The Alternating Knot 10₈₈

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Acknowledgement

KnotPlot

PD Presentation: X₄₂₅₁ X_20,14,1,13 X₈₃₉₄ X_2,9,3,10 X_14,7,15,8 X_18,15,19,16 X_12,6,13,5 X_10,18,11,17 X_16,12,17,11 X_6,19,7,20

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

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

Minimum Braid Representative:

Length is 10, width is 5
Braid index is 5

A Morse Link Presentation:

3D Invariants:

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

NegativeAmphicheiral 1 3 3 / NotAvailable 1

Alexander Polynomial: - t^-3 + 8t^-2 - 24t^-1 + 35 - 24t + 8t² - t³

Conway Polynomial: 1 - z² + 2z⁴ - z⁶

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

Determinant and Signature: {101, 0}

Jones Polynomial: - q^-5 + 4q^-4 - 8q^-3 + 13q^-2 - 16q^-1 + 17 - 16q + 13q² - 8q³ + 4q⁴ - q⁵

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

A2 (sl(3)) Invariant: - q^-16 + q^-14 + 2q^-12 - 3q^-10 + 3q^-8 - 2q^-4 + 3q^-2 - 3 + 3q² - 2q⁴ + 3q⁸ - 3q¹⁰ + 2q¹² + q¹⁴ - q¹⁶

HOMFLY-PT Polynomial: - a^-4z² + a^-2 + 2a^-2z² + 2a^-2z⁴ - 1 - 3z² - 2z⁴ - z⁶ + a² + 2a²z² + 2a²z⁴ - a⁴z²

Kauffman Polynomial: - a^-5z³ + a^-5z⁵ + 3a^-4z² - 6a^-4z⁴ + 4a^-4z⁶ - a^-3z + 6a^-3z³ - 11a^-3z⁵ + 7a^-3z⁷ - a^-2 + 7a^-2z² - 10a^-2z⁴ - 2a^-2z⁶ + 6a^-2z⁸ - 4a^-1z + 19a^-1z³ - 32a^-1z⁵ + 14a^-1z⁷ + 2a^-1z⁹ - 1 + 8z² - 8z⁴ - 12z⁶ + 12z⁸ - 4az + 19az³ - 32az⁵ + 14az⁷ + 2az⁹ - a² + 7a²z² - 10a²z⁴ - 2a²z⁶ + 6a²z⁸ - a³z + 6a³z³ - 11a³z⁵ + 7a³z⁷ + 3a⁴z² - 6a⁴z⁴ + 4a⁴z⁶ - a⁵z³ + a⁵z⁵

V₂ and V₃, the type 2 and 3 Vassiliev invariants: {-1, 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=0 is the signature of 10₈₈. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

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

j = 11 1

j = 9 3

j = 7 5 1

j = 5 8 3

j = 3 8 5

j = 1 9 8

j = -1 8 9

j = -3 5 8

j = -5 3 8

j = -7 1 5

j = -9 3

j = -11 1

n Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q^-15 - 4q^-14 + 3q^-13 + 12q^-12 - 29q^-11 + 7q^-10 + 59q^-9 - 84q^-8 - 12q^-7 + 150q^-6 - 138q^-5 - 65q^-4 + 239q^-3 - 153q^-2 - 123q^-1 + 275 - 123q - 153q² + 239q³ - 65q⁴ - 138q⁵ + 150q⁶ - 12q⁷ - 84q⁸ + 59q⁹ + 7q¹⁰ - 29q¹¹ + 12q¹² + 3q¹³ - 4q¹⁴ + q¹⁵

3 - q^-30 + 4q^-29 - 3q^-28 - 7q^-27 + 4q^-26 + 24q^-25 - 8q^-24 - 64q^-23 + 12q^-22 + 130q^-21 + 15q^-20 - 245q^-19 - 84q^-18 + 391q^-17 + 229q^-16 - 551q^-15 - 453q^-14 + 674q^-13 + 771q^-12 - 760q^-11 - 1111q^-10 + 745q^-9 + 1471q^-8 - 664q^-7 - 1781q^-6 + 513q^-5 + 2025q^-4 - 325q^-3 - 2172q^-2 + 110q^-1 + 2223 + 110q - 2172q² - 325q³ + 2025q⁴ + 513q⁵ - 1781q⁶ - 664q⁷ + 1471q⁸ + 745q⁹ - 1111q¹⁰ - 760q¹¹ + 771q¹² + 674q¹³ - 453q¹⁴ - 551q¹⁵ + 229q¹⁶ + 391q¹⁷ - 84q¹⁸ - 245q¹⁹ + 15q²⁰ + 130q²¹ + 12q²² - 64q²³ - 8q²⁴ + 24q²⁵ + 4q²⁶ - 7q²⁷ - 3q²⁸ + 4q²⁹ - q³⁰

