The Alternating Knot 10₁₀₆

Visit 10₁₀₆'s page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 10₁₀₆'s page at Knotilus!

Acknowledgement

KnotPlot

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

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

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

Chiral 2 4 3 / NotAvailable 1

Alexander Polynomial: - t^-4 + 4t^-3 - 9t^-2 + 15t^-1 - 17 + 15t - 9t² + 4t³ - t⁴

Conway Polynomial: 1 - z² - 5z⁴ - 4z⁶ - z⁸

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

Determinant and Signature: {75, 2}

Jones Polynomial: q^-3 - 3q^-2 + 6q^-1 - 9 + 12q - 12q² + 12q³ - 10q⁴ + 6q⁵ - 3q⁶ + q⁷

Other knots (up to mirrors) with the same Jones Polynomial: {10₅₉, ...}

A2 (sl(3)) Invariant: q^-8 - q^-6 + 2q^-4 - q^-2 + 2q² - 2q⁴ + 4q⁶ - 2q⁸ + q¹⁰ - q¹² - 2q¹⁴ + 2q¹⁶ - q¹⁸ + q²⁰

HOMFLY-PT Polynomial: a^-4 + 5a^-4z² + 4a^-4z⁴ + a^-4z⁶ - 2a^-2 - 11a^-2z² - 13a^-2z⁴ - 6a^-2z⁶ - a^-2z⁸ + 2 + 5z² + 4z⁴ + z⁶

Kauffman Polynomial: - a^-8z² + a^-8z⁴ + a^-7z - 3a^-7z³ + 3a^-7z⁵ + 2a^-6z² - 5a^-6z⁴ + 5a^-6z⁶ + a^-5z + 3a^-5z³ - 7a^-5z⁵ + 6a^-5z⁷ + a^-4 - 3a^-4z² + 4a^-4z⁴ - 6a^-4z⁶ + 5a^-4z⁸ - a^-3z + 8a^-3z³ - 13a^-3z⁵ + 4a^-3z⁷ + 2a^-3z⁹ + 2a^-2 - 13a^-2z² + 22a^-2z⁴ - 23a^-2z⁶ + 9a^-2z⁸ - 2a^-1z + 9a^-1z³ - 12a^-1z⁵ + a^-1z⁷ + 2a^-1z⁹ + 2 - 5z² + 9z⁴ - 11z⁶ + 4z⁸ - az + 7az³ - 9az⁵ + 3az⁷ + 2a²z² - 3a²z⁴ + a²z⁶

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 10₁₀₆. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

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

j = 15 1

j = 13 2

j = 11 4 1

j = 9 6 2

j = 7 6 4

j = 5 6 6

j = 3 6 6

j = 1 4 7

j = -1 2 5

j = -3 1 4

j = -5 2

j = -7 1

n Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q^-10 - 3q^-9 + q^-8 + 10q^-7 - 16q^-6 - 6q^-5 + 40q^-4 - 30q^-3 - 36q^-2 + 81q^-1 - 28 - 80q + 111q² - 9q³ - 115q⁴ + 115q⁵ + 15q⁶ - 124q⁷ + 92q⁸ + 29q⁹ - 97q¹⁰ + 53q¹¹ + 26q¹² - 50q¹³ + 20q¹⁴ + 11q¹⁵ - 16q¹⁶ + 6q¹⁷ + 2q¹⁸ - 3q¹⁹ + q²⁰

3 q^-21 - 3q^-20 + q^-19 + 5q^-18 + 2q^-17 - 16q^-16 - 8q^-15 + 32q^-14 + 29q^-13 - 50q^-12 - 70q^-11 + 54q^-10 + 137q^-9 - 38q^-8 - 208q^-7 - 21q^-6 + 279q^-5 + 110q^-4 - 323q^-3 - 225q^-2 + 336q^-1 + 343 - 307q - 466q² + 266q³ + 555q⁴ - 186q⁵ - 641q⁶ + 116q⁷ + 685q⁸ - 22q⁹ - 720q¹⁰ - 52q¹¹ + 702q¹² + 137q¹³ - 661q¹⁴ - 199q¹⁵ + 574q¹⁶ + 241q¹⁷ - 459q¹⁸ - 253q¹⁹ + 338q²⁰ + 223q²¹ - 212q²² - 182q²³ + 125q²⁴ + 121q²⁵ - 65q²⁶ - 68q²⁷ + 32q²⁸ + 33q²⁹ - 18q³⁰ - 14q³¹ + 12q³² + 5q³³ - 8q³⁴ + 2q³⁶ + 2q³⁷ - 3q³⁸ + q³⁹

4 q^-36 - 3q^-35 + q^-34 + 5q^-33 - 3q^-32 + 2q^-31 - 19q^-30 + 4q^-29 + 35q^-28 + 4q^-27 + 9q^-26 - 99q^-25 - 45q^-24 + 104q^-23 + 106q^-22 + 144q^-21 - 247q^-20 - 306q^-19 + q^-18 + 256q^-17 + 662q^-16 - 118q^-15 - 671q^-14 - 596q^-13 - 55q^-12 + 1358q^-11 + 661q^-10 - 465q^-9 - 1376q^-8 - 1226q^-7 + 1428q^-6 + 1676q^-5 + 719q^-4 - 1476q^-3 - 2753q^-2 + 475q^-1 + 2064 + 2356q - 586q² - 3785q³ - 991q⁴ + 1580q⁵ + 3674q⁶ + 790q⁷ - 4070q⁸ - 2326q⁹ + 658q¹⁰ + 4448q¹¹ + 2088q¹² - 3872q¹³ - 3315q¹⁴ - 323q¹⁵ + 4755q¹⁶ + 3153q¹⁷ - 3291q¹⁸ - 3934q¹⁹ - 1347q²⁰ + 4475q²¹ + 3918q²² - 2190q²³ - 3903q²⁴ - 2323q²⁵ + 3345q²⁶ + 4003q²⁷ - 714q²⁸ - 2922q²⁹ - 2740q³⁰ + 1639q³¹ + 3071q³² + 403q³³ - 1359q³⁴ - 2190q³⁵ + 272q³⁶ + 1586q³⁷ + 609q³⁸ - 181q³⁹ - 1125q⁴⁰ - 183q⁴¹ + 483q⁴² + 263q⁴³ + 180q⁴⁴ - 358q⁴⁵ - 103q⁴⁶ + 79q⁴⁷ + 11q⁴⁸ + 116q⁴⁹ - 80q⁵⁰ - 6q⁵¹ + 15q⁵² - 33q⁵³ + 36q⁵⁴ - 18q⁵⁵ + 7q⁵⁶ + 7q⁵⁷ - 14q⁵⁸ + 8q⁵⁹ - 4q⁶⁰ + 2q⁶¹ + 2q⁶² - 3q⁶³ + q⁶⁴

