55q^-54 - 27q^-53 + 55q^-52 - 28q^-51 - 232q^-50 + 198q^-49 + 145q^-48 + 269q^-47 - 269q^-46 - 983q^-45 + 249q^-44 + 862q^-43 + 1360q^-42 - 448q^-41 - 2941q^-40 - 811q^-39 + 1805q^-38 + 4182q^-37 + 740q^-36 - 5594q^-35 - 3948q^-34 + 1345q^-33 + 7928q^-32 + 4236q^-31 - 6883q^-30 - 8040q^-29 - 1497q^-28 + 10226q^-27 + 8647q^-26 - 5643q^-25 - 10643q^-24 - 5380q^-23 + 9912q^-22 + 11658q^-21 - 2928q^-20 - 10782q^-19 - 8382q^-18 + 7828q^-17 + 12510q^-16 - 107q^-15 - 9201q^-14 - 10029q^-13 + 4979q^-12 + 11806q^-11 + 2511q^-10 - 6624q^-9 - 10619q^-8 + 1610q^-7 + 9831q^-6 + 4841q^-5 - 3125q^-4 - 9881q^-3 - 1879q^-2 + 6391q^-1 + 5964 + 751q - 7212q² - 4099q³ + 2117q⁴ + 4829q⁵ + 3334q⁶ - 3267q⁷ - 3791q⁸ - 1021q⁹ + 2083q¹⁰ + 3313q¹¹ - 203q¹² - 1744q¹³ - 1661q¹⁴ - 66q¹⁵ + 1639q¹⁶ + 662q¹⁷ - 131q¹⁸ - 786q¹⁹ - 528q²⁰ + 343q²¹ + 300q²² + 215q²³ - 122q²⁴ - 216q²⁵ + 4q²⁶ + 24q²⁷ + 74q²⁸ + 8q²⁹ - 34q³⁰ - 2q³¹ - 5q³² + 9q³³ + 2q³⁴ - 4q³⁵ + q³⁶

5 q^-95 - 4q^-94 + 4q^-93 + 4q^-92 - 10q^-91 + 5q^-90 + 4q^-89 - 12q^-88 + 6q^-87 + 12q^-86 - 14q^-85 + 13q^-84 + 24q^-83 - 42q^-82 - 63q^-81 - 24q^-80 + 75q^-79 + 197q^-78 + 175q^-77 - 149q^-76 - 599q^-75 - 570q^-74 + 239q^-73 + 1304q^-72 + 1550q^-71 + 83q^-70 - 2539q^-69 - 3738q^-68 - 1180q^-67 + 4075q^-66 + 7475q^-65 + 4292q^-64 - 5182q^-63 - 13275q^-62 - 10469q^-61 + 4587q^-60 + 20444q^-59 + 20677q^-58 - 299q^-57 - 27565q^-56 - 34970q^-55 - 9270q^-54 + 32318q^-53 + 51857q^-52 + 24779q^-51 - 32134q^-50 - 68728q^-49 - 45362q^-48 + 25518q^-47 + 82476q^-46 + 68371q^-45 - 12476q^-44 - 90423q^-43 - 90533q^-42 - 5289q^-41 + 91499q^-40 + 108768q^-39 + 24914q^-38 - 86308q^-37 - 121029q^-36 - 43529q^-35 + 76515q^-34 + 127042q^-33 + 59151q^-32 - 64564q^-31 - 127652q^-30 - 70720q^-29 + 52038q^-28 + 124391q^-27 + 78788q^-26 - 40095q^-25 - 118820q^-24 - 84100q^-23 + 28669q^-22 + 111762q^-21 + 87985q^-20 - 17383q^-19 - 103446q^-18 - 90914q^-17 + 5243q^-16 + 93558q^-15 + 93116q^-14 + 7789q^-13 - 81195q^-12 - 93607q^-11 - 22045q^-10 + 65983q^-9 + 91612q^-8 + 35814q^-7 - 47681q^-6 - 85286q^-5 - 48142q^-4 + 27311q^-3 + 74484q^-2 + 56169q^-1 - 6675 - 58666q - 58780q² - 11723q³ + 39932q⁴ + 54628q⁵ + 25193q⁶ - 20301q⁷ - 44778q⁸ - 32018q⁹ + 3192q¹⁰ + 31088q¹¹ + 31894q¹² + 9094q¹³ - 16852q¹⁴ - 26243q¹⁵ - 15096q¹⁶ + 4795q¹⁷ + 17665q¹⁸ + 15598q¹⁹ + 2976q²⁰ - 9117q²¹ - 12227q²² - 6270q²³ + 2554q²⁴ + 7571q²⁵ + 6200q²⁶ + 999q²⁷ - 3460q²⁸ - 4334q²⁹ - 2127q³⁰ + 798q³¹ + 2342q³² + 1831q³³ + 285q³⁴ - 893q³⁵ - 1050q³⁶ - 505q³⁷ + 160q³⁸ + 487q³⁹ + 334q⁴⁰ + 30q⁴¹ - 147q⁴² - 138q⁴³ - 68q⁴⁴ + 29q⁴⁵ + 67q⁴⁶ + 19q⁴⁷ - 10q⁴⁸ - 8q⁴⁹ - 8q⁵⁰ - 5q⁵¹ + 9q⁵² + 2q⁵³ - 4q⁵⁴ + q⁵⁵