4 q^-50 - 4q^-49 + 3q^-48 + 7q^-47 - 9q^-46 + q^-45 - 23q^-44 + 28q^-43 + 57q^-42 - 50q^-41 - 41q^-40 - 138q^-39 + 126q^-38 + 337q^-37 - 50q^-36 - 251q^-35 - 716q^-34 + 162q^-33 + 1214q^-32 + 557q^-31 - 431q^-30 - 2424q^-29 - 760q^-28 + 2541q^-27 + 2732q^-26 + 658q^-25 - 5136q^-24 - 3919q^-23 + 2771q^-22 + 6350q^-21 + 4546q^-20 - 7050q^-19 - 9106q^-18 + 165q^-17 + 9436q^-16 + 10854q^-15 - 6275q^-14 - 14088q^-13 - 4965q^-12 + 10078q^-11 + 17176q^-10 - 2981q^-9 - 16783q^-8 - 10469q^-7 + 8268q^-6 + 21342q^-5 + 1199q^-4 - 16789q^-3 - 14594q^-2 + 5083q^-1 + 22721 + 5083q - 14594q² - 16789q³ + 1199q⁴ + 21342q⁵ + 8268q⁶ - 10469q⁷ - 16783q⁸ - 2981q⁹ + 17176q¹⁰ + 10078q¹¹ - 4965q¹² - 14088q¹³ - 6275q¹⁴ + 10854q¹⁵ + 9436q¹⁶ + 165q¹⁷ - 9106q¹⁸ - 7050q¹⁹ + 4546q²⁰ + 6350q²¹ + 2771q²² - 3919q²³ - 5136q²⁴ + 658q²⁵ + 2732q²⁶ + 2541q²⁷ - 760q²⁸ - 2424q²⁹ - 431q³⁰ + 557q³¹ + 1214q³² + 162q³³ - 716q³⁴ - 251q³⁵ - 50q³⁶ + 337q³⁷ + 126q³⁸ - 138q³⁹ - 41q⁴⁰ - 50q⁴¹ + 57q⁴² + 28q⁴³ - 23q⁴⁴ + q⁴⁵ - 9q⁴⁶ + 7q⁴⁷ + 3q⁴⁸ - 4q⁴⁹ + q⁵⁰

5 - q^-75 + 4q^-74 - 3q^-73 - 7q^-72 + 9q^-71 + 4q^-70 - 2q^-69 + 3q^-68 - 21q^-67 - 34q^-66 + 40q^-65 + 84q^-64 + 33q^-63 - 59q^-62 - 199q^-61 - 197q^-60 + 113q^-59 + 531q^-58 + 543q^-57 - 109q^-56 - 1040q^-55 - 1392q^-54 - 320q^-53 + 1822q^-52 + 3134q^-51 + 1551q^-50 - 2527q^-49 - 5861q^-48 - 4569q^-47 + 2283q^-46 + 9758q^-45 + 10091q^-44 + 71q^-43 - 13769q^-42 - 18660q^-41 - 6376q^-40 + 16455q^-39 + 29950q^-38 + 17815q^-37 - 15371q^-36 - 42477q^-35 - 34960q^-34 + 8400q^-33 + 53555q^-32 + 56888q^-31 + 6187q^-30 - 60539q^-29 - 81348q^-28 - 27953q^-27 + 60590q^-26 + 105028q^-25 + 55924q^-24 - 52997q^-23 - 125046q^-22 - 86597q^-21 + 37910q^-20 + 138741q^-19 + 117296q^-18 - 17294q^-17 - 145636q^-16 - 144641q^-15 - 6338q^-14 + 145775q^-13 + 167068q^-12 + 30435q^-11 - 140607q^-10 - 183710q^-9 - 53108q^-8 + 131586q^-7 + 194854q^-6 + 73319q^-5 - 119974q^-4 - 201061q^-3 - 91032q^-2 + 106443q^-1 + 203085 + 106443q - 91032q² - 201061q³ - 119974q⁴ + 73319q⁵ + 194854q⁶ + 131586q⁷ - 53108q⁸ - 183710q⁹ - 140607q¹⁰ + 30435q¹¹ + 167068q¹² + 145775q¹³ - 6338q¹⁴ - 144641q¹⁵ - 145636q¹⁶ - 17294q¹⁷ + 117296q¹⁸ + 138741q¹⁹ + 37910q²⁰ - 86597q²¹ - 125046q²² - 52997q²³ + 55924q²⁴ + 105028q²⁵ + 60590q²⁶ - 27953q²⁷ - 81348q²⁸ - 60539q²⁹ + 6187q³⁰ + 56888q³¹ + 53555q³² + 8400q³³ - 34960q³⁴ - 42477q³⁵ - 15371q³⁶ + 17815q³⁷ + 29950q³⁸ + 16455q³⁹ - 6376q⁴⁰ - 18660q⁴¹ - 13769q⁴² + 71q⁴³ + 10091q⁴⁴ + 9758q⁴⁵ + 2283q⁴⁶ - 4569q⁴⁷ - 5861q⁴⁸ - 2527q⁴⁹ + 1551q⁵⁰ + 3134q⁵¹ + 1822q⁵² - 320q⁵³ - 1392q⁵⁴ - 1040q⁵⁵ - 109q⁵⁶ + 543q⁵⁷ + 531q⁵⁸ + 113q⁵⁹ - 197q⁶⁰ - 199q⁶¹ - 59q⁶² + 33q⁶³ + 84q⁶⁴ + 40q⁶⁵ - 34q⁶⁶ - 21q⁶⁷ + 3q⁶⁸ - 2q⁶⁹ + 4q⁷⁰ + 9q⁷¹ - 7q⁷² - 3q⁷³ + 4q⁷⁴ - q⁷⁵