5 q^-55 - 3q^-54 + q^-53 + 5q^-52 - 3q^-51 - 3q^-50 - q^-49 - 7q^-48 + 6q^-47 + 30q^-46 + 8q^-45 - 30q^-44 - 44q^-43 - 51q^-42 + 13q^-41 + 129q^-40 + 161q^-39 + 23q^-38 - 190q^-37 - 349q^-36 - 258q^-35 + 173q^-34 + 637q^-33 + 695q^-32 + 120q^-31 - 806q^-30 - 1364q^-29 - 879q^-28 + 577q^-27 + 2022q^-26 + 2148q^-25 + 372q^-24 - 2223q^-23 - 3604q^-22 - 2263q^-21 + 1400q^-20 + 4783q^-19 + 4808q^-18 + 728q^-17 - 4791q^-16 - 7409q^-15 - 4234q^-14 + 3172q^-13 + 9184q^-12 + 8417q^-11 + 370q^-10 - 9270q^-9 - 12480q^-8 - 5437q^-7 + 7248q^-6 + 15406q^-5 + 11293q^-4 - 3184q^-3 - 16563q^-2 - 16914q^-1 - 2452 + 15680q + 21675q² + 8761q³ - 13129q⁴ - 24862q⁵ - 15056q⁶ + 9237q⁷ + 26781q⁸ + 20669q⁹ - 4965q¹⁰ - 27304q¹¹ - 25352q¹² + 508q¹³ + 27188q¹⁴ + 29154q¹⁵ + 3400q¹⁶ - 26480q¹⁷ - 32163q¹⁸ - 7101q¹⁹ + 25730q²⁰ + 34685q²¹ + 10333q²² - 24629q²³ - 36800q²⁴ - 13687q²⁵ + 23267q²⁶ + 38498q²⁷ + 17017q²⁸ - 20998q²⁹ - 39478q³⁰ - 20646q³¹ + 17757q³² + 39330q³³ + 24043q³⁴ - 13227q³⁵ - 37498q³⁶ - 26867q³⁷ + 7703q³⁸ + 33794q³⁹ + 28336q⁴⁰ - 1847q⁴¹ - 28195q⁴² - 27875q⁴³ - 3601q⁴⁴ + 21340q⁴⁵ + 25440q⁴⁶ + 7554q⁴⁷ - 14202q⁴⁸ - 21086q⁴⁹ - 9682q⁵⁰ + 7759q⁵¹ + 15893q⁵² + 9759q⁵³ - 2917q⁵⁴ - 10609q⁵⁵ - 8339q⁵⁶ - 120q⁵⁷ + 6197q⁵⁸ + 6166q⁵⁹ + 1477q⁶⁰ - 3051q⁶¹ - 3972q⁶² - 1665q⁶³ + 1148q⁶⁴ + 2217q⁶⁵ + 1318q⁶⁶ - 226q⁶⁷ - 1079q⁶⁸ - 832q⁶⁹ - 85q⁷⁰ + 431q⁷¹ + 437q⁷² + 139q⁷³ - 125q⁷⁴ - 211q⁷⁵ - 104q⁷⁶ + 40q⁷⁷ + 72q⁷⁸ + 42q⁷⁹ + 19q⁸⁰ - 24q⁸¹ - 40q⁸² + 6q⁸³ + 10q⁸⁴ - 5q⁸⁵ + 13q⁸⁶ + 2q⁸⁷ - 12q⁸⁸ + 2q⁸⁹ + 4q⁹⁰ - 4q⁹¹ + 2q⁹² + 2q⁹³ - 3q⁹⁴ + q⁹⁵

6 q^-78 - 3q^-77 + q^-76 + 5q^-75 - 3q^-74 - 3q^-73 - 6q^-72 + 11q^-71 - 5q^-70 + q^-69 + 33q^-68 - 11q^-67 - 30q^-66 - 59q^-65 + 13q^-64 + 6q^-63 + 47q^-62 + 185q^-61 + 66q^-60 - 74q^-59 - 319q^-58 - 225q^-57 - 238q^-56 + 45q^-55 + 722q^-54 + 805q^-53 + 561q^-52 - 454q^-51 - 978q^-50 - 1815q^-49 - 1519q^-48 + 479q^-47 + 2270q^-46 + 3554q^-45 + 2354q^-44 + 531q^-43 - 3788q^-42 - 6566q^-41 - 5222q^-40 - 934q^-39 + 5627q^-38 + 9374q^-37 + 10759q^-36 + 2983q^-35 - 7671q^-34 - 15386q^-33 - 16143q^-32 - 6745q^-31 + 7218q^-30 + 23915q^-29 + 25058q^-28 + 12786q^-27 - 9285q^-26 - 30212q^-25 - 36779q^-24 - 24266q^-23 + 11380q^-22 + 39966q^-21 + 51715q^-20 + 33412q^-19 - 7668q^-18 - 51748q^-17 - 73168q^-16 - 44472q^-15 + 7775q^-14 + 66637q^-13 + 90754q^-12 + 63403q^-11 - 10346q^-10 - 89378q^-9 - 111777q^-8 - 75642q^-7 + 18206q^-6 + 108419q^-5 + 142545q^-4 + 82927q^-3 - 38505q^-2 - 135504q^-1 - 163856 - 80763q + 59118q² + 177059q³ + 178671q⁴ + 59611q⁵ - 96083q⁶ - 209015q⁷ - 180593q⁸ - 32147q⁹ + 154435q¹⁰ + 235038q¹¹ + 157340q¹² - 21066q¹³ - 204506q¹⁴ - 245647q¹⁵ - 121471q¹⁶ + 102255q¹⁷ + 249440q¹⁸ + 225079q¹⁹ + 51001q²⁰ - 176062q²¹ - 275844q²² - 185128q²³ + 52604q²⁴ + 244394q²⁵ + 264708q²⁶ + 102736q²⁷ - 148963q²⁸ - 290420q²⁹ - 227455q³⁰ + 16089q³¹ + 238656q³² + 293319q³³ + 142238q³⁴ - 126959q³⁵ - 302667q³⁶ - 265211q³⁷ - 20236q³⁸ + 229429q³⁹ + 320128q⁴⁰ + 187362q⁴¹ - 91916q⁴² - 303644q⁴³ - 304347q⁴⁴ - 75539q⁴⁵ + 193561q⁴⁶ + 330107q⁴⁷ + 240285q⁴⁸ - 24320q⁴⁹ - 265509q⁵⁰ - 323146q⁵¹ - 146450q⁵² + 112745q⁵³ + 291214q⁵⁴ + 271422q⁵⁵ + 64689q⁵⁶ - 173037q⁵⁷ - 286019q⁵⁸ - 194346q⁵⁹ + 7610q⁶⁰ + 193161q⁶¹ + 242596q⁶² + 127552q⁶³ - 57259q⁶⁴ - 189105q⁶⁵ - 179901q⁶⁶ - 67014q⁶⁷ + 76607q⁶⁸ + 156965q⁶⁹ + 126524q⁷⁰ + 22920q⁷¹ - 80373q⁷² - 113168q⁷³ - 77961q⁷⁴ - 143q⁷⁵ + 65878q⁷⁶ + 78397q⁷⁷ + 41837q⁷⁸ - 12841q⁷⁹ - 44951q⁸⁰ - 47657q⁸¹ - 20950q⁸² + 13335q⁸³ + 30532q⁸⁴ + 25722q⁸⁵ + 6863q⁸⁶ - 8813q⁸⁷ - 17618q⁸⁸ - 13525q⁸⁹ - 1774q⁹⁰ + 6888q⁹¹ + 8980q⁹² + 5087q⁹³ + 814q⁹⁴ - 3902q⁹⁵ - 4763q⁹⁶ - 1967q⁹⁷ + 591q⁹⁸ + 1917q⁹⁹ + 1522q¹⁰⁰ + 1132q¹⁰¹ - 432q¹⁰² - 1177q¹⁰³ - 592q¹⁰⁴ - 123q¹⁰⁵ + 248q¹⁰⁶ + 220q¹⁰⁷ + 433q¹⁰⁸ + 27q¹⁰⁹ - 256q¹¹⁰ - 95q¹¹¹ - 50q¹¹² + 19q¹¹³ - 17q¹¹⁴ + 122q¹¹⁵ + 23q¹¹⁶ - 61q¹¹⁷ - 2q¹¹⁸ - 9q¹¹⁹ + 8q¹²⁰ - 21q¹²¹ + 26q¹²² + 8q¹²³ - 17q¹²⁴ + 4q¹²⁵ - 2q¹²⁶ + 4q¹²⁷ - 4q¹²⁸ + 2q¹²⁹ + 2q¹³⁰ - 3q¹³¹ + q¹³²