6 q^-132 - 4q^-131 + 4q^-130 + 4q^-129 - 10q^-128 + 5q^-127 + 8q^-125 - 25q^-124 + 7q^-123 + 50q^-122 - 46q^-121 + 13q^-120 - 2q^-119 - 13q^-118 - 111q^-117 + 14q^-116 + 271q^-115 + q^-114 + 73q^-113 - 102q^-112 - 360q^-111 - 684q^-110 - 62q^-109 + 1223q^-108 + 1022q^-107 + 1034q^-106 - 352q^-105 - 2474q^-104 - 4185q^-103 - 2050q^-102 + 3737q^-101 + 6820q^-100 + 8263q^-99 + 2593q^-98 - 8747q^-97 - 19362q^-96 - 16995q^-95 + 2124q^-94 + 23198q^-93 + 39215q^-92 + 29186q^-91 - 9448q^-90 - 57726q^-89 - 76085q^-88 - 38504q^-87 + 34107q^-86 + 112817q^-85 + 129144q^-84 + 50738q^-83 - 93140q^-82 - 205301q^-81 - 189786q^-80 - 41565q^-79 + 187090q^-78 + 331258q^-77 + 264909q^-76 - 14632q^-75 - 338667q^-74 - 475002q^-73 - 313121q^-72 + 119496q^-71 + 536866q^-70 + 638692q^-69 + 294233q^-68 - 303221q^-67 - 756111q^-66 - 756425q^-65 - 202630q^-64 + 548003q^-63 + 992144q^-62 + 772158q^-61 - 50q^-60 - 819959q^-59 - 1155290q^-58 - 680215q^-57 + 287898q^-56 + 1109630q^-55 + 1180554q^-54 + 440482q^-53 - 617229q^-52 - 1307243q^-51 - 1070398q^-50 - 92361q^-49 + 971098q^-48 + 1345149q^-47 + 789264q^-46 - 307772q^-45 - 1221615q^-44 - 1236819q^-43 - 390898q^-42 + 734163q^-41 + 1301981q^-40 + 951368q^-39 - 58709q^-38 - 1046663q^-37 - 1236161q^-36 - 551677q^-35 + 529585q^-34 + 1185314q^-33 + 996800q^-32 + 106366q^-31 - 879991q^-30 - 1185242q^-29 - 650722q^-28 + 356307q^-27 + 1063599q^-26 + 1022958q^-25 + 263526q^-24 - 702379q^-23 - 1125649q^-22 - 764887q^-21 + 141376q^-20 + 901461q^-19 + 1045038q^-18 + 469179q^-17 - 443143q^-16 - 1003598q^-15 - 882116q^-14 - 152922q^-13 + 625443q^-12 + 986413q^-11 + 683430q^-10 - 79473q^-9 - 736858q^-8 - 899048q^-7 - 458883q^-6 + 223348q^-5 + 751954q^-4 + 774867q^-3 + 294204q^-2 - 322429q^-1 - 707778 - 617876q - 184699q² + 349841q³ + 627460q⁴ + 496027q⁵ + 93427q⁶ - 333339q⁷ - 513072q⁸ - 399680q⁹ - 47292q¹⁰ + 289815q¹¹ + 418383q¹² + 301992q¹³ + 26480q¹⁴ - 221638q¹⁵ - 333551q¹⁶ - 230694q¹⁷ - 20203q¹⁸ + 169572q¹⁹ + 243138q²⁰ + 174222q²¹ + 28148q²² - 123546q²³ - 173776q²⁴ - 128643q²⁵ - 24126q²⁶ + 75089q²⁷ + 118620q²⁸ + 96454q²⁹ + 20145q³⁰ - 44054q³¹ - 76490q³² - 63064q³³ - 21686q³⁴ + 23245q³⁵ + 48612q³⁶ + 38980q³⁷ + 16692q³⁸ - 10630q³⁹ - 25415q⁴⁰ - 25362q⁴¹ - 11611q⁴² + 5075q⁴³ + 12290q⁴⁴ + 13764q⁴⁵ + 7162q⁴⁶ - 460q⁴⁷ - 6627q⁴⁸ - 6886q⁴⁹ - 3424q⁵⁰ - 522q⁵¹ + 2596q⁵² + 3033q⁵³ + 2170q⁵⁴ + 9q⁵⁵ - 987q⁵⁶ - 1006q⁵⁷ - 927q⁵⁸ - 144q⁵⁹ + 297q⁶⁰ + 540q⁶¹ + 188q⁶² + 28q⁶³ - 34q⁶⁴ - 162q⁶⁵ - 81q⁶⁶ - 19q⁶⁷ + 72q⁶⁸ + 12q⁶⁹ + q⁷⁰ + 16q⁷¹ - 14q⁷² - 8q⁷³ - 5q⁷⁴ + 9q⁷⁵ + 2q⁷⁶ - 4q⁷⁷ + q⁷⁸