6 q^-105 - 4q^-104 + 3q^-103 + 7q^-102 - 9q^-101 - 4q^-100 - 3q^-99 + 22q^-98 - 10q^-97 - 2q^-96 + 44q^-95 - 68q^-94 - 58q^-93 - 20q^-92 + 145q^-91 + 95q^-90 + 46q^-89 + 125q^-88 - 408q^-87 - 531q^-86 - 317q^-85 + 615q^-84 + 967q^-83 + 1024q^-82 + 876q^-81 - 1619q^-80 - 3217q^-79 - 3248q^-78 + 420q^-77 + 4033q^-76 + 6961q^-75 + 7385q^-74 - 1329q^-73 - 11203q^-72 - 17531q^-71 - 10354q^-70 + 4368q^-69 + 23577q^-68 + 35863q^-67 + 19355q^-66 - 14959q^-65 - 52800q^-64 - 59173q^-63 - 30996q^-62 + 33402q^-61 + 99926q^-60 + 104116q^-59 + 38455q^-58 - 79161q^-57 - 162940q^-56 - 165555q^-55 - 42543q^-54 + 151254q^-53 + 271312q^-52 + 233769q^-51 + 13377q^-50 - 249520q^-49 - 414572q^-48 - 308643q^-47 + 46760q^-46 + 420480q^-45 + 576578q^-44 + 343984q^-43 - 145228q^-42 - 645910q^-41 - 753595q^-40 - 338824q^-39 + 352823q^-38 + 902403q^-37 + 878082q^-36 + 270412q^-35 - 643372q^-34 - 1182741q^-33 - 942447q^-32 - 39231q^-31 + 986653q^-30 + 1396470q^-29 + 904710q^-28 - 317470q^-27 - 1371425q^-26 - 1527787q^-25 - 639520q^-24 + 759338q^-23 + 1682298q^-22 + 1515191q^-21 + 201994q^-20 - 1266624q^-19 - 1892397q^-18 - 1214631q^-17 + 352347q^-16 + 1696603q^-15 + 1921584q^-14 + 704085q^-13 - 991256q^-12 - 2009256q^-11 - 1615471q^-10 - 55878q^-9 + 1547088q^-8 + 2110070q^-7 + 1076094q^-6 - 686020q^-5 - 1969547q^-4 - 1844533q^-3 - 390299q^-2 + 1337527q^-1 + 2158285 + 1337527q - 390299q² - 1844533q³ - 1969547q⁴ - 686020q⁵ + 1076094q⁶ + 2110070q⁷ + 1547088q⁸ - 55878q⁹ - 1615471q¹⁰ - 2009256q¹¹ - 991256q¹² + 704085q¹³ + 1921584q¹⁴ + 1696603q¹⁵ + 352347q¹⁶ - 1214631q¹⁷ - 1892397q¹⁸ - 1266624q¹⁹ + 201994q²⁰ + 1515191q²¹ + 1682298q²² + 759338q²³ - 639520q²⁴ - 1527787q²⁵ - 1371425q²⁶ - 317470q²⁷ + 904710q²⁸ + 1396470q²⁹ + 986653q³⁰ - 39231q³¹ - 942447q³² - 1182741q³³ - 643372q³⁴ + 270412q³⁵ + 878082q³⁶ + 902403q³⁷ + 352823q³⁸ - 338824q³⁹ - 753595q⁴⁰ - 645910q⁴¹ - 145228q⁴² + 343984q⁴³ + 576578q⁴⁴ + 420480q⁴⁵ + 46760q⁴⁶ - 308643q⁴⁷ - 414572q⁴⁸ - 249520q⁴⁹ + 13377q⁵⁰ + 233769q⁵¹ + 271312q⁵² + 151254q⁵³ - 42543q⁵⁴ - 165555q⁵⁵ - 162940q⁵⁶ - 79161q⁵⁷ + 38455q⁵⁸ + 104116q⁵⁹ + 99926q⁶⁰ + 33402q⁶¹ - 30996q⁶² - 59173q⁶³ - 52800q⁶⁴ - 14959q⁶⁵ + 19355q⁶⁶ + 35863q⁶⁷ + 23577q⁶⁸ + 4368q⁶⁹ - 10354q⁷⁰ - 17531q⁷¹ - 11203q⁷² - 1329q⁷³ + 7385q⁷⁴ + 6961q⁷⁵ + 4033q⁷⁶ + 420q⁷⁷ - 3248q⁷⁸ - 3217q⁷⁹ - 1619q⁸⁰ + 876q⁸¹ + 1024q⁸² + 967q⁸³ + 615q⁸⁴ - 317q⁸⁵ - 531q⁸⁶ - 408q⁸⁷ + 125q⁸⁸ + 46q⁸⁹ + 95q⁹⁰ + 145q⁹¹ - 20q⁹² - 58q⁹³ - 68q⁹⁴ + 44q⁹⁵ - 2q⁹⁶ - 10q⁹⁷ + 22q⁹⁸ - 3q⁹⁹ - 4q¹⁰⁰ - 9q¹⁰¹ + 7q¹⁰² + 3q¹⁰³ - 4q¹⁰⁴ + q¹⁰⁵