7 q^-105 - 3q^-104 + q^-103 + 5q^-102 - 3q^-101 - 3q^-100 - 6q^-99 + 6q^-98 + 13q^-97 - 10q^-96 + 4q^-95 + 14q^-94 - 12q^-93 - 25q^-92 - 50q^-91 - 3q^-90 + 80q^-89 + 42q^-88 + 72q^-87 + 72q^-86 - 39q^-85 - 133q^-84 - 342q^-83 - 279q^-82 + 57q^-81 + 254q^-80 + 594q^-79 + 759q^-78 + 486q^-77 + 9q^-76 - 1124q^-75 - 1896q^-74 - 1647q^-73 - 932q^-72 + 955q^-71 + 2938q^-70 + 4047q^-69 + 4038q^-68 + 1092q^-67 - 3291q^-66 - 6931q^-65 - 9129q^-64 - 6742q^-63 - 510q^-62 + 7483q^-61 + 15587q^-60 + 17180q^-59 + 10979q^-58 - 1470q^-57 - 18079q^-56 - 29129q^-55 - 29534q^-54 - 17075q^-53 + 8795q^-52 + 35257q^-51 + 51533q^-50 + 48852q^-49 + 20093q^-48 - 22386q^-47 - 63937q^-46 - 86703q^-45 - 70946q^-44 - 21262q^-43 + 48502q^-42 + 112121q^-41 + 132398q^-40 + 97985q^-39 + 12256q^-38 - 98091q^-37 - 178849q^-36 - 194403q^-35 - 122869q^-34 + 21845q^-33 + 175514q^-32 + 276254q^-31 + 265771q^-30 + 125444q^-29 - 89784q^-28 - 299918q^-27 - 402334q^-26 - 325142q^-25 - 90214q^-24 + 224363q^-23 + 478052q^-22 + 533251q^-21 + 349424q^-20 - 29284q^-19 - 444320q^-18 - 690381q^-17 - 641639q^-16 - 273304q^-15 + 272259q^-14 + 738301q^-13 + 904069q^-12 + 641466q^-11 + 34253q^-10 - 640987q^-9 - 1074371q^-8 - 1012324q^-7 - 438644q^-6 + 393151q^-5 + 1107960q^-4 + 1321628q^-3 + 883728q^-2 - 21214q^-1 - 990806 - 1520985q - 1307002q² - 424382q³ + 739118q⁴ + 1585901q⁵ + 1656665q⁶ + 887810q⁷ - 390664q⁸ - 1522563q⁹ - 1903592q¹⁰ - 1316759q¹¹ - 4130q¹² + 1355917q¹³ + 2039434q¹⁴ + 1677613q¹⁵ + 400243q¹⁶ - 1124778q¹⁷ - 2079769q¹⁸ - 1953966q¹⁹ - 759370q²⁰ + 868410q²¹ + 2047854q²² + 2147322q²³ + 1062582q²⁴ - 619843q²⁵ - 1975843q²⁶ - 2272141q²⁷ - 1300813q²⁸ + 402489q²⁹ + 1889193q³⁰ + 2348252q³¹ + 1481447q³² - 226337q³³ - 1811411q³⁴ - 2398596q³⁵ - 1616620q³⁶ + 91642q³⁷ + 1753220q³⁸ + 2442790q³⁹ + 1727454q⁴⁰ + 12519q⁴¹ - 1718909q⁴² - 2495996q⁴³ - 1833381q⁴⁴ - 104858q⁴⁵ + 1698635q⁴⁶ + 2564661q⁴⁷ + 1955137q⁴⁸ + 209368q⁴⁹ - 1675195q⁵⁰ - 2643587q⁵¹ - 2103041q⁵² - 351029q⁵³ + 1619337q⁵⁴ + 2714539q⁵⁵ + 2278314q⁵⁶ + 548598q⁵⁷ - 1501612q⁵⁸ - 2746890q⁵⁹ - 2462126q⁶⁰ - 806612q⁶¹ + 1294080q⁶² + 2702078q⁶³ + 2621929q⁶⁴ + 1110528q⁶⁵ - 986729q⁶⁶ - 2546245q⁶⁷ - 2711259q⁶⁸ - 1422084q⁶⁹ + 591376q⁷⁰ + 2258827q⁷¹ + 2685590q⁷² + 1689580q⁷³ - 146835q⁷⁴ - 1848435q⁷⁵ - 2516424q⁷⁶ - 1856384q⁷⁷ - 286538q⁷⁸ + 1352236q⁷⁹ + 2201356q⁸⁰ + 1882426q⁸¹ + 645896q⁸² - 832447q⁸³ - 1775160q⁸⁴ - 1756429q⁸⁵ - 877312q⁸⁶ + 359492q⁸⁷ + 1294634q⁸⁸ + 1500630q⁸⁹ + 960681q⁹⁰ + 9456q⁹¹ - 830937q⁹² - 1167237q⁹³ - 906664q⁹⁴ - 242462q⁹⁵ + 441871q⁹⁶ + 817587q⁹⁷ + 756244q⁹⁸ + 343645q⁹⁹ - 161749q¹⁰⁰ - 507743q¹⁰¹ - 562357q¹⁰² - 342023q¹⁰³ - 5088q¹⁰⁴ + 270836q¹⁰⁵ + 372645q¹⁰⁶ + 280008q¹⁰⁷ + 79590q¹⁰⁸ - 115150q¹⁰⁹ - 219105q¹¹⁰ - 197895q¹¹¹ - 92961q¹¹² + 29786q¹¹³ + 112686q¹¹⁴ + 122556q¹¹⁵ + 76035q¹¹⁶ + 7271q¹¹⁷ - 49181q¹¹⁸ - 67228q¹¹⁹ - 51260q¹²⁰ - 16781q¹²¹ + 17333q¹²² + 32602q¹²³ + 29634q¹²⁴ + 14530q¹²⁵ - 3792q¹²⁶ - 13838q¹²⁷ - 15347q¹²⁸ - 9640q¹²⁹ - 389q¹³⁰ + 5297q¹³¹ + 7195q¹³² + 5268q¹³³ + 963q¹³⁴ - 1609q¹³⁵ - 3006q¹³⁶ - 2773q¹³⁷ - 824q¹³⁸ + 472q¹³⁹ + 1389q¹⁴⁰ + 1291q¹⁴¹ + 252q¹⁴² - 55q¹⁴³ - 414q¹⁴⁴ - 606q¹⁴⁵ - 245q¹⁴⁶ - 57q¹⁴⁷ + 293q¹⁴⁸ + 323q¹⁴⁹ - 23q¹⁵⁰ + 7q¹⁵¹ - 25q¹⁵² - 107q¹⁵³ - 45q¹⁵⁴ - 62q¹⁵⁵ + 62q¹⁵⁶ + 92q¹⁵⁷ - 33q¹⁵⁸ - q¹⁵⁹ + q¹⁶⁰ - 12q¹⁶¹ + 3q¹⁶² - 23q¹⁶³ + 10q¹⁶⁴ + 21q¹⁶⁵ - 11q¹⁶⁶ - q¹⁶⁷ - 2q¹⁶⁹ + 4q¹⁷⁰ - 4q¹⁷¹ + 2q¹⁷² + 2q¹⁷³ - 3q¹⁷⁴ + q¹⁷⁵