7 q^-175 - 4q^-174 + 4q^-173 + 4q^-172 - 10q^-171 + 5q^-170 + 4q^-168 - 5q^-167 - 24q^-166 + 45q^-165 + 18q^-164 - 46q^-163 - 3q^-162 - 24q^-161 + 10q^-160 - 9q^-159 - 46q^-158 + 217q^-157 + 142q^-156 - 128q^-155 - 178q^-154 - 377q^-153 - 160q^-152 + 57q^-151 + 264q^-150 + 1178q^-149 + 1004q^-148 - 110q^-147 - 1317q^-146 - 2880q^-145 - 2372q^-144 - 342q^-143 + 2599q^-142 + 6857q^-141 + 7231q^-140 + 2703q^-139 - 5541q^-138 - 15802q^-137 - 18275q^-136 - 9964q^-135 + 8031q^-134 + 31897q^-133 + 43440q^-132 + 32043q^-131 - 4964q^-130 - 58733q^-129 - 94081q^-128 - 84181q^-127 - 18052q^-126 + 90431q^-125 + 181237q^-124 + 195147q^-123 + 94482q^-122 - 108131q^-121 - 312094q^-120 - 397793q^-119 - 273890q^-118 + 62828q^-117 + 464786q^-116 + 718788q^-115 + 627756q^-114 + 130472q^-113 - 580276q^-112 - 1153869q^-111 - 1217100q^-110 - 583518q^-109 + 539213q^-108 + 1634346q^-107 + 2065397q^-106 + 1411284q^-105 - 178950q^-104 - 2017173q^-103 - 3113040q^-102 - 2673341q^-101 - 667941q^-100 + 2089008q^-99 + 4192792q^-98 + 4328336q^-97 + 2112926q^-96 - 1624008q^-95 - 5051850q^-94 - 6200227q^-93 - 4138335q^-92 + 459641q^-91 + 5404833q^-90 + 7997555q^-89 + 6572543q^-88 + 1430079q^-87 - 5030815q^-86 - 9388717q^-85 - 9107156q^-84 - 3892291q^-83 + 3853496q^-82 + 10093465q^-81 + 11374372q^-80 + 6628537q^-79 - 1977404q^-78 - 9977978q^-77 - 13056008q^-76 - 9267914q^-75 - 331461q^-74 + 9089906q^-73 + 13965385q^-72 + 11473738q^-71 + 2731817q^-70 - 7635703q^-69 - 14093722q^-68 - 13033080q^-67 - 4900343q^-66 + 5911388q^-65 + 13587239q^-64 + 13889997q^-63 + 6614112q^-62 - 4204675q^-61 - 12681350q^-60 - 14137150q^-59 - 7793083q^-58 + 2728723q^-57 + 11626559q^-56 + 13955752q^-55 + 8482179q^-54 - 1581260q^-53 - 10613901q^-52 - 13551342q^-51 - 8817560q^-50 + 748305q^-49 + 9750846q^-48 + 13098623q^-47 + 8964002q^-46 - 138945q^-45 - 9050877q^-44 - 12704262q^-43 - 9074976q^-42 - 379459q^-41 + 8457565q^-40 + 12405864q^-39 + 9258553q^-38 + 937575q^-37 - 7870129q^-36 - 12173638q^-35 - 9565849q^-34 - 1642486q^-33 + 7172709q^-32 + 11928935q^-31 + 9987193q^-30 + 2558510q^-29 - 6254716q^-28 - 11562618q^-27 - 10458859q^-26 - 3692443q^-25 + 5035675q^-24 + 10947851q^-23 + 10863732q^-22 + 4993754q^-21 - 3476329q^-20 - 9971490q^-19 - 11055527q^-18 - 6340984q^-17 + 1610856q^-16 + 8550235q^-15 + 10866184q^-14 + 7560095q^-13 + 457160q^-12 - 6671885q^-11 - 10161317q^-10 - 8438363q^-9 - 2531357q^-8 + 4415087q^-7 + 8859134q^-6 + 8770447q^-5 + 4368608q^-4 - 1962291q^-3 - 6996260q^-2 - 8413324q^-1 - 5700197 - 414462q + 4724899q² + 7335380q³ + 6319177q⁴ + 2410339q⁵ - 2320647q⁶ - 5656063q⁷ - 6129725q⁸ - 3754692q⁹ + 115205q¹⁰ + 3630757q¹¹ + 5201831q¹² + 4295576q¹³ + 1577925q¹⁴ - 1601328q¹⁵ - 3761567q¹⁶ - 4048701q¹⁷ - 2556796q¹⁸ - 94472q¹⁹ + 2139708q²⁰ + 3199093q²¹ + 2785688q²² + 1219794q²³ - 674546q²⁴ - 2047942q²⁵ - 2405172q²⁶ - 1699282q²⁷ - 379123q²⁸ + 916490q²⁹ + 1670699q³⁰ + 1625024q³¹ + 922337q³² - 49172q³³ - 869708q³⁴ - 1207684q³⁵ - 1007750q³⁶ - 440031q³⁷ + 222203q³⁸ + 682179q³⁹ + 796327q⁴⁰ + 581175q⁴¹ + 165868q⁴² - 237719q⁴³ - 475811q⁴⁴ - 488677q⁴⁵ - 302773q⁴⁶ - 38424q⁴⁷ + 189710q⁴⁸ + 304034q⁴⁹ + 273463q⁵⁰ + 146476q⁵¹ - 10319q⁵² - 132631q⁵³ - 173979q⁵⁴ - 142103q⁵⁵ - 64103q⁵⁶ + 22968q⁵⁷ + 78711q⁵⁸ + 92746q⁵⁹ + 68514q⁶⁰ + 22324q⁶¹ - 18261q⁶² - 42440q⁶³ - 44958q⁶⁴ - 28685q⁶⁵ - 7272q⁶⁶ + 11831q⁶⁷ + 21313q⁶⁸ + 18808q⁶⁹ + 10906q⁷⁰ + 1164q⁷¹ - 6386q⁷² - 8625q⁷³ - 7603q⁷⁴ - 3595q⁷⁵ + 771q⁷⁶ + 2759q⁷⁷ + 3310q⁷⁸ + 2326q⁷⁹ + 772q⁸⁰ - 299q⁸¹ - 1177q⁸² - 1187q⁸³ - 501q⁸⁴ - 69q⁸⁵ + 277q⁸⁶ + 317q⁸⁷ + 203q⁸⁸ + 180q⁸⁹ - 35q⁹⁰ - 140q⁹¹ - 81q⁹² - 32q⁹³ + 24q⁹⁴ + 17q⁹⁵ - 6q⁹⁶ + 27q⁹⁷ + 10q⁹⁸ - 14q⁹⁹ - 8q¹⁰⁰ - 5q¹⁰¹ + 9q¹⁰² + 2q¹⁰³ - 4q¹⁰⁴ + q¹⁰⁵

Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):

In[1]:=

<< KnotTheory`

Loading KnotTheory` (version of August 30, 2005, 10:15:35)...