7 - q^-140 + 4q^-139 - 3q^-138 - 7q^-137 + 9q^-136 + 4q^-135 + 3q^-134 - 17q^-133 - 15q^-132 + 33q^-131 - 8q^-130 - 16q^-129 + 42q^-128 + 30q^-127 + 13q^-126 - 121q^-125 - 168q^-124 + 67q^-123 + 72q^-122 + 153q^-121 + 345q^-120 + 206q^-119 + 28q^-118 - 744q^-117 - 1273q^-116 - 608q^-115 + 135q^-114 + 1493q^-113 + 2823q^-112 + 2540q^-111 + 1128q^-110 - 2911q^-109 - 7248q^-108 - 7552q^-107 - 4466q^-106 + 3929q^-105 + 14027q^-104 + 18818q^-103 + 16073q^-102 - 110q^-101 - 24199q^-100 - 41450q^-99 - 42981q^-98 - 16792q^-97 + 30828q^-96 + 75952q^-95 + 97722q^-94 + 66617q^-93 - 17353q^-92 - 118190q^-91 - 190921q^-90 - 173528q^-89 - 48185q^-88 + 141342q^-87 + 318844q^-86 + 366831q^-85 + 218101q^-84 - 93530q^-83 - 454180q^-82 - 657775q^-81 - 546567q^-80 - 108698q^-79 + 519225q^-78 + 1018197q^-77 + 1077733q^-76 + 567378q^-75 - 394097q^-74 - 1360104q^-73 - 1798579q^-72 - 1366682q^-71 - 78293q^-70 + 1520419q^-69 + 2618368q^-68 + 2535184q^-67 + 1042851q^-66 - 1289355q^-65 - 3349882q^-64 - 3995819q^-63 - 2580161q^-62 + 454017q^-61 + 3732829q^-60 + 5554146q^-59 + 4652596q^-58 + 1132131q^-57 - 3491647q^-56 - 6921359q^-55 - 7080887q^-54 - 3483098q^-53 + 2414596q^-52 + 7768501q^-51 + 9563059q^-50 + 6458604q^-49 - 417981q^-48 - 7822357q^-47 - 11746214q^-46 - 9770005q^-45 - 2404931q^-44 + 6931124q^-43 + 13296490q^-42 + 13053332q^-41 + 5814614q^-40 - 5113233q^-39 - 14002256q^-38 - 15949372q^-37 - 9459051q^-36 + 2549624q^-35 + 13795700q^-34 + 18181824q^-33 + 12981758q^-32 + 470475q^-31 - 12777028q^-30 - 19616672q^-29 - 16076808q^-28 - 3614657q^-27 + 11154661q^-26 + 20259777q^-25 + 18557545q^-24 + 6592876q^-23 - 9195152q^-22 - 20238112q^-21 - 20364308q^-20 - 9198979q^-19 + 7151404q^-18 + 19740523q^-17 + 21549728q^-16 + 11338004q^-15 - 5213041q^-14 - 18966439q^-13 - 22240226q^-12 - 13018279q^-11 + 3484032q^-10 + 18080389q^-9 + 22588382q^-8 + 14321596q^-7 - 1979218q^-6 - 17180052q^-5 - 22733021q^-4 - 15371342q^-3 + 639284q^-2 + 16291800q^-1 + 22770653 + 16291800q + 639284q² - 15371342q³ - 22733021q⁴ - 17180052q⁵ - 1979218q⁶ + 14321596q⁷ + 22588382q⁸ + 18080389q⁹ + 3484032q¹⁰ - 13018279q¹¹ - 22240226q¹² - 18966439q¹³ - 5213041q¹⁴ + 11338004q¹⁵ + 21549728q¹⁶ + 19740523q¹⁷ + 7151404q¹⁸ - 9198979q¹⁹ - 20364308q²⁰ - 20238112q²¹ - 9195152q²² + 6592876q²³ + 18557545q²⁴ + 20259777q²⁵ + 11154661q²⁶ - 3614657q²⁷ - 16076808q²⁸ - 19616672q²⁹ - 12777028q³⁰ + 470475q³¹ + 12981758q³² + 18181824q³³ + 13795700q³⁴ + 2549624q³⁵ - 9459051q³⁶ - 15949372q³⁷ - 14002256q³⁸ - 5113233q³⁹ + 5814614q⁴⁰ + 13053332q⁴¹ + 13296490q⁴² + 6931124q⁴³ - 2404931q⁴⁴ - 9770005q⁴⁵ - 11746214q⁴⁶ - 7822357q⁴⁷ - 417981q⁴⁸ + 6458604q⁴⁹ + 9563059q⁵⁰ + 7768501q⁵¹ + 2414596q⁵² - 3483098q⁵³ - 7080887q⁵⁴ - 6921359q⁵⁵ - 3491647q⁵⁶ + 1132131q⁵⁷ + 4652596q⁵⁸ + 5554146q⁵⁹ + 3732829q⁶⁰ + 454017q⁶¹ - 2580161q⁶² - 3995819q⁶³ - 3349882q⁶⁴ - 1289355q⁶⁵ + 1042851q⁶⁶ + 2535184q⁶⁷ + 2618368q⁶⁸ + 1520419q⁶⁹ - 78293q⁷⁰ - 1366682q⁷¹ - 1798579q⁷² - 1360104q⁷³ - 394097q⁷⁴ + 567378q⁷⁵ + 1077733q⁷⁶ + 1018197q⁷⁷ + 519225q⁷⁸ - 108698q⁷⁹ - 546567q⁸⁰ - 657775q⁸¹ - 454180q⁸² - 93530q⁸³ + 218101q⁸⁴ + 366831q⁸⁵ + 318844q⁸⁶ + 141342q⁸⁷ - 48185q⁸⁸ - 173528q⁸⁹ - 190921q⁹⁰ - 118190q⁹¹ - 17353q⁹² + 66617q⁹³ + 97722q⁹⁴ + 75952q⁹⁵ + 30828q⁹⁶ - 16792q⁹⁷ - 42981q⁹⁸ - 41450q⁹⁹ - 24199q¹⁰⁰ - 110q¹⁰¹ + 16073q¹⁰² + 18818q¹⁰³ + 14027q¹⁰⁴ + 3929q¹⁰⁵ - 4466q¹⁰⁶ - 7552q¹⁰⁷ - 7248q¹⁰⁸ - 2911q¹⁰⁹ + 1128q¹¹⁰ + 2540q¹¹¹ + 2823q¹¹² + 1493q¹¹³ + 135q¹¹⁴ - 608q¹¹⁵ - 1273q¹¹⁶ - 744q¹¹⁷ + 28q¹¹⁸ + 206q¹¹⁹ + 345q¹²⁰ + 153q¹²¹ + 72q¹²² + 67q¹²³ - 168q¹²⁴ - 121q¹²⁵ + 13q¹²⁶ + 30q¹²⁷ + 42q¹²⁸ - 16q¹²⁹ - 8q¹³⁰ + 33q¹³¹ - 15q¹³² - 17q¹³³ + 3q¹³⁴ + 4q¹³⁵ + 9q¹³⁶ - 7q¹³⁷ - 3q¹³⁸ + 4q¹³⁹ - q¹⁴⁰

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, 88]]