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, 106]]

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

In[3]:=
GaussCode[Knot[10, 106]]

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

In[4]:=
DTCode[Knot[10, 106]]

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

In[5]:=
br = BR[Knot[10, 106]]

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

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

Out[6]=
{3, 10}

In[7]:=
BraidIndex[Knot[10, 106]]

Out[7]=
3

In[8]:=
Show[DrawMorseLink[Knot[10, 106]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[10, 106]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

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

In[10]:=
alex = Alexander[Knot[10, 106]][t]

Out[10]=
-4 4 9 15 2 3 4 -17 - t + -- - -- + -- + 15 t - 9 t + 4 t - t 3 2 t t t

In[11]:=
Conway[Knot[10, 106]][z]

Out[11]=
2 4 6 8 1 - z - 5 z - 4 z - z

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

Out[12]=
{Knot[10, 106]}

In[13]:=
{KnotDet[Knot[10, 106]], KnotSignature[Knot[10, 106]]}

Out[13]=
{75, 2}

In[14]:=
Jones[Knot[10, 106]][q]

Out[14]=
-3 3 6 2 3 4 5 6 7 -9 + q - -- + - + 12 q - 12 q + 12 q - 10 q + 6 q - 3 q + q 2 q q

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

Out[15]=
{Knot[10, 59], Knot[10, 106]}

In[16]:=
A2Invariant[Knot[10, 106]][q]

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

In[17]:=
HOMFLYPT[Knot[10, 106]][a, z]

Out[17]=
2 2 4 4 6 6 8 -4 2 2 5 z 11 z 4 4 z 13 z 6 z 6 z z 2 + a - -- + 5 z + ---- - ----- + 4 z + ---- - ----- + z + -- - ---- - -- 2 4 2 4 2 4 2 2 a a a a a a a a

In[18]:=
Kauffman[Knot[10, 106]][a, z]

Out[18]=
2 2 2 2 -4 2 z z z 2 z 2 z 2 z 3 z 13 z 2 + a + -- + -- + -- - -- - --- - a z - 5 z - -- + ---- - ---- - ----- + 2 7 5 3 a 8 6 4 2 a a a a a a a a 3 3 3 3 4 4 4 2 2 3 z 3 z 8 z 9 z 3 4 z 5 z 4 z > 2 a z - ---- + ---- + ---- + ---- + 7 a z + 9 z + -- - ---- + ---- + 7 5 3 a 8 6 4 a a a a a a 4 5 5 5 5 6 22 z 2 4 3 z 7 z 13 z 12 z 5 6 5 z > ----- - 3 a z + ---- - ---- - ----- - ----- - 9 a z - 11 z + ---- - 2 7 5 3 a 6 a a a a a 6 6 7 7 7 8 8 6 z 23 z 2 6 6 z 4 z z 7 8 5 z 9 z > ---- - ----- + a z + ---- + ---- + -- + 3 a z + 4 z + ---- + ---- + 4 2 5 3 a 4 2 a a a a a a 9 9 2 z 2 z > ---- + ---- 3 a a

In[19]:=
{Vassiliev[2][Knot[10, 106]], Vassiliev[3][Knot[10, 106]]}

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

In[20]:=
Kh[Knot[10, 106]][q, t]

