The Alternating Knot 10₁₁₂

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Acknowledgement

KnotPlot

PD Presentation: X₆₂₇₁ X₈₃₉₄ X_18,11,19,12 X_20,13,1,14 X_2,16,3,15 X_4,17,5,18 X_12,19,13,20 X_10,6,11,5 X_14,7,15,8 X_16,10,17,9

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

DT (Dowker-Thistlethwaite) Code: 6 8 10 14 16 18 20 2 4 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 - 11t^-2 + 17t^-1 - 19 + 17t - 11t² + 5t³ - t⁴

Conway Polynomial: 1 + 2z² - z⁴ - 3z⁶ - z⁸

Other knots with the same Alexander/Conway Polynomial: {K11a184, ...}

Determinant and Signature: {87, -2}

Jones Polynomial: q^-7 - 4q^-6 + 7q^-5 - 11q^-4 + 14q^-3 - 14q^-2 + 14q^-1 - 10 + 7q - 4q² + q³

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

A2 (sl(3)) Invariant: q^-20 - 2q^-18 + q^-16 - 3q^-14 - q^-12 + 2q^-10 - q^-8 + 6q^-6 - q^-4 + 3q^-2 - 2q² + q⁴ - 2q⁶ + q⁸

HOMFLY-PT Polynomial: - 1 + z² + 3z⁴ + z⁶ + 4a² - 7a²z⁴ - 5a²z⁶ - a²z⁸ - 2a⁴ + a⁴z² + 3a⁴z⁴ + a⁴z⁶

Kauffman Polynomial: - 2a^-2z⁴ + a^-2z⁶ + 6a^-1z³ - 11a^-1z⁵ + 4a^-1z⁷ - 1 - 3z² + 15z⁴ - 18z⁶ + 6z⁸ + 13az³ - 16az⁵ + 3az⁹ - 4a² - 3a²z² + 28a²z⁴ - 35a²z⁶ + 13a²z⁸ + 2a³z + 9a³z³ - 17a³z⁵ + 4a³z⁷ + 3a³z⁹ - 2a⁴ + a⁴z² + 3a⁴z⁴ - 9a⁴z⁶ + 7a⁴z⁸ + 2a⁵z - a⁵z³ - 8a⁵z⁵ + 8a⁵z⁷ + a⁶z² - 7a⁶z⁴ + 7a⁶z⁶ - 3a⁷z³ + 4a⁷z⁵ + a⁸z⁴

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

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 10₁₁₂. 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 4 1

j = 1 6 3

j = -1 8 4

j = -3 7 7

j = -5 7 7

j = -7 4 7

j = -9 3 7

j = -11 1 4

j = -13 3

j = -15 1

n Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q^-20 - 4q^-19 + 3q^-18 + 9q^-17 - 22q^-16 + 14q^-15 + 27q^-14 - 65q^-13 + 33q^-12 + 66q^-11 - 123q^-10 + 38q^-9 + 117q^-8 - 158q^-7 + 19q^-6 + 148q^-5 - 148q^-4 - 12q^-3 + 143q^-2 - 103q^-1 - 36 + 106q - 47q² - 41q³ + 54q⁴ - 8q⁵ - 23q⁶ + 14q⁷ + 2q⁸ - 4q⁹ + q¹⁰

3 q^-39 - 4q^-38 + 3q^-37 + 5q^-36 - 2q^-35 - 12q^-34 + 3q^-33 + 23q^-32 - 19q^-31 - 24q^-30 + 37q^-29 + 47q^-28 - 94q^-27 - 75q^-26 + 163q^-25 + 155q^-24 - 259q^-23 - 268q^-22 + 332q^-21 + 433q^-20 - 380q^-19 - 610q^-18 + 375q^-17 + 769q^-16 - 310q^-15 - 908q^-14 + 231q^-13 + 966q^-12 - 100q^-11 - 1001q^-10 - 7q^-9 + 956q^-8 + 148q^-7 - 906q^-6 - 246q^-5 + 785q^-4 + 368q^-3 - 666q^-2 - 434q^-1 + 494 + 487q - 326q² - 476q³ + 158q⁴ + 416q⁵ - 20q⁶ - 323q⁷ - 63q⁸ + 210q⁹ + 95q¹⁰ - 113q¹¹ - 83q¹² + 43q¹³ + 58q¹⁴ - 13q¹⁵ - 26q¹⁶ + 9q¹⁸ + 2q¹⁹ - 4q²⁰ + q²¹

4 q^-64 - 4q^-63 + 3q^-62 + 5q^-61 - 6q^-60 + 8q^-59 - 23q^-58 + 9q^-57 + 17q^-56 - 19q^-55 + 59q^-54 - 62q^-53 - 17q^-51 - 95q^-50 + 230q^-49 + 22q^-48 + 51q^-47 - 252q^-46 - 530q^-45 + 437q^-44 + 533q^-43 + 631q^-42 - 576q^-41 - 1820q^-40 - 2q^-39 + 1347q^-38 + 2403q^-37 - 122q^-36 - 3733q^-35 - 1845q^-34 + 1370q^-33 + 4953q^-32 + 1881q^-31 - 4890q^-30 - 4476q^-29 - 196q^-28 + 6680q^-27 + 4635q^-26 - 4374q^-25 - 6251q^-24 - 2562q^-23 + 6689q^-22 + 6586q^-21 - 2802q^-20 - 6449q^-19 - 4440q^-18 + 5500q^-17 + 7191q^-16 - 1089q^-15 - 5562q^-14 - 5519q^-13 + 3792q^-12 + 6878q^-11 + 585q^-10 - 4077q^-9 - 6042q^-8 + 1676q^-7 + 5840q^-6 + 2227q^-5 - 1973q^-4 - 5829q^-3 - 662q^-2 + 3856q^-1 + 3222 + 491q - 4366q² - 2298q³ + 1209q⁴ + 2753q⁵ + 2231q⁶ - 1952q⁷ - 2284q⁸ - 819q⁹ + 1115q¹⁰ + 2270q¹¹ + 6q¹² - 1023q¹³ - 1220q¹⁴ - 235q¹⁵ + 1140q¹⁶ + 537q¹⁷ + 21q¹⁸ - 576q¹⁹ - 476q²⁰ + 230q²¹ + 235q²² + 210q²³ - 83q²⁴ - 192q²⁵ - 6q²⁶ + 16q²⁷ + 70q²⁸ + 11q²⁹ - 32q³⁰ - 3q³¹ - 5q³² + 9q³³ + 2q³⁴ - 4q³⁵ + q³⁶