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

In[3]:=
GaussCode[Knot[10, 88]]

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

In[4]:=
DTCode[Knot[10, 88]]

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

In[5]:=
br = BR[Knot[10, 88]]

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

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

Out[6]=
{5, 10}

In[7]:=
BraidIndex[Knot[10, 88]]

Out[7]=
5

In[8]:=
Show[DrawMorseLink[Knot[10, 88]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[10, 88]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

Out[9]=
{NegativeAmphicheiral, 1, 3, 3, NotAvailable, 1}

In[10]:=
alex = Alexander[Knot[10, 88]][t]

Out[10]=
-3 8 24 2 3 35 - t + -- - -- - 24 t + 8 t - t 2 t t

In[11]:=
Conway[Knot[10, 88]][z]

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

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

Out[12]=
{Knot[10, 88]}

In[13]:=
{KnotDet[Knot[10, 88]], KnotSignature[Knot[10, 88]]}

Out[13]=
{101, 0}

In[14]:=
Jones[Knot[10, 88]][q]

Out[14]=
-5 4 8 13 16 2 3 4 5 17 - q + -- - -- + -- - -- - 16 q + 13 q - 8 q + 4 q - q 4 3 2 q q q q

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

Out[15]=
{Knot[10, 88]}

In[16]:=
A2Invariant[Knot[10, 88]][q]

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

In[17]:=
HOMFLYPT[Knot[10, 88]][a, z]

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

In[18]:=
Kauffman[Knot[10, 88]][a, z]

Out[18]=
2 2 -2 2 z 4 z 3 2 3 z 7 z 2 2 -1 - a - a - -- - --- - 4 a z - a z + 8 z + ---- + ---- + 7 a z + 3 a 4 2 a a a 3 3 3 4 4 2 z 6 z 19 z 3 3 3 5 3 4 6 z > 3 a z - -- + ---- + ----- + 19 a z + 6 a z - a z - 8 z - ---- - 5 3 a 4 a a a 4 5 5 5 10 z 2 4 4 4 z 11 z 32 z 5 3 5 > ----- - 10 a z - 6 a z + -- - ----- - ----- - 32 a z - 11 a z + 2 5 3 a a a a 6 6 7 7 5 5 6 4 z 2 z 2 6 4 6 7 z 14 z 7 > a z - 12 z + ---- - ---- - 2 a z + 4 a z + ---- + ----- + 14 a z + 4 2 3 a a a a 8 9 3 7 8 6 z 2 8 2 z 9 > 7 a z + 12 z + ---- + 6 a z + ---- + 2 a z 2 a a

In[19]:=
{Vassiliev[2][Knot[10, 88]], Vassiliev[3][Knot[10, 88]]}

Out[19]=
{-1, 0}

In[20]:=
Kh[Knot[10, 88]][q, t]

Out[20]=
9 1 3 1 5 3 8 5 8 8 - + 9 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ---- + --- + q 11 5 9 4 7 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t q t q t 3 3 2 5 2 5 3 7 3 7 4 9 4 > 8 q t + 8 q t + 5 q t + 8 q t + 3 q t + 5 q t + q t + 3 q t + 11 5 > q t

In[21]:=
ColouredJones[Knot[10, 88], 2][q]

Out[21]=
-15 4 3 12 29 7 59 84 12 150 138 65 275 + q - --- + --- + --- - --- + --- + -- - -- - -- + --- - --- - -- + 14 13 12 11 10 9 8 7 6 5 4 q q q q q q q q q q q 239 153 123 2 3 4 5 6 > --- - --- - --- - 123 q - 153 q + 239 q - 65 q - 138 q + 150 q - 3 2 q q q 7 8 9 10 11 12 13 14 15 > 12 q - 84 q + 59 q + 7 q - 29 q + 12 q + 3 q - 4 q + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 10₈₈