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

In[21]:=
ColouredJones[Knot[10, 106], 2][q]

Out[21]=
-10 3 -8 10 16 6 40 30 36 81 2 -28 + q - -- + q + -- - -- - -- + -- - -- - -- + -- - 80 q + 111 q - 9 7 6 5 4 3 2 q q q q q q q q 3 4 5 6 7 8 9 10 11 > 9 q - 115 q + 115 q + 15 q - 124 q + 92 q + 29 q - 97 q + 53 q + 12 13 14 15 16 17 18 19 20 > 26 q - 50 q + 20 q + 11 q - 16 q + 6 q + 2 q - 3 q + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 10₁₀₆

10₁₀₅
10₁₀₇

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-10 - 3q^-9 + q^-8 + 10q^-7 - 16q^-6 - 6q^-5 + 40q^-4 - 30q^-3 - 36q^-2 + 81q^-1 - 28 - 80q + 111q² - 9q³ - 115q⁴ + 115q⁵ + 15q⁶ - 124q⁷ + 92q⁸ + 29q⁹ - 97q¹⁰ + 53q¹¹ + 26q¹² - 50q¹³ + 20q¹⁴ + 11q¹⁵ - 16q¹⁶ + 6q¹⁷ + 2q¹⁸ - 3q¹⁹ + q²⁰
3	q^-21 - 3q^-20 + q^-19 + 5q^-18 + 2q^-17 - 16q^-16 - 8q^-15 + 32q^-14 + 29q^-13 - 50q^-12 - 70q^-11 + 54q^-10 + 137q^-9 - 38q^-8 - 208q^-7 - 21q^-6 + 279q^-5 + 110q^-4 - 323q^-3 - 225q^-2 + 336q^-1 + 343 - 307q - 466q² + 266q³ + 555q⁴ - 186q⁵ - 641q⁶ + 116q⁷ + 685q⁸ - 22q⁹ - 720q¹⁰ - 52q¹¹ + 702q¹² + 137q¹³ - 661q¹⁴ - 199q¹⁵ + 574q¹⁶ + 241q¹⁷ - 459q¹⁸ - 253q¹⁹ + 338q²⁰ + 223q²¹ - 212q²² - 182q²³ + 125q²⁴ + 121q²⁵ - 65q²⁶ - 68q²⁷ + 32q²⁸ + 33q²⁹ - 18q³⁰ - 14q³¹ + 12q³² + 5q³³ - 8q³⁴ + 2q³⁶ + 2q³⁷ - 3q³⁸ + q³⁹
4	q^-36 - 3q^-35 + q^-34 + 5q^-33 - 3q^-32 + 2q^-31 - 19q^-30 + 4q^-29 + 35q^-28 + 4q^-27 + 9q^-26 - 99q^-25 - 45q^-24 + 104q^-23 + 106q^-22 + 144q^-21 - 247q^-20 - 306q^-19 + q^-18 + 256q^-17 + 662q^-16 - 118q^-15 - 671q^-14 - 596q^-13 - 55q^-12 + 1358q^-11 + 661q^-10 - 465q^-9 - 1376q^-8 - 1226q^-7 + 1428q^-6 + 1676q^-5 + 719q^-4 - 1476q^-3 - 2753q^-2 + 475q^-1 + 2064 + 2356q - 586q² - 3785q³ - 991q⁴ + 1580q⁵ + 3674q⁶ + 790q⁷ - 4070q⁸ - 2326q⁹ + 658q¹⁰ + 4448q¹¹ + 2088q¹² - 3872q¹³ - 3315q¹⁴ - 323q¹⁵ + 4755q¹⁶ + 3153q¹⁷ - 3291q¹⁸ - 3934q¹⁹ - 1347q²⁰ + 4475q²¹ + 3918q²² - 2190q²³ - 3903q²⁴ - 2323q²⁵ + 3345q²⁶ + 4003q²⁷ - 714q²⁸ - 2922q²⁹ - 2740q³⁰ + 1639q³¹ + 3071q³² + 403q³³ - 1359q³⁴ - 2190q³⁵ + 272q³⁶ + 1586q³⁷ + 609q³⁸ - 181q³⁹ - 1125q⁴⁰ - 183q⁴¹ + 483q⁴² + 263q⁴³ + 180q⁴⁴ - 358q⁴⁵ - 103q⁴⁶ + 79q⁴⁷ + 11q⁴⁸ + 116q⁴⁹ - 80q⁵⁰ - 6q⁵¹ + 15q⁵² - 33q⁵³ + 36q⁵⁴ - 18q⁵⁵ + 7q⁵⁶ + 7q⁵⁷ - 14q⁵⁸ + 8q⁵⁹ - 4q⁶⁰ + 2q⁶¹ + 2q⁶² - 3q⁶³ + q⁶⁴
5	q^-55 - 3q^-54 + q^-53 + 5q^-52 - 3q^-51 - 3q^-50 - q^-49 - 7q^-48 + 6q^-47 + 30q^-46 + 8q^-45 - 30q^-44 - 44q^-43 - 51q^-42 + 13q^-41 + 129q^-40 + 161q^-39 + 23q^-38 - 190q^-37 - 349q^-36 - 258q^-35 + 173q^-34 + 637q^-33 + 695q^-32 + 120q^-31 - 806q^-30 - 1364q^-29 - 879q^-28 + 577q^-27 + 2022q^-26 + 2148q^-25 + 372q^-24 - 2223q^-23 - 3604q^-22 - 2263q^-21 + 1400q^-20 + 4783q^-19 + 4808q^-18 + 728q^-17 - 4791q^-16 - 7409q^-15 - 4234q^-14 + 3172q^-13 + 9184q^-12 + 8417q^-11 + 370q^-10 - 9270q^-9 - 12480q^-8 - 5437q^-7 + 7248q^-6 + 15406q^-5 + 11293q^-4 - 3184q^-3 - 16563q^-2 - 16914q^-1 - 2452 + 15680q + 21675q² + 8761q³ - 13129q⁴ - 24862q⁵ - 15056q⁶ + 9237q⁷ + 26781q⁸ + 20669q⁹ - 4965q¹⁰ - 27304q¹¹ - 25352q¹² + 508q¹³ + 27188q¹⁴ + 29154q¹⁵ + 3400q¹⁶ - 26480q¹⁷ - 32163q¹⁸ - 7101q¹⁹ + 25730q²⁰ + 34685q²¹ + 10333q²² - 24629q²³ - 36800q²⁴ - 13687q²⁵ + 23267q²⁶ + 38498q²⁷ + 17017q²⁸ - 