In[2]:=
PD[Knot[10, 116]]

Out[2]=
PD[X[6, 2, 7, 1], X[16, 3, 17, 4], X[14, 7, 15, 8], X[8, 15, 9, 16], > X[10, 18, 11, 17], X[18, 6, 19, 5], X[20, 13, 1, 14], X[12, 19, 13, 20], > X[2, 10, 3, 9], X[4, 11, 5, 12]]

In[3]:=
GaussCode[Knot[10, 116]]

Out[3]=
GaussCode[1, -9, 2, -10, 6, -1, 3, -4, 9, -5, 10, -8, 7, -3, 4, -2, 5, -6, 8, > -7]

In[4]:=
DTCode[Knot[10, 116]]

Out[4]=
DTCode[6, 16, 18, 14, 2, 4, 20, 8, 10, 12]

In[5]:=
br = BR[Knot[10, 116]]

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

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

Out[6]=
{3, 10}

In[7]:=
BraidIndex[Knot[10, 116]]

Out[7]=
3

In[8]:=
Show[DrawMorseLink[Knot[10, 116]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[10, 116]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

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

In[10]:=
alex = Alexander[Knot[10, 116]][t]

Out[10]=
-4 5 12 19 2 3 4 -21 - t + -- - -- + -- + 19 t - 12 t + 5 t - t 3 2 t t t

In[11]:=
Conway[Knot[10, 116]][z]

Out[11]=
4 6 8 1 - 2 z - 3 z - z

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

Out[12]=
{Knot[10, 116], Knot[11, Alternating, 7], Knot[11, Alternating, 33], > Knot[11, Alternating, 82]}

In[13]:=
{KnotDet[Knot[10, 116]], KnotSignature[Knot[10, 116]]}

Out[13]=
{95, -2}

In[14]:=
Jones[Knot[10, 116]][q]

Out[14]=
-7 4 8 12 15 16 15 2 3 -11 + q - -- + -- - -- + -- - -- + -- + 8 q - 4 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[10, 116]}

In[16]:=
A2Invariant[Knot[10, 116]][q]

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

In[17]:=
HOMFLYPT[Knot[10, 116]][a, z]

Out[17]=
2 2 2 4 2 4 2 4 4 4 6 2 6 1 + 2 z - 4 a z + 2 a z + 3 z - 8 a z + 3 a z + z - 5 a z + 4 6 2 8 > a z - a z

In[18]:=
Kauffman[Knot[10, 116]][a, z]

Out[18]=
2 3 z 3 5 2 z 2 2 4 2 6 2 6 z 1 - - - 3 a z - 3 a z - a z - z + -- - 3 a z + a z + 2 a z + ---- + a 2 a a 4 3 3 3 5 3 7 3 4 2 z 2 4 4 4 > 17 a z + 19 a z + 6 a z - 2 a z + 9 z - ---- + 19 a z - a z - 2 a 5 6 4 8 4 10 z 5 3 5 5 5 7 5 6 > 8 a z + a z - ----- - 22 a z - 29 a z - 13 a z + 4 a z - 15 z + a 6 7 z 2 6 4 6 6 6 4 z 7 3 7 5 7 > -- - 32 a z - 8 a z + 8 a z + ---- + 3 a z + 9 a z + 10 a z + 2 a a 8 2 8 4 8 9 3 9 > 6 z + 14 a z + 8 a z + 3 a z + 3 a z

In[19]:=
{Vassiliev[2][Knot[10, 116]], Vassiliev[3][Knot[10, 116]]}

Out[19]=
{0, 0}

In[20]:=
Kh[Knot[10, 116]][q, t]

Out[20]=
7 9 1 3 1 5 3 7 5 8 -- + - + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 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 7 8 8 5 t 2 3 2 3 3 5 3 > ----- + ---- + ---- + --- + 6 q t + 3 q t + 5 q t + q t + 3 q t + 5 2 5 3 q q t q t q t 7 4 > q t

In[21]:=
ColouredJones[Knot[10, 116], 2][q]

Out[21]=
-20 4 4 8 26 20 30 81 46 80 155 52 -42 + q - --- + --- + --- - --- + --- + --- - --- + --- + --- - --- + -- + 19 18 17 16 15 14 13 12 11 10 9 q q q q q q q q q q q 143 199 27 182 186 12 176 129 2 3 > --- - --- + -- + --- - --- - -- + --- - --- + 129 q - 58 q - 47 q + 8 7 6 5 4 3 2 q q q q q q q q 4 5 6 7 8 9 10 > 63 q - 10 q - 25 q + 15 q + 2 q - 4 q + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 10₁₁₆