10₈₇
10₈₉

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-15 - 4q^-14 + 3q^-13 + 12q^-12 - 29q^-11 + 7q^-10 + 59q^-9 - 84q^-8 - 12q^-7 + 150q^-6 - 138q^-5 - 65q^-4 + 239q^-3 - 153q^-2 - 123q^-1 + 275 - 123q - 153q² + 239q³ - 65q⁴ - 138q⁵ + 150q⁶ - 12q⁷ - 84q⁸ + 59q⁹ + 7q¹⁰ - 29q¹¹ + 12q¹² + 3q¹³ - 4q¹⁴ + q¹⁵
3	- q^-30 + 4q^-29 - 3q^-28 - 7q^-27 + 4q^-26 + 24q^-25 - 8q^-24 - 64q^-23 + 12q^-22 + 130q^-21 + 15q^-20 - 245q^-19 - 84q^-18 + 391q^-17 + 229q^-16 - 551q^-15 - 453q^-14 + 674q^-13 + 771q^-12 - 760q^-11 - 1111q^-10 + 745q^-9 + 1471q^-8 - 664q^-7 - 1781q^-6 + 513q^-5 + 2025q^-4 - 325q^-3 - 2172q^-2 + 110q^-1 + 2223 + 110q - 2172q² - 325q³ + 2025q⁴ + 513q⁵ - 1781q⁶ - 664q⁷ + 1471q⁸ + 745q⁹ - 1111q¹⁰ - 760q¹¹ + 771q¹² + 674q¹³ - 453q¹⁴ - 551q¹⁵ + 229q¹⁶ + 391q¹⁷ - 84q¹⁸ - 245q¹⁹ + 15q²⁰ + 130q²¹ + 12q²² - 64q²³ - 8q²⁴ + 24q²⁵ + 4q²⁶ - 7q²⁷ - 3q²⁸ + 4q²⁹ - q³⁰
4	q^-50 - 4q^-49 + 3q^-48 + 7q^-47 - 9q^-46 + q^-45 - 23q^-44 + 28q^-43 + 57q^-42 - 50q^-41 - 41q^-40 - 138q^-39 + 126q^-38 + 337q^-37 - 50q^-36 - 251q^-35 - 716q^-34 + 162q^-33 + 1214q^-32 + 557q^-31 - 431q^-30 - 2424q^-29 - 760q^-28 + 2541q^-27 + 2732q^-26 + 658q^-25 - 5136q^-24 - 3919q^-23 + 2771q^-22 + 6350q^-21 + 4546q^-20 - 7050q^-19 - 9106q^-18 + 165q^-17 + 9436q^-16 + 10854q^-15 - 6275q^-14 - 14088q^-13 - 4965q^-12 + 10078q^-11 + 17176q^-10 - 2981q^-9 - 16783q^-8 - 10469q^-7 + 8268q^-6 + 21342q^-5 + 1199q^-4 - 16789q^-3 - 14594q^-2 + 5083q^-1 + 22721 + 5083q - 14594q² - 16789q³ + 1199q⁴ + 21342q⁵ + 8268q⁶ - 10469q⁷ - 16783q⁸ - 2981q⁹ + 17176q¹⁰ + 10078q¹¹ - 4965q¹² - 14088q¹³ - 6275q¹⁴ + 10854q¹⁵ + 9436q¹⁶ + 165q¹⁷ - 9106q¹⁸ - 7050q¹⁹ + 4546q²⁰ + 6350q²¹ + 2771q²² - 3919q²³ - 5136q²⁴ + 658q²⁵ + 2732q²⁶ + 2541q²⁷ - 760q²⁸ - 2424q²⁹ - 431q³⁰ + 557q³¹ + 1214q³² + 162q³³ - 716q³⁴ - 251q³⁵ - 50q³⁶ + 337q³⁷ + 126q³⁸ - 138q³⁹ - 41q⁴⁰ - 50q⁴¹ + 57q⁴² + 28q⁴³ - 23q⁴⁴ + q⁴⁵ - 9q⁴⁶ + 7q⁴⁷ + 3q⁴⁸ - 4q⁴⁹ + q⁵⁰
5	- q^-75 + 4q^-74 - 3q^-73 - 7q^-72 + 9q^-71 + 4q^-70 - 2q^-69 + 3q^-68 - 21q^-67 - 34q^-66 + 40q^-65 + 84q^-64 + 33q^-63 - 59q^-62 - 199q^-61 - 197q^-60 + 113q^-59 + 531q^-58 + 543q^-57 - 109q^-56 - 1040q^-55 - 1392q^-54 - 320q^-53 + 1822q^-52 + 3134q^-51 + 1551q^-50 - 2527q^-49 - 5861q^-48 - 4569q^-47 + 2283q^-46 + 9758q^-45 + 10091q^-44 + 71q^-43 - 13769q^-42 - 18660q^-41 - 6376q^-40 + 16455q^-39 + 29950q^-38 + 17815q^-37 - 15371q^-36 - 42477q^-35 - 34960q^-34 + 8400q^-33 + 53555q^-32 + 56888q^-31 + 6187q^-30 - 60539q^-29 - 81348q^-28 - 27953q^-27 + 60590q^-26 + 105028q^-25 + 55924q^-24 - 52997q^-23 - 125046q^-22 - 86597q^-21 + 37910q^-20 + 138741q^-19 + 117296q^-18 - 17294q^-17 - 145636q^-16 - 144641q^-15 - 6338q^-14 + 145775q^-13 + 167068q^-12 + 30435q^-11 - 140607q^-10 - 183710q^-9 - 53108q^-8 + 131586q^-7 + 194854q^-6 + 73319q^-5 - 119974q^-4 - 201061q^-3 - 91032q^-2 + 106443q^-1 + 203085 + 106443q - 91032q² - 201061q³ - 119974q⁴ + 73319q⁵ + 194854q⁶ + 131586q⁷ - 53108q⁸ - 183710q⁹ - 140607q¹⁰ + 30435q¹¹ + 167068q¹² + 145775q¹³ - 6338q¹⁴ - 144641q¹⁵ - 145636q¹⁶ - 17294q¹⁷ + 117296q¹⁸ + 138741q¹⁹ + 37910q²⁰ - 86597q²¹ - 125046q²² - 52997q²³ + 55924q²⁴ + 105028q²⁵ + 60590q²⁶ - 27953q²⁷ - 81348q²⁸ - 60539q²⁹ + 6187q³⁰ + 56888q³¹ + 53555q³² + 8400q³³ - 34960q³⁴ - 42477q³⁵ - 15371q³⁶ + 17815q³⁷ + 29950q³⁸ + 16455q³⁹ - 6376q⁴⁰ - 18660q⁴¹ - 13769q⁴² + 71q⁴³ + 10091q⁴⁴ + 9758q⁴⁵ + 2283q⁴⁶ - 4569q⁴⁷ - 5861q⁴⁸ - 2527q⁴⁹ + 1551q⁵⁰ + 3134q⁵¹ + 1822q⁵² - 320q⁵³ - 1392q⁵⁴ - 1040q⁵⁵ - 109q⁵⁶ + 543q⁵⁷ + 531q⁵⁸ + 113q⁵⁹ - 197q⁶⁰ - 199q⁶¹ - 59q⁶² + 33q⁶³ + 84q⁶⁴ + 40q⁶⁵ - 34q⁶⁶ - 21q⁶⁷ + 3q⁶⁸ - 2q⁶⁹ + 4q⁷⁰ + 9q⁷¹ - 7q⁷² - 3q⁷³ + 4q⁷⁴ - q⁷⁵