20998q²⁹ - 39478q³⁰ - 20646q³¹ + 17757q³² + 39330q³³ + 24043q³⁴ - 13227q³⁵ - 37498q³⁶ - 26867q³⁷ + 7703q³⁸ + 33794q³⁹ + 28336q⁴⁰ - 1847q⁴¹ - 28195q⁴² - 27875q⁴³ - 3601q⁴⁴ + 21340q⁴⁵ + 25440q⁴⁶ + 7554q⁴⁷ - 14202q⁴⁸ - 21086q⁴⁹ - 9682q⁵⁰ + 7759q⁵¹ + 15893q⁵² + 9759q⁵³ - 2917q⁵⁴ - 10609q⁵⁵ - 8339q⁵⁶ - 120q⁵⁷ + 6197q⁵⁸ + 6166q⁵⁹ + 1477q⁶⁰ - 3051q⁶¹ - 3972q⁶² - 1665q⁶³ + 1148q⁶⁴ + 2217q⁶⁵ + 1318q⁶⁶ - 226q⁶⁷ - 1079q⁶⁸ - 832q⁶⁹ - 85q⁷⁰ + 431q⁷¹ + 437q⁷² + 139q⁷³ - 125q⁷⁴ - 211q⁷⁵ - 104q⁷⁶ + 40q⁷⁷ + 72q⁷⁸ + 42q⁷⁹ + 19q⁸⁰ - 24q⁸¹ - 40q⁸² + 6q⁸³ + 10q⁸⁴ - 5q⁸⁵ + 13q⁸⁶ + 2q⁸⁷ - 12q⁸⁸ + 2q⁸⁹ + 4q⁹⁰ - 4q⁹¹ + 2q⁹² + 2q⁹³ - 3q⁹⁴ + q⁹⁵
6	q^-78 - 3q^-77 + q^-76 + 5q^-75 - 3q^-74 - 3q^-73 - 6q^-72 + 11q^-71 - 5q^-70 + q^-69 + 33q^-68 - 11q^-67 - 30q^-66 - 59q^-65 + 13q^-64 + 6q^-63 + 47q^-62 + 185q^-61 + 66q^-60 - 74q^-59 - 319q^-58 - 225q^-57 - 238q^-56 + 45q^-55 + 722q^-54 + 805q^-53 + 561q^-52 - 454q^-51 - 978q^-50 - 1815q^-49 - 1519q^-48 + 479q^-47 + 2270q^-46 + 3554q^-45 + 2354q^-44 + 531q^-43 - 3788q^-42 - 6566q^-41 - 5222q^-40 - 934q^-39 + 5627q^-38 + 9374q^-37 + 10759q^-36 + 2983q^-35 - 7671q^-34 - 15386q^-33 - 16143q^-32 - 6745q^-31 + 7218q^-30 + 23915q^-29 + 25058q^-28 + 12786q^-27 - 9285q^-26 - 30212q^-25 - 36779q^-24 - 24266q^-23 + 11380q^-22 + 39966q^-21 + 51715q^-20 + 33412q^-19 - 7668q^-18 - 51748q^-17 - 73168q^-16 - 44472q^-15 + 7775q^-14 + 66637q^-13 + 90754q^-12 + 63403q^-11 - 10346q^-10 - 89378q^-9 - 111777q^-8 - 75642q^-7 + 18206q^-6 + 108419q^-5 + 142545q^-4 + 82927q^-3 - 38505q^-2 - 135504q^-1 - 163856 - 80763q + 59118q² + 177059q³ + 178671q⁴ + 59611q⁵ - 96083q⁶ - 209015q⁷ - 180593q⁸ - 32147q⁹ + 154435q¹⁰ + 235038q¹¹ + 157340q¹² - 21066q¹³ - 204506q¹⁴ - 245647q¹⁵ - 121471q¹⁶ + 102255q¹⁷ + 249440q¹⁸ + 225079q¹⁹ + 51001q²⁰ - 176062q²¹ - 275844q²² - 185128q²³ + 52604q²⁴ + 244394q²⁵ + 264708q²⁶ + 102736q²⁷ - 148963q²⁸ - 290420q²⁹ - 227455q³⁰ + 16089q³¹ + 238656q³² + 293319q³³ + 142238q³⁴ - 126959q³⁵ - 302667q³⁶ - 265211q³⁷ - 20236q³⁸ + 229429q³⁹ + 320128q⁴⁰ + 187362q⁴¹ - 91916q⁴² - 303644q⁴³ - 304347q⁴⁴ - 75539q⁴⁵ + 193561q⁴⁶ + 330107q⁴⁷ + 240285q⁴⁸ - 24320q⁴⁹ - 265509q⁵⁰ - 323146q⁵¹ - 146450q⁵² + 112745q⁵³ + 291214q⁵⁴ + 271422q⁵⁵ + 64689q⁵⁶ - 173037q⁵⁷ - 286019q⁵⁸ - 194346q⁵⁹ + 7610q⁶⁰ + 193161q⁶¹ + 242596q⁶² + 127552q⁶³ - 57259q⁶⁴ - 189105q⁶⁵ - 179901q⁶⁶ - 67014q⁶⁷ + 76607q⁶⁸ + 156965q⁶⁹ + 126524q⁷⁰ + 22920q⁷¹ - 80373q⁷² - 113168q⁷³ - 77961q⁷⁴ - 143q⁷⁵ + 65878q⁷⁶ + 78397q⁷⁷ + 41837q⁷⁸ - 12841q⁷⁹ - 44951q⁸⁰ - 47657q⁸¹ - 20950q⁸² + 13335q⁸³ + 30532q⁸⁴ + 25722q⁸⁵ + 6863q⁸⁶ - 8813q⁸⁷ - 17618q⁸⁸ - 13525q⁸⁹ - 1774q⁹⁰ + 6888q⁹¹ + 8980q⁹² + 5087q⁹³ + 814q⁹⁴ - 3902q⁹⁵ - 4763q⁹⁶ - 1967q⁹⁷ + 591q⁹⁸ + 1917q⁹⁹ + 1522q¹⁰⁰ + 1132q¹⁰¹ - 432q¹⁰² - 1177q¹⁰³ - 592q¹⁰⁴ - 123q¹⁰⁵ + 248q¹⁰⁶ + 220q¹⁰⁷ + 433q¹⁰⁸ + 27q¹⁰⁹ - 256q¹¹⁰ - 95q¹¹¹ - 50q¹¹² + 19q¹¹³ - 17q¹¹⁴ + 122q¹¹⁵ + 23q¹¹⁶ - 61q¹¹⁷ - 2q¹¹⁸ - 9q¹¹⁹ + 8q¹²⁰ - 21q¹²¹ + 26q¹²² + 8q¹²³ - 17q¹²⁴ + 4q¹²⁵ - 2q¹²⁶ + 4q¹²⁷ - 4q¹²⁸ + 2q¹²⁹ + 2q¹³⁰ - 3q¹³¹ + q¹³²