10₁₁₅
10₁₁₇

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-20 - 4q^-19 + 4q^-18 + 8q^-17 - 26q^-16 + 20q^-15 + 30q^-14 - 81q^-13 + 46q^-12 + 80q^-11 - 155q^-10 + 52q^-9 + 143q^-8 - 199q^-7 + 27q^-6 + 182q^-5 - 186q^-4 - 12q^-3 + 176q^-2 - 129q^-1 - 42 + 129q - 58q² - 47q³ + 63q⁴ - 10q⁵ - 25q⁶ + 15q⁷ + 2q⁸ - 4q⁹ + q¹⁰
3	q^-39 - 4q^-38 + 4q^-37 + 4q^-36 - 6q^-35 - 11q^-34 + 15q^-33 + 24q^-32 - 43q^-31 - 34q^-30 + 86q^-29 + 68q^-28 - 168q^-27 - 135q^-26 + 280q^-25 + 258q^-24 - 403q^-23 - 446q^-22 + 504q^-21 + 687q^-20 - 553q^-19 - 945q^-18 + 532q^-17 + 1178q^-16 - 439q^-15 - 1360q^-14 + 305q^-13 + 1457q^-12 - 134q^-11 - 1485q^-10 - 41q^-9 + 1436q^-8 + 223q^-7 - 1341q^-6 - 376q^-5 + 1172q^-4 + 532q^-3 - 984q^-2 - 620q^-1 + 732 + 681q - 492q² - 647q³ + 248q⁴ + 562q⁵ - 63q⁶ - 421q⁷ - 59q⁸ + 274q⁹ + 100q¹⁰ - 143q¹¹ - 92q¹² + 55q¹³ + 63q¹⁴ - 16q¹⁵ - 28q¹⁶ + q¹⁷ + 9q¹⁸ + 2q¹⁹ - 4q²⁰ + q²¹
4	q^-64 - 4q^-63 + 4q^-62 + 4q^-61 - 10q^-60 + 9q^-59 - 16q^-58 + 19q^-57 + 11q^-56 - 52q^-55 + 55q^-54 - 27q^-53 + 55q^-52 - 28q^-51 - 232q^-50 + 198q^-49 + 145q^-48 + 269q^-47 - 269q^-46 - 983q^-45 + 249q^-44 + 862q^-43 + 1360q^-42 - 448q^-41 - 2941q^-40 - 811q^-39 + 1805q^-38 + 4182q^-37 + 740q^-36 - 5594q^-35 - 3948q^-34 + 1345q^-33 + 7928q^-32 + 4236q^-31 - 6883q^-30 - 8040q^-29 - 1497q^-28 + 10226q^-27 + 8647q^-26 - 5643q^-25 - 10643q^-24 - 5380q^-23 + 9912q^-22 + 11658q^-21 - 2928q^-20 - 10782q^-19 - 8382q^-18 + 7828q^-17 + 12510q^-16 - 107q^-15 - 9201q^-14 - 10029q^-13 + 4979q^-12 + 11806q^-11 + 2511q^-10 - 6624q^-9 - 10619q^-8 + 1610q^-7 + 9831q^-6 + 4841q^-5 - 3125q^-4 - 9881q^-3 - 1879q^-2 + 6391q^-1 + 5964 + 751q - 7212q² - 4099q³ + 2117q⁴ + 4829q⁵ + 3334q⁶ - 3267q⁷ - 3791q⁸ - 1021q⁹ + 2083q¹⁰ + 3313q¹¹ - 203q¹² - 1744q¹³ - 1661q¹⁴ - 66q¹⁵ + 1639q¹⁶ + 662q¹⁷ - 131q¹⁸ - 786q¹⁹ - 528q²⁰ + 343q²¹ + 300q²² + 215q²³ - 122q²⁴ - 216q²⁵ + 4q²⁶ + 24q²⁷ + 74q²⁸ + 8q²⁹ - 34q³⁰ - 2q³¹ - 5q³² + 9q³³ + 2q³⁴ - 4q³⁵ + q³⁶
5	q^-95 - 4q^-94 + 4q^-93 + 4q^-92 - 10q^-91 + 5q^-90 + 4q^-89 - 12q^-88 + 6q^-87 + 12q^-86 - 14q^-85 + 13q^-84 + 24q^-83 - 42q^-82 - 63q^-81 - 24q^-80 + 75q^-79 + 197q^-78 + 175q^-77 - 149q^-76 - 599q^-75 - 570q^-74 + 239q^-73 + 1304q^-72 + 1550q^-71 + 83q^-70 - 2539q^-69 - 3738q^-68 - 1180q^-67 + 4075q^-66 + 7475q^-65 + 4292q^-64 - 5182q^-63 - 13275q^-62 - 10469q^-61 + 4587q^-60 + 20444q^-59 + 20677q^-58 - 299q^-57 - 27565q^-56 - 34970q^-55 - 9270q^-54 + 32318q^-53 + 51857q^-52 + 24779q^-51 - 32134q^-50 - 68728q^-49 - 45362q^-48 + 25518q^-47 + 82476q^-46 + 68371q^-45 - 12476q^-44 - 90423q^-43 - 90533q^-42 - 5289q^-41 + 91499q^-40 + 108768q^-39 + 24914q^-38 - 86308q^-37 - 121029q^-36 - 43529q^-35 + 76515q^-34 + 127042q^-33 + 59151q^-32 - 64564q^-31 - 127652q^-30 - 70720q^-29 + 52038q^-28 + 124391q^-27 + 78788q^-26 - 40095q^-25 - 118820q^-24 - 84100q^-23 + 28669q^-22 + 111762q^-21 + 87985q^-20 - 17383q^-19 - 103446q^-18 - 90914q^-17 + 5243q^-16 + 93558q^-15 + 93116q^-14 + 7789q^-13 - 81195q^-12 - 93607q^-11 - 22045q^-10 + 65983q^-9 + 91612q^-8 + 35814q^-7 - 47681q^-6 - 85286q^-5 - 48142q^-4 + 27311q^-3 + 74484q^-2 + 56169q^-1 - 6675 - 58666q - 58780q² - 11723q³ + 39932q⁴ + 54628q⁵ + 25193q⁶ - 20301q⁷ - 44778q⁸ - 32018q⁹ + 3192q¹⁰ + 31088q¹¹ + 31894q¹² + 9094q¹³ - 16852q¹⁴ - 26243q¹⁵ - 15096q¹⁶ + 4795q¹⁷ + 17665q¹⁸ + 15598q¹⁹ + 2976q²⁰ - 9117q²¹ - 12227q²² - 6270q²³ + 2554q²⁴ + 7571q²⁵ + 6200q²⁶ + 999q²⁷ - 3460q²⁸ - 4334q²⁹ - 2127q³⁰ + 798q³¹ + 2342q³² + 1831q³³ + 285q³⁴ - 893q³⁵ - 1050q³⁶ - 505q³⁷ + 160q³⁸ + 487q³⁹ + 334q⁴⁰ + 30q⁴¹ - 147q⁴² - 138q⁴³ - 68q⁴⁴ + 29q⁴⁵ + 67q⁴⁶ + 19q⁴⁷ - 10q⁴⁸ - 8q⁴⁹ - 8q⁵⁰ - 5q⁵¹ + 9q⁵² + 2q⁵³ - 4q⁵⁴ + q⁵⁵