5 q^-95 - 4q^-94 + 3q^-93 + 5q^-92 - 6q^-91 + 4q^-90 - 3q^-89 - 17q^-88 + 3q^-87 + 27q^-86 + 12q^-85 + 17q^-84 - 15q^-83 - 88q^-82 - 72q^-81 + 35q^-80 + 169q^-79 + 204q^-78 + 49q^-77 - 296q^-76 - 557q^-75 - 301q^-74 + 495q^-73 + 1132q^-72 + 904q^-71 - 429q^-70 - 2126q^-69 - 2334q^-68 + 46q^-67 + 3401q^-66 + 4668q^-65 + 1606q^-64 - 4597q^-63 - 8507q^-62 - 4934q^-61 + 4988q^-60 + 13263q^-59 + 10885q^-58 - 3367q^-57 - 18472q^-56 - 19347q^-55 - 1315q^-54 + 22500q^-53 + 29845q^-52 + 9608q^-51 - 23983q^-50 - 40637q^-49 - 21187q^-48 + 21632q^-47 + 50059q^-46 + 34582q^-45 - 15438q^-44 - 56204q^-43 - 47851q^-42 + 6144q^-41 + 58319q^-40 + 59125q^-39 + 4579q^-38 - 56686q^-37 - 66993q^-36 - 14921q^-35 + 52009q^-34 + 71274q^-33 + 23873q^-32 - 46097q^-31 - 72336q^-30 - 30398q^-29 + 39436q^-28 + 71176q^-27 + 35262q^-26 - 33287q^-25 - 68688q^-24 - 38415q^-23 + 27032q^-22 + 65425q^-21 + 41233q^-20 - 20995q^-19 - 61496q^-18 - 43492q^-17 + 13975q^-16 + 56709q^-15 + 45857q^-14 - 6411q^-13 - 50446q^-12 - 47264q^-11 - 2411q^-10 + 42408q^-9 + 47609q^-8 + 11130q^-7 - 32362q^-6 - 45362q^-5 - 19455q^-4 + 20767q^-3 + 40565q^-2 + 25475q^-1 - 8697 - 32575q - 28442q² - 2495q³ + 22575q⁴ + 27476q⁵ + 11014q⁶ - 11687q⁷ - 22871q⁸ - 15820q⁹ + 1851q¹⁰ + 15821q¹¹ + 16469q¹² + 5306q¹³ - 8006q¹⁴ - 13720q¹⁵ - 8945q¹⁶ + 1320q¹⁷ + 9027q¹⁸ + 9219q¹⁹ + 3047q²⁰ - 4190q²¹ - 7184q²² - 4706q²³ + 445q²⁴ + 4276q²⁵ + 4402q²⁶ + 1492q²⁷ - 1751q²⁸ - 2976q²⁹ - 1937q³⁰ + 104q³¹ + 1564q³² + 1534q³³ + 472q³⁴ - 545q³⁵ - 848q³⁶ - 524q³⁷ + 44q³⁸ + 394q³⁹ + 317q⁴⁰ + 65q⁴¹ - 115q⁴² - 133q⁴³ - 74q⁴⁴ + 22q⁴⁵ + 63q⁴⁶ + 22q⁴⁷ - 8q⁴⁸ - 9q⁴⁹ - 8q⁵⁰ - 5q⁵¹ + 9q⁵² + 2q⁵³ - 4q⁵⁴ + q⁵⁵