7	q^-105 - 3q^-104 + q^-103 + 5q^-102 - 3q^-101 - 3q^-100 - 6q^-99 + 6q^-98 + 13q^-97 - 10q^-96 + 4q^-95 + 14q^-94 - 12q^-93 - 25q^-92 - 50q^-91 - 3q^-90 + 80q^-89 + 42q^-88 + 72q^-87 + 72q^-86 - 39q^-85 - 133q^-84 - 342q^-83 - 279q^-82 + 57q^-81 + 254q^-80 + 594q^-79 + 759q^-78 + 486q^-77 + 9q^-76 - 1124q^-75 - 1896q^-74 - 1647q^-73 - 932q^-72 + 955q^-71 + 2938q^-70 + 4047q^-69 + 4038q^-68 + 1092q^-67 - 3291q^-66 - 6931q^-65 - 9129q^-64 - 6742q^-63 - 510q^-62 + 7483q^-61 + 15587q^-60 + 17180q^-59 + 10979q^-58 - 1470q^-57 - 18079q^-56 - 29129q^-55 - 29534q^-54 - 17075q^-53 + 8795q^-52 + 35257q^-51 + 51533q^-50 + 48852q^-49 + 20093q^-48 - 22386q^-47 - 63937q^-46 - 86703q^-45 - 70946q^-44 - 21262q^-43 + 48502q^-42 + 112121q^-41 + 132398q^-40 + 97985q^-39 + 12256q^-38 - 98091q^-37 - 178849q^-36 - 194403q^-35 - 122869q^-34 + 21845q^-33 + 175514q^-32 + 276254q^-31 + 265771q^-30 + 125444q^-29 - 89784q^-28 - 299918q^-27 - 402334q^-26 - 325142q^-25 - 90214q^-24 + 224363q^-23 + 478052q^-22 + 533251q^-21 + 349424q^-20 - 29284q^-19 - 444320q^-18 - 690381q^-17 - 641639q^-16 - 273304q^-15 + 272259q^-14 + 738301q^-13 + 904069q^-12 + 641466q^-11 + 34253q^-10 - 640987q^-9 - 1074371q^-8 - 1012324q^-7 - 438644q^-6 + 393151q^-5 + 1107960q^-4 + 1321628q^-3 + 883728q^-2 - 21214q^-1 - 990806 - 1520985q - 1307002q² - 424382q³ + 739118q⁴ + 1585901q⁵ + 1656665q⁶ + 887810q⁷ - 390664q⁸ - 1522563q⁹ - 1903592q¹⁰ - 1316759q¹¹ - 4130q¹² + 1355917q¹³ + 2039434q¹⁴ + 1677613q¹⁵ + 400243q¹⁶ - 1124778q¹⁷ - 2079769q¹⁸ - 1953966q¹⁹ - 759370q²⁰ + 868410q²¹ + 2047854q²² + 2147322q²³ + 1062582q²⁴ - 619843q²⁵ - 1975843q²⁶ - 2272141q²⁷ - 1300813q²⁸ + 402489q²⁹ + 1889193q³⁰ + 2348252q³¹ + 1481447q³² - 226337q³³ - 1811411q³⁴ - 2398596q³⁵ - 1616620q³⁶ + 91642q³⁷ + 1753220q³⁸ + 2442790q³⁹ + 1727454q⁴⁰ + 12519q⁴¹ - 1718909q⁴² - 2495996q⁴³ - 1833381q⁴⁴ - 104858q⁴⁵ + 1698635q⁴⁶ + 2564661q⁴⁷ + 1955137q⁴⁸ + 209368q⁴⁹ - 1675195q⁵⁰ - 2643587q⁵¹ - 2103041q⁵² - 351029q⁵³ + 1619337q⁵⁴ + 2714539q⁵⁵ + 2278314q⁵⁶ + 548598q⁵⁷ - 1501612q⁵⁸ - 2746890q⁵⁹ - 2462126q⁶⁰ - 806612q⁶¹ + 1294080q⁶² + 2702078q⁶³ + 2621929q⁶⁴ + 1110528q⁶⁵ - 986729q⁶⁶ - 2546245q⁶⁷ - 2711259q⁶⁸ - 1422084q⁶⁹ + 591376q⁷⁰ + 2258827q⁷¹ + 2685590q⁷² + 1689580q⁷³ - 146835q⁷⁴ - 1848435q⁷⁵ - 2516424q⁷⁶ - 1856384q⁷⁷ - 286538q⁷⁸ + 1352236q⁷⁹ + 2201356q⁸⁰ + 1882426q⁸¹ + 645896q⁸² - 832447q⁸³ - 1775160q⁸⁴ - 1756429q⁸⁵ - 877312q⁸⁶ + 359492q⁸⁷ + 1294634q⁸⁸ + 1500630q⁸⁹ + 960681q⁹⁰ + 9456q⁹¹ - 830937q⁹² - 1167237q⁹³ - 906664q⁹⁴ - 242462q⁹⁵ + 441871q⁹⁶ + 817587q⁹⁷ + 756244q⁹⁸ + 343645q⁹⁹ - 161749q¹⁰⁰ - 507743q¹⁰¹ - 562357q¹⁰² - 342023q¹⁰³ - 5088q¹⁰⁴ + 270836q¹⁰⁵ + 372645q¹⁰⁶ + 280008q¹⁰⁷ + 79590q¹⁰⁸ - 115150q¹⁰⁹ - 219105q¹¹⁰ - 197895q¹¹¹ - 92961q¹¹² + 29786q¹¹³ + 112686q¹¹⁴ + 122556q¹¹⁵ + 76035q¹¹⁶ + 7271q¹¹⁷ - 49181q¹¹⁸ - 67228q¹¹⁹ - 51260q¹²⁰ - 16781q¹²¹ + 17333q¹²² + 32602q¹²³ + 29634q¹²⁴ + 14530q¹²⁵ - 3792q¹²⁶ - 13838q¹²⁷ - 15347q¹²⁸ - 9640q¹²⁹ - 389q¹³⁰ + 5297q¹³¹ + 7195q¹³² + 5268q¹³³ + 963q¹³⁴ - 1609q¹³⁵ - 3006q¹³⁶ - 2773q¹³⁷ - 824q¹³⁸ + 472q¹³⁹ + 1389q¹⁴⁰ + 1291q¹⁴¹ + 252q¹⁴² - 55q¹⁴³ - 414q¹⁴⁴ - 606q¹⁴⁵ - 245q¹⁴⁶ - 57q¹⁴⁷ + 293q¹⁴⁸ + 323q¹⁴⁹ - 23q¹⁵⁰ + 7q¹⁵¹ - 25q¹⁵² - 107q¹⁵³ - 45q¹⁵⁴ - 62q¹⁵⁵ + 62q¹⁵⁶ + 92q¹⁵⁷ - 33q¹⁵⁸ - q¹⁵⁹ + q¹⁶⁰ - 12q¹⁶¹ + 3q¹⁶² - 23q¹⁶³ + 10q¹⁶⁴ + 21q¹⁶⁵ - 11q¹⁶⁶ - q¹⁶⁷ - 2q¹⁶⁹ + 4q¹⁷⁰ - 4q¹⁷¹ + 2q¹⁷² + 2q¹⁷³ - 3q¹⁷⁴ + q¹⁷⁵