6	q^-132 - 4q^-131 + 4q^-130 + 4q^-129 - 10q^-128 + 5q^-127 + 8q^-125 - 25q^-124 + 7q^-123 + 50q^-122 - 46q^-121 + 13q^-120 - 2q^-119 - 13q^-118 - 111q^-117 + 14q^-116 + 271q^-115 + q^-114 + 73q^-113 - 102q^-112 - 360q^-111 - 684q^-110 - 62q^-109 + 1223q^-108 + 1022q^-107 + 1034q^-106 - 352q^-105 - 2474q^-104 - 4185q^-103 - 2050q^-102 + 3737q^-101 + 6820q^-100 + 8263q^-99 + 2593q^-98 - 8747q^-97 - 19362q^-96 - 16995q^-95 + 2124q^-94 + 23198q^-93 + 39215q^-92 + 29186q^-91 - 9448q^-90 - 57726q^-89 - 76085q^-88 - 38504q^-87 + 34107q^-86 + 112817q^-85 + 129144q^-84 + 50738q^-83 - 93140q^-82 - 205301q^-81 - 189786q^-80 - 41565q^-79 + 187090q^-78 + 331258q^-77 + 264909q^-76 - 14632q^-75 - 338667q^-74 - 475002q^-73 - 313121q^-72 + 119496q^-71 + 536866q^-70 + 638692q^-69 + 294233q^-68 - 303221q^-67 - 756111q^-66 - 756425q^-65 - 202630q^-64 + 548003q^-63 + 992144q^-62 + 772158q^-61 - 50q^-60 - 819959q^-59 - 1155290q^-58 - 680215q^-57 + 287898q^-56 + 1109630q^-55 + 1180554q^-54 + 440482q^-53 - 617229q^-52 - 1307243q^-51 - 1070398q^-50 - 92361q^-49 + 971098q^-48 + 1345149q^-47 + 789264q^-46 - 307772q^-45 - 1221615q^-44 - 1236819q^-43 - 390898q^-42 + 734163q^-41 + 1301981q^-40 + 951368q^-39 - 58709q^-38 - 1046663q^-37 - 1236161q^-36 - 551677q^-35 + 529585q^-34 + 1185314q^-33 + 996800q^-32 + 106366q^-31 - 879991q^-30 - 1185242q^-29 - 650722q^-28 + 356307q^-27 + 1063599q^-26 + 1022958q^-25 + 263526q^-24 - 702379q^-23 - 1125649q^-22 - 764887q^-21 + 141376q^-20 + 901461q^-19 + 1045038q^-18 + 469179q^-17 - 443143q^-16 - 1003598q^-15 - 882116q^-14 - 152922q^-13 + 625443q^-12 + 986413q^-11 + 683430q^-10 - 79473q^-9 - 736858q^-8 - 899048q^-7 - 458883q^-6 + 223348q^-5 + 751954q^-4 + 774867q^-3 + 294204q^-2 - 322429q^-1 - 707778 - 617876q - 184699q² + 349841q³ + 627460q⁴ + 496027q⁵ + 93427q⁶ - 333339q⁷ - 513072q⁸ - 399680q⁹ - 47292q¹⁰ + 289815q¹¹ + 418383q¹² + 301992q¹³ + 26480q¹⁴ - 221638q¹⁵ - 333551q¹⁶ - 230694q¹⁷ - 20203q¹⁸ + 169572q¹⁹ + 243138q²⁰ + 174222q²¹ + 28148q²² - 123546q²³ - 173776q²⁴ - 128643q²⁵ - 24126q²⁶ + 75089q²⁷ + 118620q²⁸ + 96454q²⁹ + 20145q³⁰ - 44054q³¹ - 76490q³² - 63064q³³ - 21686q³⁴ + 23245q³⁵ + 48612q³⁶ + 38980q³⁷ + 16692q³⁸ - 10630q³⁹ - 25415q⁴⁰ - 25362q⁴¹ - 11611q⁴² + 5075q⁴³ + 12290q⁴⁴ + 13764q⁴⁵ + 7162q⁴⁶ - 460q⁴⁷ - 6627q⁴⁸ - 6886q⁴⁹ - 3424q⁵⁰ - 522q⁵¹ + 2596q⁵² + 3033q⁵³ + 2170q⁵⁴ + 9q⁵⁵ - 987q⁵⁶ - 1006q⁵⁷ - 927q⁵⁸ - 144q⁵⁹ + 297q⁶⁰ + 540q⁶¹ + 188q⁶² + 28q⁶³ - 34q⁶⁴ - 162q⁶⁵ - 81q⁶⁶ - 19q⁶⁷ + 72q⁶⁸ + 12q⁶⁹ + q⁷⁰ + 16q⁷¹ - 14q⁷² - 8q⁷³ - 5q⁷⁴ + 9q⁷⁵ + 2q⁷⁶ - 4q⁷⁷ + q⁷⁸