6 q^-132 - 4q^-131 + 3q^-130 + 5q^-129 - 6q^-128 + 4q^-127 - 7q^-126 + 3q^-125 - 23q^-124 + 13q^-123 + 58q^-122 - 20q^-121 + 12q^-120 - 46q^-119 - 49q^-118 - 125q^-117 + 43q^-116 + 292q^-115 + 98q^-114 + 113q^-113 - 182q^-112 - 399q^-111 - 699q^-110 - 44q^-109 + 1020q^-108 + 949q^-107 + 938q^-106 - 316q^-105 - 1815q^-104 - 3173q^-103 - 1348q^-102 + 2604q^-101 + 4536q^-100 + 5297q^-99 + 1258q^-98 - 5644q^-97 - 12103q^-96 - 9363q^-95 + 2588q^-94 + 14378q^-93 + 22294q^-92 + 14459q^-91 - 8179q^-90 - 34699q^-89 - 40520q^-88 - 15829q^-87 + 24702q^-86 + 64252q^-85 + 65883q^-84 + 17368q^-83 - 61507q^-82 - 112645q^-81 - 91715q^-80 - 4372q^-79 + 115245q^-78 + 177133q^-77 + 122951q^-76 - 36886q^-75 - 198805q^-74 - 246291q^-73 - 135601q^-72 + 102154q^-71 + 304969q^-70 + 322841q^-69 + 108338q^-68 - 208035q^-67 - 415401q^-66 - 369190q^-65 - 44939q^-64 + 342567q^-63 + 529521q^-62 + 355455q^-61 - 75312q^-60 - 481338q^-59 - 596395q^-58 - 287062q^-57 + 235112q^-56 + 621571q^-55 + 581643q^-54 + 142621q^-53 - 403747q^-52 - 701406q^-51 - 497136q^-50 + 52303q^-49 + 575432q^-48 + 686103q^-47 + 324454q^-46 - 259600q^-45 - 678658q^-44 - 595466q^-43 - 97920q^-42 + 469528q^-41 + 680395q^-40 + 414046q^-39 - 139053q^-38 - 604029q^-37 - 607338q^-36 - 181645q^-35 + 374947q^-34 + 634805q^-33 + 445995q^-32 - 57346q^-31 - 531071q^-30 - 595039q^-29 - 238220q^-28 + 292811q^-27 + 587365q^-26 + 473028q^-25 + 25931q^-24 - 450631q^-23 - 582336q^-22 - 310007q^-21 + 183718q^-20 + 519968q^-19 + 505015q^-18 + 142871q^-17 - 323845q^-16 - 541057q^-15 - 392844q^-14 + 24539q^-13 + 391334q^-12 + 501456q^-11 + 275403q^-10 - 133940q^-9 - 423976q^-8 - 431978q^-7 - 153273q^-6 + 188234q^-5 + 406834q^-4 + 353982q^-3 + 75072q^-2 - 220510q^-1 - 363557 - 265788q - 33219q² + 214477q³ + 308987q⁴ + 206600q⁵ + 217q⁶ - 188271q⁷ - 241469q⁸ - 166458q⁹ + 7525q¹⁰ + 152823q¹¹ + 192532q¹² + 124933q¹³ - 3599q¹⁴ - 107807q¹⁵ - 152582q¹⁶ - 98846q¹⁷ - 4510q¹⁸ + 77740q¹⁹ + 108954q²⁰ + 79351q²¹ + 17336q²² - 53589q²³ - 78088q²⁴ - 63016q²⁵ - 18441q²⁶ + 28028q²⁷ + 54149q²⁸ + 51592q²⁹ + 17450q³⁰ - 13872q³¹ - 35784q³² - 35541q³³ - 18963q³⁴ + 5342q³⁵ + 23945q³⁶ + 23370q³⁷ + 14894q³⁸ - 932q³⁹ - 12353q⁴⁰ - 16538q⁴¹ - 10547q⁴² - 35q⁴³ + 6049q⁴⁴ + 9489q⁴⁵ + 6581q⁴⁶ + 1697q⁴⁷ - 3686q⁴⁸ - 5067q⁴⁹ - 3293q⁵⁰ - 1324q⁵¹ + 1470q⁵² + 2348q⁵³ + 2093q⁵⁴ + 367q⁵⁵ - 620q⁵⁶ - 836q⁵⁷ - 903q⁵⁸ - 250q⁵⁹ + 190q⁶⁰ + 491q⁶¹ + 210q⁶² + 56q⁶³ - 21q⁶⁴ - 153q⁶⁵ - 86q⁶⁶ - 26q⁶⁷ + 68q⁶⁸ + 15q⁶⁹ + 3q⁷⁰ + 15q⁷¹ - 14q⁷² - 8q⁷³ - 5q⁷⁴ + 9q⁷⁵ + 2q⁷⁶ - 4q⁷⁷ + q⁷⁸

7 q^-175 - 4q^-174 + 3q^-173 + 5q^-172 - 6q^-171 + 4q^-170 - 7q^-169 - q^-168 - 3q^-167 - 13q^-166 + 44q^-165 + 26q^-164 - 25q^-163 - 9q^-162 - 59q^-161 - 45q^-160 - 23q^-159 - 17q^-158 + 237q^-157 + 207q^-156 + q^-155 - 94q^-154 - 390q^-153 - 358q^-152 - 214q^-151 + 11q^-150 + 895q^-149 + 1047q^-148 + 445q^-147 - 298q^-146 - 1680q^-145 - 1866q^-144 - 1068q^-143 + 370q^-142 + 3300q^-141 + 4247q^-140 + 2497q^-139 - 1259q^-138 - 6990q^-137 - 8897q^-136 - 5527q^-135 + 2242q^-134 + 13821q^-133 + 19478q^-132 + 14365q^-131 - 2055q^-130 - 26819q^-129 - 41684q^-128 - 35039q^-127 - 4126q^-126 + 44758q^-125 + 81789q^-124 + 81618q^-123 + 30928q^-122 - 62536q^-121 - 147523q^-120 - 171068q^-119 - 99750q^-118 + 60507q^-117 + 232766q^-116 + 321322q^-115 + 247773q^-114 - 1268q^-113 - 318086q^-112 - 538782q^-111 - 508679q^-110 - 169154q^-109 + 350733q^-108 + 801291q^-107 + 906860q^-106 + 512977q^-105 - 255713q^-104 - 1050542q^-103 - 1427253q^-102 - 1072392q^-101 - 55431q^-100 + 1188097q^-99 + 2004985q^-98 + 1846579q^-97 + 650464q^-96 - 1100025q^-95 - 2524928q^-94 - 2768946q^-93 - 1543588q^-92 + 692290q^-91 + 2847577q^-90 + 3709833q^-89 + 2672182q^-88 + 70836q^-87 - 2849932q^-86 - 4509092q^-85 - 3903117q^-84 - 1138645q^-83 + 2473724q^-82 + 5018303q^-81 + 5059882q^-80 + 2384254q^-79 - 1743975q^-78 - 5151849q^-77 - 5980942q^-76 - 3634733q^-75 + 771904q^-74 + 4910362q^-73 + 6558152q^-72 + 4722091q^-71 + 287316q^-70 - 4374923q^-69 - 6768298q^-68 - 5532402q^-67 - 1277096q^-66 + 3679256q^-65 + 6666957q^-64 + 6023339q^-63 + 2081549q^-62 - 2958913q^-61 - 6356639q^-60 - 6228347q^-59 - 2654449q^-58 + 2321108q^-57 + 5956977q^-56 + 6225844q^-55 + 3006066q^-54 - 1818171q^-53 - 5558109q^-52 - 6111765q^-51 - 3198548q^-50 + 1449833q^-49 + 5218300q^-48 + 5969456q^-47 + 3304370q^-46 - 1177198q^-45 - 4945050q^-44 - 5851981q^-43 - 3399370q^-42 + 940126q^-41 + 4718685q^-40 + 5780653q^-39 + 3532924q^-38 - 677796q^-37 - 4491455q^-36 - 5745173q^-35 - 3735338q^-34 + 338862q^-33 + 4213556q^-32 + 5712096q^-31 + 4003226q^-30 + 110075q^-29 - 3829595q^-28 - 5632970q^-27 - 4314371q^-26 - 677090q^-25 + 3301323q^-24 + 5448137q^-23 + 4615074q^-22 + 1345575q^-21 - 2600896q^-20 - 5101061q^-19 - 4840920q^-18 - 2065024q^-17 + 1737138q^-16 + 4544110q^-15 + 4907479q^-14 + 2758223q^-13 - 745625q^-12 - 3758569q^-11 - 4744622q^-10 - 3324924q^-9 - 288310q^-8 + 2766712q^-7 + 4297269q^-6 + 3659966q^-5 + 1255163q^-4 - 1639523q^-3 - 3565159q^-2 - 3681823q^-1 - 2022845 + 498263q + 2601296q² + 3353864q³ + 2479981q⁴ + 513794q⁵ - 1522784q⁶ - 2712791q⁷ - 2561391q⁸ - 1256298q⁹ + 484452q¹⁰ + 1862041q¹¹ + 2276578q¹² + 1638995q¹³ + 358674q¹⁴ - 958013q¹⁵ - 1716158q¹⁶ - 1646414q¹⁷ - 892107q¹⁸ + 165462q¹⁹ + 1026443q²⁰ + 1348069q²¹ + 1079864q²² + 388842q²³ - 371829q²⁴ - 876873q²⁵ - 969702q²⁶ - 651507q²⁷ - 116839q²⁸ + 385742q²⁹ + 674467q³⁰ + 650984q³¹ + 377537q³² + 1427q³³ - 329249q³⁴ - 479667q³⁵ - 424559q³⁶ - 219684q³⁷ + 44859q³⁸ + 248280q³⁹ + 328024q⁴⁰ + 276101q⁴¹ + 122230q⁴² - 51085q⁴³ - 178291q⁴⁴ - 221936q⁴⁵ - 170667q⁴⁶ - 66572q⁴⁷ + 45576q⁴⁸ + 125723q⁴⁹ + 141541q⁵⁰ + 103269q⁵¹ + 32573q⁵² - 40221q⁵³ - 81674q⁵⁴ - 86351q⁵⁵ - 58147q⁵⁶ - 11109q⁵⁷ + 29082q⁵⁸ + 51247q⁵⁹ + 49490q⁶⁰ + 27496q⁶¹ + 1745q⁶² - 19717q⁶³ - 29328q⁶⁴ - 24453q⁶⁵ - 12339q⁶⁶ + 2272q⁶⁷ + 12544q⁶⁸ + 14273q⁶⁹ + 10866q⁷⁰ + 4089q⁷¹ - 2726q⁷² - 6050q⁷³ - 6650q⁷⁴ - 4106q⁷⁵ - 476q⁷⁶ + 1680q⁷⁷ + 2755q⁷⁸ + 2323q⁷⁹ + 1068q⁸⁰ + 19q⁸¹ - 937q⁸² - 1114q⁸³ - 575q⁸⁴ - 170q⁸⁵ + 213q⁸⁶ + 307q⁸⁷ + 218q⁸⁸ + 189q⁸⁹ - 18q⁹⁰ - 130q⁹¹ - 86q⁹² - 39q⁹³ + 20q⁹⁴ + 20q⁹⁵ - 4q⁹⁶ + 26q⁹⁷ + 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, 112]]