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	PD[Knot[10, 106]]
Out[2]=	PD[X[6, 2, 7, 1], X[16, 8, 17, 7], X[10, 3, 11, 4], X[2, 15, 3, 16], > X[14, 5, 15, 6], X[4, 11, 5, 12], X[18, 10, 19, 9], X[20, 14, 1, 13], > X[8, 18, 9, 17], X[12, 20, 13, 19]]
In[3]:=	GaussCode[Knot[10, 106]]
Out[3]=	GaussCode[1, -4, 3, -6, 5, -1, 2, -9, 7, -3, 6, -10, 8, -5, 4, -2, 9, -7, 10, > -8]
In[4]:=	DTCode[Knot[10, 106]]
Out[4]=	DTCode[6, 10, 14, 16, 18, 4, 20, 2, 8, 12]
In[5]:=	br = BR[Knot[10, 106]]
Out[5]=	BR[3, {1, 1, 1, -2, 1, -2, 1, 1, -2, -2}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{3, 10}
In[7]:=	BraidIndex[Knot[10, 106]]
Out[7]=	3
In[8]:=	Show[DrawMorseLink[Knot[10, 106]]]

Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 106]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Chiral, 2, 4, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 106]][t]
Out[10]=	-4 4 9 15 2 3 4 -17 - t + -- - -- + -- + 15 t - 9 t + 4 t - t 3 2 t t t
In[11]:=	Conway[Knot[10, 106]][z]
Out[11]=	2 4 6 8 1 - z - 5 z - 4 z - z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 106]}
In[13]:=	{KnotDet[Knot[10, 106]], KnotSignature[Knot[10, 106]]}
Out[13]=	{75, 2}
In[14]:=	Jones[Knot[10, 106]][q]
Out[14]=	-3 3 6 2 3 4 5 6 7 -9 + q - -- + - + 12 q - 12 q + 12 q - 10 q + 6 q - 3 q + q 2 q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 59], Knot[10, 106]}
In[16]:=	A2Invariant[Knot[10, 106]][q]
Out[16]=	-8 -6 2 -2 2 4 6 8 10 12 14 16 q - q + -- - q + 2 q - 2 q + 4 q - 2 q + q - q - 2 q + 2 q - 4 q 18 20 > q + q
In[17]:=	HOMFLYPT[Knot[10, 106]][a, z]
Out[17]=	2 2 4 4 6 6 8 -4 2 2 5 z 11 z 4 4 z 13 z 6 z 6 z z 2 + a - -- + 5 z + ---- - ----- + 4 z + ---- - ----- + z + -- - ---- - -- 2 4 2 4 2 4 2 2 a a a a a a a a
In[18]:=	Kauffman[Knot[10, 106]][a, z]
Out[18]=	2 2 2 2 -4 2 z z z 2 z 2 z 2 z 3 z 13 z 2 + a + -- + -- + -- - -- - --- - a z - 5 z - -- + ---- - ---- - ----- + 2 7 5 3 a 8 6 4 2 a a a a a a a a 3 3 3 3 4 4 4 2 2 3 z 3 z 8 z 9 z 3 4 z 5 z 4 z > 2 a z - ---- + ---- + ---- + ---- + 7 a z + 9 z + -- - ---- + ---- + 7 5 3 a 8 6 4 a a a a a a 4 5 5 5 5 6 22 z 2 4 3 z 7 z 13 z 12 z 5 6 5 z > ----- - 3 a z + ---- - ---- - ----- - ----- - 9 a z - 11 z + ---- - 2 7 5 3 a 6 a a a a a 6 6 7 7 7 8 8 6 z 23 z 2 6 6 z 4 z z 7 8 5 z 9 z > ---- - ----- + a z + ---- + ---- + -- + 3 a z + 4 z + ---- + ---- + 4 2 5 3 a 4 2 a a a a a a 9 9 2 z 2 z > ---- + ---- 3 a a
In[19]:=	{Vassiliev[2][Knot[10, 106]], Vassiliev[3][Knot[10, 106]]}
Out[19]=	{-1, -1}
In[20]:=	Kh[Knot[10, 106]][q, t]
Out[20]=	3 1 2 1 4 2 5 4 q 3 7 q + 6 q + ----- + ----- + ----- + ----- + ---- + --- + --- + 6 q t + 7 4 5 3 3 3 3 2 2 q t t q t q t q t q t q t 5 5 2 7 2 7 3 9 3 9 4 11 4 > 6 q t + 6 q t + 6 q t + 4 q t + 6 q t + 2 q t + 4 q t + 11 5 13 5 15 6 > q t + 2 q t + q t
In[21]:=	ColouredJones[Knot[10, 106], 2][q]
Out[21]=	-10 3 -8 10 16 6 40 30 36 81 2 -28 + q - -- + q + -- - -- - -- + -- - -- - -- + -- - 80 q + 111 q - 9 7 6 5 4 3 2 q q q q q q q q 3 4 5 6 7 8 9 10 11 > 9 q - 115 q + 115 q + 15 q - 124 q + 92 q + 29 q - 97 q + 53 q + 12 13 14 15 16 17 18 19 20 > 26 q - 50 q + 20 q + 11 q - 16 q + 6 q + 2 q - 3 q + q

The Alternating Knot 10106

The Alternating Knot 10₁₀₆