7	q^-175 - 4q^-174 + 4q^-173 + 4q^-172 - 10q^-171 + 5q^-170 + 4q^-168 - 5q^-167 - 24q^-166 + 45q^-165 + 18q^-164 - 46q^-163 - 3q^-162 - 24q^-161 + 10q^-160 - 9q^-159 - 46q^-158 + 217q^-157 + 142q^-156 - 128q^-155 - 178q^-154 - 377q^-153 - 160q^-152 + 57q^-151 + 264q^-150 + 1178q^-149 + 1004q^-148 - 110q^-147 - 1317q^-146 - 2880q^-145 - 2372q^-144 - 342q^-143 + 2599q^-142 + 6857q^-141 + 7231q^-140 + 2703q^-139 - 5541q^-138 - 15802q^-137 - 18275q^-136 - 9964q^-135 + 8031q^-134 + 31897q^-133 + 43440q^-132 + 32043q^-131 - 4964q^-130 - 58733q^-129 - 94081q^-128 - 84181q^-127 - 18052q^-126 + 90431q^-125 + 181237q^-124 + 195147q^-123 + 94482q^-122 - 108131q^-121 - 312094q^-120 - 397793q^-119 - 273890q^-118 + 62828q^-117 + 464786q^-116 + 718788q^-115 + 627756q^-114 + 130472q^-113 - 580276q^-112 - 1153869q^-111 - 1217100q^-110 - 583518q^-109 + 539213q^-108 + 1634346q^-107 + 2065397q^-106 + 1411284q^-105 - 178950q^-104 - 2017173q^-103 - 3113040q^-102 - 2673341q^-101 - 667941q^-100 + 2089008q^-99 + 4192792q^-98 + 4328336q^-97 + 2112926q^-96 - 1624008q^-95 - 5051850q^-94 - 6200227q^-93 - 4138335q^-92 + 459641q^-91 + 5404833q^-90 + 7997555q^-89 + 6572543q^-88 + 1430079q^-87 - 5030815q^-86 - 9388717q^-85 - 9107156q^-84 - 3892291q^-83 + 3853496q^-82 + 10093465q^-81 + 11374372q^-80 + 6628537q^-79 - 1977404q^-78 - 9977978q^-77 - 13056008q^-76 - 9267914q^-75 - 331461q^-74 + 9089906q^-73 + 13965385q^-72 + 11473738q^-71 + 2731817q^-70 - 7635703q^-69 - 14093722q^-68 - 13033080q^-67 - 4900343q^-66 + 5911388q^-65 + 13587239q^-64 + 13889997q^-63 + 6614112q^-62 - 4204675q^-61 - 12681350q^-60 - 14137150q^-59 - 7793083q^-58 + 2728723q^-57 + 11626559q^-56 + 13955752q^-55 + 8482179q^-54 - 1581260q^-53 - 10613901q^-52 - 13551342q^-51 - 8817560q^-50 + 748305q^-49 + 9750846q^-48 + 13098623q^-47 + 8964002q^-46 - 138945q^-45 - 9050877q^-44 - 12704262q^-43 - 9074976q^-42 - 379459q^-41 + 8457565q^-40 + 12405864q^-39 + 9258553q^-38 + 937575q^-37 - 7870129q^-36 - 12173638q^-35 - 9565849q^-34 - 1642486q^-33 + 7172709q^-32 + 11928935q^-31 + 9987193q^-30 + 2558510q^-29 - 6254716q^-28 - 11562618q^-27 - 10458859q^-26 - 3692443q^-25 + 5035675q^-24 + 10947851q^-23 + 10863732q^-22 + 4993754q^-21 - 3476329q^-20 - 9971490q^-19 - 11055527q^-18 - 6340984q^-17 + 1610856q^-16 + 8550235q^-15 + 10866184q^-14 + 7560095q^-13 + 457160q^-12 - 6671885q^-11 - 10161317q^-10 - 8438363q^-9 - 2531357q^-8 + 4415087q^-7 + 8859134q^-6 + 8770447q^-5 + 4368608q^-4 - 1962291q^-3 - 6996260q^-2 - 8413324q^-1 - 5700197 - 414462q + 4724899q² + 7335380q³ + 6319177q⁴ + 2410339q⁵ - 2320647q⁶ - 5656063q⁷ - 6129725q⁸ - 3754692q⁹ + 115205q¹⁰ + 3630757q¹¹ + 5201831q¹² + 4295576q¹³ + 1577925q¹⁴ - 1601328q¹⁵ - 3761567q¹⁶ - 4048701q¹⁷ - 2556796q¹⁸ - 94472q¹⁹ + 2139708q²⁰ + 3199093q²¹ + 2785688q²² + 1219794q²³ - 674546q²⁴ - 2047942q²⁵ - 2405172q²⁶ - 1699282q²⁷ - 379123q²⁸ + 916490q²⁹ + 1670699q³⁰ + 1625024q³¹ + 922337q³² - 49172q³³ - 869708q³⁴ - 1207684q³⁵ - 1007750q³⁶ - 440031q³⁷ + 222203q³⁸ + 682179q³⁹ + 796327q⁴⁰ + 581175q⁴¹ + 165868q⁴² - 237719q⁴³ - 475811q⁴⁴ - 488677q⁴⁵ - 302773q⁴⁶ - 38424q⁴⁷ + 189710q⁴⁸ + 304034q⁴⁹ + 273463q⁵⁰ + 146476q⁵¹ - 10319q⁵² - 132631q⁵³ - 173979q⁵⁴ - 142103q⁵⁵ - 64103q⁵⁶ + 22968q⁵⁷ + 78711q⁵⁸ + 92746q⁵⁹ + 68514q⁶⁰ + 22324q⁶¹ - 18261q⁶² - 42440q⁶³ - 44958q⁶⁴ - 28685q⁶⁵ - 7272q⁶⁶ + 11831q⁶⁷ + 21313q⁶⁸ + 18808q⁶⁹ + 10906q⁷⁰ + 1164q⁷¹ - 6386q⁷² - 8625q⁷³ - 7603q⁷⁴ - 3595q⁷⁵ + 771q⁷⁶ + 2759q⁷⁷ + 3310q⁷⁸ + 2326q⁷⁹ + 772q⁸⁰ - 299q⁸¹ - 1177q⁸² - 1187q⁸³ - 501q⁸⁴ - 69q⁸⁵ + 277q⁸⁶ + 317q⁸⁷ + 203q⁸⁸ + 180q⁸⁹ - 35q⁹⁰ - 140q⁹¹ - 81q⁹² - 32q⁹³ + 24q⁹⁴ + 17q⁹⁵ - 6q⁹⁶ + 27q⁹⁷ + 10q⁹⁸ - 14q⁹⁹ - 8q¹⁰⁰ - 5q¹⁰¹ + 9q¹⁰² + 2q¹⁰³ - 4q¹⁰⁴ + q¹⁰⁵