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

In[3]:=
GaussCode[Knot[10, 112]]

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

In[4]:=
DTCode[Knot[10, 112]]

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

In[5]:=
br = BR[Knot[10, 112]]

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

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

Out[6]=
{3, 10}

In[7]:=
BraidIndex[Knot[10, 112]]

Out[7]=
3

In[8]:=
Show[DrawMorseLink[Knot[10, 112]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[10, 112]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

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

In[10]:=
alex = Alexander[Knot[10, 112]][t]

Out[10]=
-4 5 11 17 2 3 4 -19 - t + -- - -- + -- + 17 t - 11 t + 5 t - t 3 2 t t t

In[11]:=
Conway[Knot[10, 112]][z]

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

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

Out[12]=
{Knot[10, 112], Knot[11, Alternating, 184]}

In[13]:=
{KnotDet[Knot[10, 112]], KnotSignature[Knot[10, 112]]}

Out[13]=
{87, -2}

In[14]:=
Jones[Knot[10, 112]][q]

Out[14]=
-7 4 7 11 14 14 14 2 3 -10 + q - -- + -- - -- + -- - -- + -- + 7 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, 112]}

In[16]:=
A2Invariant[Knot[10, 112]][q]

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

In[17]:=
HOMFLYPT[Knot[10, 112]][a, z]

Out[17]=
2 4 2 4 2 4 2 4 4 4 6 2 6 -1 + 4 a - 2 a + z + a z + 3 z - 7 a z + 3 a z + z - 5 a z + 4 6 2 8 > a z - a z

In[18]:=
Kauffman[Knot[10, 112]][a, z]

Out[18]=
3 2 4 3 5 2 2 2 4 2 6 2 6 z -1 - 4 a - 2 a + 2 a z + 2 a z - 3 z - 3 a z + a z + a z + ---- + a 4 3 3 3 5 3 7 3 4 2 z 2 4 4 4 > 13 a z + 9 a z - a z - 3 a z + 15 z - ---- + 28 a z + 3 a z - 2 a 5 6 4 8 4 11 z 5 3 5 5 5 7 5 6 > 7 a z + a z - ----- - 16 a z - 17 a z - 8 a z + 4 a z - 18 z + a 6 7 z 2 6 4 6 6 6 4 z 3 7 5 7 8 > -- - 35 a z - 9 a z + 7 a z + ---- + 4 a z + 8 a z + 6 z + 2 a a 2 8 4 8 9 3 9 > 13 a z + 7 a z + 3 a z + 3 a z

In[19]:=
{Vassiliev[2][Knot[10, 112]], Vassiliev[3][Knot[10, 112]]}

Out[19]=
{2, -2}

In[20]:=
Kh[Knot[10, 112]][q, t]