6	q^-105 - 4q^-104 + 3q^-103 + 7q^-102 - 9q^-101 - 4q^-100 - 3q^-99 + 22q^-98 - 10q^-97 - 2q^-96 + 44q^-95 - 68q^-94 - 58q^-93 - 20q^-92 + 145q^-91 + 95q^-90 + 46q^-89 + 125q^-88 - 408q^-87 - 531q^-86 - 317q^-85 + 615q^-84 + 967q^-83 + 1024q^-82 + 876q^-81 - 1619q^-80 - 3217q^-79 - 3248q^-78 + 420q^-77 + 4033q^-76 + 6961q^-75 + 7385q^-74 - 1329q^-73 - 11203q^-72 - 17531q^-71 - 10354q^-70 + 4368q^-69 + 23577q^-68 + 35863q^-67 + 19355q^-66 - 14959q^-65 - 52800q^-64 - 59173q^-63 - 30996q^-62 + 33402q^-61 + 99926q^-60 + 104116q^-59 + 38455q^-58 - 79161q^-57 - 162940q^-56 - 165555q^-55 - 42543q^-54 + 151254q^-53 + 271312q^-52 + 233769q^-51 + 13377q^-50 - 249520q^-49 - 414572q^-48 - 308643q^-47 + 46760q^-46 + 420480q^-45 + 576578q^-44 + 343984q^-43 - 145228q^-42 - 645910q^-41 - 753595q^-40 - 338824q^-39 + 352823q^-38 + 902403q^-37 + 878082q^-36 + 270412q^-35 - 643372q^-34 - 1182741q^-33 - 942447q^-32 - 39231q^-31 + 986653q^-30 + 1396470q^-29 + 904710q^-28 - 317470q^-27 - 1371425q^-26 - 1527787q^-25 - 639520q^-24 + 759338q^-23 + 1682298q^-22 + 1515191q^-21 + 201994q^-20 - 1266624q^-19 - 1892397q^-18 - 1214631q^-17 + 352347q^-16 + 1696603q^-15 + 1921584q^-14 + 704085q^-13 - 991256q^-12 - 2009256q^-11 - 1615471q^-10 - 55878q^-9 + 1547088q^-8 + 2110070q^-7 + 1076094q^-6 - 686020q^-5 - 1969547q^-4 - 1844533q^-3 - 390299q^-2 + 1337527q^-1 + 2158285 + 1337527q - 390299q² - 1844533q³ - 1969547q⁴ - 686020q⁵ + 1076094q⁶ + 2110070q⁷ + 1547088q⁸ - 55878q⁹ - 1615471q¹⁰ - 2009256q¹¹ - 991256q¹² + 704085q¹³ + 1921584q¹⁴ + 1696603q¹⁵ + 352347q¹⁶ - 1214631q¹⁷ - 1892397q¹⁸ - 1266624q¹⁹ + 201994q²⁰ + 1515191q²¹ + 1682298q²² + 759338q²³ - 639520q²⁴ - 1527787q²⁵ - 1371425q²⁶ - 317470q²⁷ + 904710q²⁸ + 1396470q²⁹ + 986653q³⁰ - 39231q³¹ - 942447q³² - 1182741q³³ - 643372q³⁴ + 270412q³⁵ + 878082q³⁶ + 902403q³⁷ + 352823q³⁸ - 338824q³⁹ - 753595q⁴⁰ - 645910q⁴¹ - 145228q⁴² + 343984q⁴³ + 576578q⁴⁴ + 420480q⁴⁵ + 46760q⁴⁶ - 308643q⁴⁷ - 414572q⁴⁸ - 249520q⁴⁹ + 13377q⁵⁰ + 233769q⁵¹ + 271312q⁵² + 151254q⁵³ - 42543q⁵⁴ - 165555q⁵⁵ - 162940q⁵⁶ - 79161q⁵⁷ + 38455q⁵⁸ + 104116q⁵⁹ + 99926q⁶⁰ + 33402q⁶¹ - 30996q⁶² - 59173q⁶³ - 52800q⁶⁴ - 14959q⁶⁵ + 19355q⁶⁶ + 35863q⁶⁷ + 23577q⁶⁸ + 4368q⁶⁹ - 10354q⁷⁰ - 17531q⁷¹ - 11203q⁷² - 1329q⁷³ + 7385q⁷⁴ + 6961q⁷⁵ + 4033q⁷⁶ + 420q⁷⁷ - 3248q⁷⁸ - 3217q⁷⁹ - 1619q⁸⁰ + 876q⁸¹ + 1024q⁸² + 967q⁸³ + 615q⁸⁴ - 317q⁸⁵ - 531q⁸⁶ - 408q⁸⁷ + 125q⁸⁸ + 46q⁸⁹ + 95q⁹⁰ + 145q⁹¹ - 20q⁹² - 58q⁹³ - 68q⁹⁴ + 44q⁹⁵ - 2q⁹⁶ - 10q⁹⁷ + 22q⁹⁸ - 3q⁹⁹ - 4q¹⁰⁰ - 9q¹⁰¹ + 7q¹⁰² + 3q¹⁰³ - 4q¹⁰⁴ + q¹⁰⁵