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	PD[Knot[10, 116]]
Out[2]=	PD[X[6, 2, 7, 1], X[16, 3, 17, 4], X[14, 7, 15, 8], X[8, 15, 9, 16], > X[10, 18, 11, 17], X[18, 6, 19, 5], X[20, 13, 1, 14], X[12, 19, 13, 20], > X[2, 10, 3, 9], X[4, 11, 5, 12]]
In[3]:=	GaussCode[Knot[10, 116]]
Out[3]=	GaussCode[1, -9, 2, -10, 6, -1, 3, -4, 9, -5, 10, -8, 7, -3, 4, -2, 5, -6, 8, > -7]
In[4]:=	DTCode[Knot[10, 116]]
Out[4]=	DTCode[6, 16, 18, 14, 2, 4, 20, 8, 10, 12]
In[5]:=	br = BR[Knot[10, 116]]
Out[5]=	BR[3, {-1, -1, 2, -1, -1, 2, -1, 2, -1, 2}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{3, 10}
In[7]:=	BraidIndex[Knot[10, 116]]
Out[7]=	3
In[8]:=	Show[DrawMorseLink[Knot[10, 116]]]

Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 116]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 4, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 116]][t]
Out[10]=	-4 5 12 19 2 3 4 -21 - t + -- - -- + -- + 19 t - 12 t + 5 t - t 3 2 t t t
In[11]:=	Conway[Knot[10, 116]][z]
Out[11]=	4 6 8 1 - 2 z - 3 z - z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 116], Knot[11, Alternating, 7], Knot[11, Alternating, 33], > Knot[11, Alternating, 82]}
In[13]:=	{KnotDet[Knot[10, 116]], KnotSignature[Knot[10, 116]]}
Out[13]=	{95, -2}
In[14]:=	Jones[Knot[10, 116]][q]
Out[14]=	-7 4 8 12 15 16 15 2 3 -11 + q - -- + -- - -- + -- - -- + -- + 8 q - 4 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[10, 116]}
In[16]:=	A2Invariant[Knot[10, 116]][q]
Out[16]=	-20 2 2 2 2 3 4 3 3 2 4 6 8 1 + q - --- + --- - --- + --- - -- + -- - -- + -- - q + 2 q - 2 q + q 18 16 14 10 8 6 4 2 q q q q q q q q
In[17]:=	HOMFLYPT[Knot[10, 116]][a, z]
Out[17]=	2 2 2 4 2 4 2 4 4 4 6 2 6 1 + 2 z - 4 a z + 2 a z + 3 z - 8 a z + 3 a z + z - 5 a z + 4 6 2 8 > a z - a z
In[18]:=	Kauffman[Knot[10, 116]][a, z]
Out[18]=	2 3 z 3 5 2 z 2 2 4 2 6 2 6 z 1 - - - 3 a z - 3 a z - a z - z + -- - 3 a z + a z + 2 a z + ---- + a 2 a a 4 3 3 3 5 3 7 3 4 2 z 2 4 4 4 > 17 a z + 19 a z + 6 a z - 2 a z + 9 z - ---- + 19 a z - a z - 2 a 5 6 4 8 4 10 z 5 3 5 5 5 7 5 6 > 8 a z + a z - ----- - 22 a z - 29 a z - 13 a z + 4 a z - 15 z + a 6 7 z 2 6 4 6 6 6 4 z 7 3 7 5 7 > -- - 32 a z - 8 a z + 8 a z + ---- + 3 a z + 9 a z + 10 a z + 2 a a 8 2 8 4 8 9 3 9 > 6 z + 14 a z + 8 a z + 3 a z + 3 a z
In[19]:=	{Vassiliev[2][Knot[10, 116]], Vassiliev[3][Knot[10, 116]]}
Out[19]=	{0, 0}
In[20]:=	Kh[Knot[10, 116]][q, t]
Out[20]=	7 9 1 3 1 5 3 7 5 8 -- + - + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 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 7 8 8 5 t 2 3 2 3 3 5 3 > ----- + ---- + ---- + --- + 6 q t + 3 q t + 5 q t + q t + 3 q t + 5 2 5 3 q q t q t q t 7 4 > q t
In[21]:=	ColouredJones[Knot[10, 116], 2][q]
Out[21]=	-20 4 4 8 26 20 30 81 46 80 155 52 -42 + q - --- + --- + --- - --- + --- + --- - --- + --- + --- - --- + -- + 19 18 17 16 15 14 13 12 11 10 9 q q q q q q q q q q q 143 199 27 182 186 12 176 129 2 3 > --- - --- + -- + --- - --- - -- + --- - --- + 129 q - 58 q - 47 q + 8 7 6 5 4 3 2 q q q q q q q q 4 5 6 7 8 9 10 > 63 q - 10 q - 25 q + 15 q + 2 q - 4 q + q

The Alternating Knot 10116

The Alternating Knot 10₁₁₆