Out[20]=
7 8 1 3 1 4 3 7 4 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 7 7 7 4 t 2 3 2 3 3 5 3 > ----- + ---- + ---- + --- + 6 q t + 3 q t + 4 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, 112], 2][q]

Out[21]=
-20 4 3 9 22 14 27 65 33 66 123 38 -36 + q - --- + --- + --- - --- + --- + --- - --- + --- + --- - --- + -- + 19 18 17 16 15 14 13 12 11 10 9 q q q q q q q q q q q 117 158 19 148 148 12 143 103 2 3 > --- - --- + -- + --- - --- - -- + --- - --- + 106 q - 47 q - 41 q + 8 7 6 5 4 3 2 q q q q q q q q 4 5 6 7 8 9 10 > 54 q - 8 q - 23 q + 14 q + 2 q - 4 q + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 10₁₁₂

10₁₁₁
10₁₁₃

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-20 - 4q^-19 + 3q^-18 + 9q^-17 - 22q^-16 + 14q^-15 + 27q^-14 - 65q^-13 + 33q^-12 + 66q^-11 - 123q^-10 + 38q^-9 + 117q^-8 - 158q^-7 + 19q^-6 + 148q^-5 - 148q^-4 - 12q^-3 + 143q^-2 - 103q^-1 - 36 + 106q - 47q² - 41q³ + 54q⁴ - 8q⁵ - 23q⁶ + 14q⁷ + 2q⁸ - 4q⁹ + q¹⁰
3	q^-39 - 4q^-38 + 3q^-37 + 5q^-36 - 2q^-35 - 12q^-34 + 3q^-33 + 23q^-32 - 19q^-31 - 24q^-30 + 37q^-29 + 47q^-28 - 94q^-27 - 75q^-26 + 163q^-25 + 155q^-24 - 259q^-23 - 268q^-22 + 332q^-21 + 433q^-20 - 380q^-19 - 610q^-18 + 375q^-17 + 769q^-16 - 310q^-15 - 908q^-14 + 231q^-13 + 966q^-12 - 100q^-11 - 1001q^-10 - 7q^-9 + 956q^-8 + 148q^-7 - 906q^-6 - 246q^-5 + 785q^-4 + 368q^-3 - 666q^-2 - 434q^-1 + 494 + 487q - 326q² - 476q³ + 158q⁴ + 416q⁵ - 20q⁶ - 323q⁷ - 63q⁸ + 210q⁹ + 95q¹⁰ - 113q¹¹ - 83q¹² + 43q¹³ + 58q¹⁴ - 13q¹⁵ - 26q¹⁶ + 9q¹⁸ + 2q¹⁹ - 4q²⁰ + q²¹
4	q^-64 - 4q^-63 + 3q^-62 + 5q^-61 - 6q^-60 + 8q^-59 - 23q^-58 + 9q^-57 + 17q^-56 - 19q^-55 + 59q^-54 - 62q^-53 - 17q^-51 - 95q^-50 + 230q^-49 + 22q^-48 + 51q^-47 - 252q^-46 - 530q^-45 + 437q^-44 + 533q^-43 + 631q^-42 - 576q^-41 - 1820q^-40 - 2q^-39 + 1347q^-38 + 2403q^-37 - 122q^-36 - 3733q^-35 - 1845q^-34 + 1370q^-33 + 4953q^-32 + 1881q^-31 - 4890q^-30 - 4476q^-29 - 196q^-28 + 6680q^-27 + 4635q^-26 - 4374q^-25 - 6251q^-24 - 2562q^-23 + 6689q^-22 + 6586q^-21 - 2802q^-20 - 6449q^-19 - 4440q^-18 + 5500q^-17 + 7191q^-16 - 1089q^-15 - 5562q^-14 - 5519q^-13 + 3792q^-12 + 6878q^-11 + 585q^-10 - 4077q^-9 - 6042q^-8 + 1676q^-7 + 5840q^-6 + 2227q^-5 - 1973q^-4 - 5829q^-3 - 662q^-2 + 3856q^-1 + 3222 + 491q - 4366q² - 2298q³ + 1209q⁴ + 2753q⁵ + 2231q⁶ - 1952q⁷ - 2284q⁸ - 819q⁹ + 1115q¹⁰ + 2270q¹¹ + 6q¹² - 1023q¹³ - 1220q¹⁴ - 235q¹⁵ + 1140q¹⁶ + 537q¹⁷ + 21q¹⁸ - 576q¹⁹ - 476q²⁰ + 230q²¹ + 235q²² + 210q²³ - 83q²⁴ - 192q²⁵ - 6q²⁶ + 16q²⁷ + 70q²⁸ + 11q²⁹ - 32q³⁰ - 3q³¹ - 5q³² + 9q³³ + 2q³⁴ - 4q³⁵ + q³⁶
5	q^-95 - 4q^-94 + 3q^-93 + 5q^-92 - 6q^-91 + 4q^-90 - 3q^-89 - 17q^-88 + 3q^-87 + 27q^-86 + 12q^-85 + 17q^-84 - 15q^-83 - 88q^-82 - 72q^-81 + 35q^-80 + 169q^-79 + 204q^-78 + 49q^-77 - 296q^-76 - 557q^-75 - 301q^-74 + 495q^-73 + 1132q^-72 + 904q^-71 - 429q^-70 - 2126q^-69 - 2334q^-68 + 46q^-67 + 3401q^-66 + 4668q^-65 + 1606q^-64 - 4597q^-63 - 8507q^-62 - 4934q^-61 + 4988q^-60 + 13263q^-59 + 10885q^-58 - 3367q^-57 - 18472q^-56 - 19347q^-55 - 1315q^-54 + 22500q^-53 + 29845q^-52 + 9608q^-51 - 23983q^-50 - 40637q^-49 - 21187q^-48 + 21632q^-47 + 50059q^-46 + 34582q^-45 - 15438q^-44 - 56204q^-43 - 47851q^-42 + 6144q^-41 + 58319q^-40 + 59125q^-39 + 4579q^-38 - 56686q^-37 - 66993q^-36 - 14921q^-35 + 52009q^-34 + 71274q^-33 + 23873q^-32 - 46097q^-31 - 72336q^-30 - 30398q^-29 + 39436q^-28 + 71176q^-27 + 35262q^-26 - 33287q^-25 - 68688q^-24 - 38415q^-23 + 27032q^-22 + 65425q^-21 + 41233q^-20 - 20995q^-19 - 61496q^-18 - 43492q^-17 + 13975q^-16 + 56709q^-15 + 45857q^-14 - 6411q^-13 - 50446q^-12 - 47264q^-11 - 2411q^-10 + 42408q^-9 + 47609q^-8 + 11130q^-7 - 32362q^-6 - 45362q^-5 - 19455q^-4 + 20767q^-3 + 40565q^-2 + 25475q^-1 - 8697 - 32575q - 28442q² - 2495q³ + 22575q⁴ + 27476q⁵ + 11014q⁶ - 11687q⁷ - 22871q⁸ - 15820q⁹ + 1851q¹⁰ + 15821q¹¹ + 16469q¹² + 5306q¹³ - 8006q¹⁴ - 13720q¹⁵ - 8945q¹⁶ + 1320q¹⁷ + 9027q¹⁸ + 9219q¹⁹ + 3047q²⁰ - 4190q²¹ - 7184q²² - 4706q²³ + 445q²⁴ + 4276q²⁵ + 4402q²⁶ + 1492q²⁷ - 1751q²⁸ - 2976q²⁹ - 1937q³⁰ + 104q³¹ + 1564q³² + 1534q³³ + 472q³⁴ - 545q³⁵ - 848q³⁶ - 524q³⁷ + 44q³⁸ + 394q³⁹ + 317q⁴⁰ + 65q⁴¹ - 115q⁴² - 133q⁴³ - 74q⁴⁴ + 22q⁴⁵ + 63q⁴⁶ + 22q⁴⁷ - 8q⁴⁸ - 9q⁴⁹ - 8q⁵⁰ - 5q⁵¹ + 9q⁵² + 2q⁵³ - 4q⁵⁴ + q⁵⁵
6	q^-132 - 4q^-131 + 3q^-130 + 5q^-129 - 6q^-128 + 4q^-127 - 7q^-126 + 3q^-125 - 23q^-124 + 13q^-123 + 58q^-122 - 20q^-121 + 12q^-120 - 46q^-119 - 49q^-118 - 125q^-117 + 43q^-116 + 292q^-115 + 98q^-114 + 113q^-113 - 182q^-112 - 399q^-111 - 699q^-110 - 44q^-109 + 1020q^-108 + 949q^-107 + 938q^-106 - 316q^-105 - 1815q^-104 - 3173q^-103 - 1348q^-102 + 2604q^-101 + 4536q^-100 + 5297q^-99 + 1258q^-98 - 5644q^-97 - 12103q^-96 - 9363q^-95 + 2588q^-94 + 14378q^-93 + 22294q^-92 + 14459q^-91 - 8179q^-90 - 34699q^-89 - 40520q^-88 - 15829q^-87 + 24702q^-86 + 64252q^-85 + 65883q^-84 + 17368q^-83 - 61507q^-82 - 112645q^-81 - 91715q^-80 - 4372q^-79 + 115245q^-78 + 177133q^-77 + 122951q^-76 - 36886q^-75 - 198805q^-74 - 246291q^-73 - 135601q^-72 + 102154q^-71 + 304969q^-70 + 322841q^-69 + 108338q^-68 - 208035q^-67 - 415401q^-66 - 369190q^-65 - 44939q^-64 + 342567q^-63 + 529521q^-62 + 355455q^-61 - 75312q^-60 - 481338q^-59 - 596395q^-58 - 287062q^-57 + 235112q^-56 + 621571q^-55 + 581643q^-54 + 142621q^-53 - 403747q^-52 - 701406q^-51 - 497136q^-50 + 52303q^-49 + 575432q^-48 + 686103q^-47 + 324454q^-46 - 259600q^-45 - 678658q^-44 - 595466q^-43 - 97920q^-42 + 469528q^-41 + 680395q^-40 + 414046q^-39 - 139053q^-38 - 604029q^-37 - 607338q^-36 - 181645q^-35 + 374947q^-34 + 634805q^-33 + 445995q^-32 - 57346q^-31 - 531071q^-30 - 595039q^-29 - 238220q^-28 + 292811q^-27 + 587365q^-26 + 473028q^-25 + 25931q^-24 - 450631q^-23 - 582336q^-22 - 310007q^-21 + 183718q^-20 + 519968q^-19 + 505015q^-18 + 142871q^-17 - 323845q^-16 - 541057q^-15 - 392844q^-14 + 24539q^-13 + 391334q^-12 + 501456q^-11 + 275403q^-10 - 133940q^-9 - 423976q^-8 - 431978q^-7 - 153273q^-6 + 188234q^-5 + 406834q^-4 + 353982q^-3 + 75072q^-2 - 220510q^-1 - 363557 - 265788q - 33219q² + 214477q³ + 308987q⁴ + 206600q⁵ + 217q⁶ - 188271q⁷ - 241469q⁸ - 166458q⁹ + 7525q¹⁰ + 152823q¹¹ + 192532q¹² + 124933q¹³ - 3599q¹⁴ - 107807q¹⁵ - 152582q¹⁶ - 98846q¹⁷ - 4510q¹⁸ + 77740q¹⁹ + 108954q²⁰ + 79351q²¹ + 17336q²² - 53589q²³ - 78088q²⁴ - 63016q²⁵ - 18441q²⁶ + 28028q²⁷ + 54149q²⁸ + 51592q²⁹ + 17450q³⁰ - 13872q³¹ - 35784q³² - 35541q³³ - 18963q³⁴ + 5342q³⁵ + 23945q³⁶ + 23370q³⁷ + 14894q³⁸ - 932q³⁹ - 12353q⁴⁰ - 16538q⁴¹ - 10547q⁴² - 35q⁴³ + 6049q⁴⁴ + 9489q⁴⁵ + 6581q⁴⁶ + 1697q⁴⁷ - 3686q⁴⁸ - 5067q⁴⁹ - 3293q⁵⁰ - 1324q⁵¹ + 1470q⁵² + 2348q⁵³ + 2093q⁵⁴ + 367q⁵⁵ - 620q⁵⁶ - 836q⁵⁷ - 903q⁵⁸ - 250q⁵⁹ + 190q⁶⁰ + 491q⁶¹ + 210q⁶² + 56q⁶³ - 21q⁶⁴ - 153q⁶⁵ - 86q⁶⁶ - 26q⁶⁷ + 68q⁶⁸ + 15q⁶⁹ + 3q⁷⁰ + 15q⁷¹ - 14q⁷² - 8q⁷³ - 5q⁷⁴ + 9q⁷⁵ + 2q⁷⁶ - 4q⁷⁷ + q⁷⁸
7	q^-175 - 4q^-174 + 3q^-173 + 5q^-172 - 6q^-171 + 4q^-170 - 7q^-169 - q^-168 - 3q^-167 - 13q^-166 + 44q^-165 + 26q^-164 - 25q^-163 - 9q^-162 - 59q^-161 - 45q^-160 - 23q^-159 - 17q^-158 + 237q^-157 + 207q^-156 + q^-155 - 94q^-154 - 390q^-153 - 358q^-152 - 214q^-151 + 11q^-150 + 895q^-149 + 1047q^-148 + 445q^-147 - 298q^-146 - 1680q^-145 - 1866q^-144 - 1068q^-143 + 370q^-142 + 3300q^-141 + 4247q^-140 + 2497q^-139 - 1259q^-138 - 6990q^-137 - 8897q^-136 - 5527q^-135 + 2242q^-134 + 13821q^-133 + 19478q^-132 + 14365q^-131 - 2055q^-130 - 26819q^-129 - 41684q^-128 - 35039q^-127 - 4126q^-126 + 44758q^-125 + 81789q^-124 + 81618q^-123 + 30928q^-122 - 62536q^-121 - 147523q^-120 - 171068q^-119 - 99750q^-118 + 60507q^-117 + 232766q^-116 + 321322q^-115 + 247773q^-114 - 1268q^-113 - 318086q^-112 - 538782q^-111 - 508679q^-110 - 169154q^-109 + 350733q^-108 + 801291q^-107 + 906860q^-106 + 512977q^-105 - 255713q^-104 - 1050542q^-103 - 1427253q^-102 - 1072392q^-101 - 55431q^-100 + 1188097q^-99 + 2004985q^-98 + 1846579q^-97 + 650464q^-96 - 1100025q^-95 - 2524928q^-94 - 2768946q^-93 - 1543588q^-92 + 692290q^-91 + 2847577q^-90 + 3709833q^-89 + 2672182q^-88 + 70836q^-87 - 2849932q^-86 - 4509092q^-85 - 3903117q^-84 - 1138645q^-83 + 2473724q^-82 + 5018303q^-81 + 5059882q^-80 + 2384254q^-79 - 1743975q^-78 - 5151849q^-77 - 5980942q^-76 - 3634733q^-75 + 771904q^-74 + 4910362q^-73 + 6558152q^-72 + 4722091q^-71 + 287316q^-70 - 4374923q^-69 - 6768298q^-68 - 5532402q^-67 - 1277096q^-66 + 3679256q^-65 + 6666957q^-64 + 6023339q^-63 + 2081549q^-62 - 2958913q^-61 - 6356639q^-60 - 6228347q^-59 - 2654449q^-58 + 2321108q^-57 + 5956977q^-56 + 6225844q^-55 + 3006066q^-54 - 1818171q^-53 - 5558109q^-52 - 6111765q^-51 - 3198548q^-50 + 1449833q^-49 + 5218300q^-48 + 5969456q^-47 + 3304370q^-46 - 1177198q^-45 - 4945050q^-44 - 5851981q^-43 - 3399370q^-42 + 940126q^-41 + 4718685q^-40 + 5780653q^-39 + 3532924q^-38 - 677796q^-37 - 4491455q^-36 - 5745173q^-35 - 3735338q^-34 + 338862q^-33 + 4213556q^-32 + 5712096q^-31 + 4003226q^-30 + 110075q^-29 - 3829595q^-28 - 5632970q^-27 - 4314371q^-26 - 677090q^-25 + 3301323q^-24 + 5448137q^-23 + 4615074q^-22 + 1345575q^-21 - 2600896q^-20 - 5101061q^-19 - 4840920q^-18 - 2065024q^-17 + 1737138q^-16 + 4544110q^-15 + 4907479q^-14 + 2758223q^-13 - 745625q^-12 - 3758569q^-11 - 4744622q^-10 - 3324924q^-9 - 288310q^-8 + 2766712q^-7 + 4297269q^-6 + 3659966q^-5 + 1255163q^-4 - 1639523q^-3 - 3565159q^-2 - 3681823q^-1 - 2022845 + 498263q + 2601296q² + 3353864q³ + 2479981q⁴ + 513794q⁵ - 1522784q⁶ - 2712791q⁷ - 2561391q⁸ - 1256298q⁹ + 484452q¹⁰ + 1862041q¹¹ + 2276578q¹² + 1638995q¹³ + 358674q¹⁴ - 958013q¹⁵ - 1716158q¹⁶ - 1646414q¹⁷ - 892107q¹⁸ + 165462q¹⁹ + 1026443q²⁰ + 1348069q²¹ + 1079864q²² + 388842q²³ - 371829q²⁴ - 876873q²⁵ - 969702q²⁶ - 651507q²⁷ - 116839q²⁸ + 385742q²⁹ + 674467q³⁰ + 650984q³¹ + 377537q³² + 1427q³³ - 329249q³⁴ - 479667q³⁵ - 424559q³⁶ - 219684q³⁷ + 44859q³⁸ + 248280q³⁹ + 328024q⁴⁰ + 276101q⁴¹ + 122230q⁴² - 51085q⁴³ - 178291q⁴⁴ - 221936q⁴⁵ - 170667q⁴⁶ - 66572q⁴⁷ + 45576q⁴⁸ + 125723q⁴⁹ + 141541q⁵⁰ + 103269q⁵¹ + 32573q⁵² - 40221q⁵³ - 81674q⁵⁴ - 86351q⁵⁵ - 58147q⁵⁶ - 11109q⁵⁷ + 29082q⁵⁸ + 51247q⁵⁹ + 49490q⁶⁰ + 27496q⁶¹ + 1745q⁶² - 19717q⁶³ - 29328q⁶⁴ - 24453q⁶⁵ - 12339q⁶⁶ + 2272q⁶⁷ + 12544q⁶⁸ + 14273q⁶⁹ + 10866q⁷⁰ + 4089q⁷¹ - 2726q⁷² - 6050q⁷³ - 6650q⁷⁴ - 4106q⁷⁵ - 476q⁷⁶ + 1680q⁷⁷ + 2755q⁷⁸ + 2323q⁷⁹ + 1068q⁸⁰ + 19q⁸¹ - 937q⁸² - 1114q⁸³ - 575q⁸⁴ - 170q⁸⁵ + 213q⁸⁶ + 307q⁸⁷ + 218q⁸⁸ + 189q⁸⁹ - 18q⁹⁰ - 130q⁹¹ - 86q⁹² - 39q⁹³ + 20q⁹⁴ + 20q⁹⁵ - 4q⁹⁶ + 26q⁹⁷ + 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, 112]]
Out[2]=	PD[X[6, 2, 7, 1], X[8, 3, 9, 4], X[18, 11, 19, 12], X[20, 13, 1, 14], > X[2, 16, 3, 15], X[4, 17, 5, 18], X[12, 19, 13, 20], X[10, 6, 11, 5], > X[14, 7, 15, 8], X[16, 10, 17, 9]]
In[3]:=	GaussCode[Knot[10, 112]]
Out[3]=	GaussCode[1, -5, 2, -6, 8, -1, 9, -2, 10, -8, 3, -7, 4, -9, 5, -10, 6, -3, 7, > -4]
In[4]:=	DTCode[Knot[10, 112]]
Out[4]=	DTCode[6, 8, 10, 14, 16, 18, 20, 2, 4, 12]
In[5]:=	br = BR[Knot[10, 112]]
Out[5]=	BR[3, {-1, -1, -1, 2, -1, 2, -1, 2, -1, 2}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{3, 10}
In[7]:=	BraidIndex[Knot[10, 112]]
Out[7]=	3
In[8]:=	Show[DrawMorseLink[Knot[10, 112]]]

Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 112]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 4, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 112]][t]
Out[10]=	-4 5 11 17 2 3 4 -19 - t + -- - -- + -- + 17 t - 11 t + 5 t - t 3 2 t t t
In[11]:=	Conway[Knot[10, 112]][z]
Out[11]=	2 4 6 8 1 + 2 z - z - 3 z - z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 112], Knot[11, Alternating, 184]}
In[13]:=	{KnotDet[Knot[10, 112]], KnotSignature[Knot[10, 112]]}
Out[13]=	{87, -2}
In[14]:=	Jones[Knot[10, 112]][q]
Out[14]=	-7 4 7 11 14 14 14 2 3 -10 + q - -- + -- - -- + -- - -- + -- + 7 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, 112]}
In[16]:=	A2Invariant[Knot[10, 112]][q]
Out[16]=	-20 2 -16 3 -12 2 -8 6 -4 3 2 4 6 q - --- + q - --- - q + --- - q + -- - q + -- - 2 q + q - 2 q + 18 14 10 6 2 q q q q q 8 > q
In[17]:=	HOMFLYPT[Knot[10, 112]][a, z]
Out[17]=	2 4 2 4 2 4 2 4 4 4 6 2 6 -1 + 4 a - 2 a + z + a z + 3 z - 7 a z + 3 a z + z - 5 a z + 4 6 2 8 > a z - a z
In[18]:=	Kauffman[Knot[10, 112]][a, z]
Out[18]=	3 2 4 3 5 2 2 2 4 2 6 2 6 z -1 - 4 a - 2 a + 2 a z + 2 a z - 3 z - 3 a z + a z + a z + ---- + a 4 3 3 3 5 3 7 3 4 2 z 2 4 4 4 > 13 a z + 9 a z - a z - 3 a z + 15 z - ---- + 28 a z + 3 a z - 2 a 5 6 4 8 4 11 z 5 3 5 5 5 7 5 6 > 7 a z + a z - ----- - 16 a z - 17 a z - 8 a z + 4 a z - 18 z + a 6 7 z 2 6 4 6 6 6 4 z 3 7 5 7 8 > -- - 35 a z - 9 a z + 7 a z + ---- + 4 a z + 8 a z + 6 z + 2 a a 2 8 4 8 9 3 9 > 13 a z + 7 a z + 3 a z + 3 a z
In[19]:=	{Vassiliev[2][Knot[10, 112]], Vassiliev[3][Knot[10, 112]]}
Out[19]=	{2, -2}
In[20]:=	Kh[Knot[10, 112]][q, t]
Out[20]=	7 8 1 3 1 4 3 7 4 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 7 7 7 4 t 2 3 2 3 3 5 3 > ----- + ---- + ---- + --- + 6 q t + 3 q t + 4 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, 112], 2][q]
Out[21]=	-20 4 3 9 22 14 27 65 33 66 123 38 -36 + q - --- + --- + --- - --- + --- + --- - --- + --- + --- - --- + -- + 19 18 17 16 15 14 13 12 11 10 9 q q q q q q q q q q q 117 158 19 148 148 12 143 103 2 3 > --- - --- + -- + --- - --- - -- + --- - --- + 106 q - 47 q - 41 q + 8 7 6 5 4 3 2 q q q q q q q q 4 5 6 7 8 9 10 > 54 q - 8 q - 23 q + 14 q + 2 q - 4 q + q

The Alternating Knot 10112

The Alternating Knot 10₁₁₂