7	- q^-140 + 4q^-139 - 3q^-138 - 7q^-137 + 9q^-136 + 4q^-135 + 3q^-134 - 17q^-133 - 15q^-132 + 33q^-131 - 8q^-130 - 16q^-129 + 42q^-128 + 30q^-127 + 13q^-126 - 121q^-125 - 168q^-124 + 67q^-123 + 72q^-122 + 153q^-121 + 345q^-120 + 206q^-119 + 28q^-118 - 744q^-117 - 1273q^-116 - 608q^-115 + 135q^-114 + 1493q^-113 + 2823q^-112 + 2540q^-111 + 1128q^-110 - 2911q^-109 - 7248q^-108 - 7552q^-107 - 4466q^-106 + 3929q^-105 + 14027q^-104 + 18818q^-103 + 16073q^-102 - 110q^-101 - 24199q^-100 - 41450q^-99 - 42981q^-98 - 16792q^-97 + 30828q^-96 + 75952q^-95 + 97722q^-94 + 66617q^-93 - 17353q^-92 - 118190q^-91 - 190921q^-90 - 173528q^-89 - 48185q^-88 + 141342q^-87 + 318844q^-86 + 366831q^-85 + 218101q^-84 - 93530q^-83 - 454180q^-82 - 657775q^-81 - 546567q^-80 - 108698q^-79 + 519225q^-78 + 1018197q^-77 + 1077733q^-76 + 567378q^-75 - 394097q^-74 - 1360104q^-73 - 1798579q^-72 - 1366682q^-71 - 78293q^-70 + 1520419q^-69 + 2618368q^-68 + 2535184q^-67 + 1042851q^-66 - 1289355q^-65 - 3349882q^-64 - 3995819q^-63 - 2580161q^-62 + 454017q^-61 + 3732829q^-60 + 5554146q^-59 + 4652596q^-58 + 1132131q^-57 - 3491647q^-56 - 6921359q^-55 - 7080887q^-54 - 3483098q^-53 + 2414596q^-52 + 7768501q^-51 + 9563059q^-50 + 6458604q^-49 - 417981q^-48 - 7822357q^-47 - 11746214q^-46 - 9770005q^-45 - 2404931q^-44 + 6931124q^-43 + 13296490q^-42 + 13053332q^-41 + 5814614q^-40 - 5113233q^-39 - 14002256q^-38 - 15949372q^-37 - 9459051q^-36 + 2549624q^-35 + 13795700q^-34 + 18181824q^-33 + 12981758q^-32 + 470475q^-31 - 12777028q^-30 - 19616672q^-29 - 16076808q^-28 - 3614657q^-27 + 11154661q^-26 + 20259777q^-25 + 18557545q^-24 + 6592876q^-23 - 9195152q^-22 - 20238112q^-21 - 20364308q^-20 - 9198979q^-19 + 7151404q^-18 + 19740523q^-17 + 21549728q^-16 + 11338004q^-15 - 5213041q^-14 - 18966439q^-13 - 22240226q^-12 - 13018279q^-11 + 3484032q^-10 + 18080389q^-9 + 22588382q^-8 + 14321596q^-7 - 1979218q^-6 - 17180052q^-5 - 22733021q^-4 - 15371342q^-3 + 639284q^-2 + 16291800q^-1 + 22770653 + 16291800q + 639284q² - 15371342q³ - 22733021q⁴ - 17180052q⁵ - 1979218q⁶ + 14321596q⁷ + 22588382q⁸ + 18080389q⁹ + 3484032q¹⁰ - 13018279q¹¹ - 22240226q¹² - 18966439q¹³ - 5213041q¹⁴ + 11338004q¹⁵ + 21549728q¹⁶ + 19740523q¹⁷ + 7151404q¹⁸ - 9198979q¹⁹ - 20364308q²⁰ - 20238112q²¹ - 9195152q²² + 6592876q²³ + 18557545q²⁴ + 20259777q²⁵ + 11154661q²⁶ - 3614657q²⁷ - 16076808q²⁸ - 19616672q²⁹ - 12777028q³⁰ + 470475q³¹ + 12981758q³² + 18181824q³³ + 13795700q³⁴ + 2549624q³⁵ - 9459051q³⁶ - 15949372q³⁷ - 14002256q³⁸ - 5113233q³⁹ + 5814614q⁴⁰ + 13053332q⁴¹ + 13296490q⁴² + 6931124q⁴³ - 2404931q⁴⁴ - 9770005q⁴⁵ - 11746214q⁴⁶ - 7822357q⁴⁷ - 417981q⁴⁸ + 6458604q⁴⁹ + 9563059q⁵⁰ + 7768501q⁵¹ + 2414596q⁵² - 3483098q⁵³ - 7080887q⁵⁴ - 6921359q⁵⁵ - 3491647q⁵⁶ + 1132131q⁵⁷ + 4652596q⁵⁸ + 5554146q⁵⁹ + 3732829q⁶⁰ + 454017q⁶¹ - 2580161q⁶² - 3995819q⁶³ - 3349882q⁶⁴ - 1289355q⁶⁵ + 1042851q⁶⁶ + 2535184q⁶⁷ + 2618368q⁶⁸ + 1520419q⁶⁹ - 78293q⁷⁰ - 1366682q⁷¹ - 1798579q⁷² - 1360104q⁷³ - 394097q⁷⁴ + 567378q⁷⁵ + 1077733q⁷⁶ + 1018197q⁷⁷ + 519225q⁷⁸ - 108698q⁷⁹ - 546567q⁸⁰ - 657775q⁸¹ - 454180q⁸² - 93530q⁸³ + 218101q⁸⁴ + 366831q⁸⁵ + 318844q⁸⁶ + 141342q⁸⁷ - 48185q⁸⁸ - 173528q⁸⁹ - 190921q⁹⁰ - 118190q⁹¹ - 17353q⁹² + 66617q⁹³ + 97722q⁹⁴ + 75952q⁹⁵ + 30828q⁹⁶ - 16792q⁹⁷ - 42981q⁹⁸ - 41450q⁹⁹ - 24199q¹⁰⁰ - 110q¹⁰¹ + 16073q¹⁰² + 18818q¹⁰³ + 14027q¹⁰⁴ + 3929q¹⁰⁵ - 4466q¹⁰⁶ - 7552q¹⁰⁷ - 7248q¹⁰⁸ - 2911q¹⁰⁹ + 1128q¹¹⁰ + 2540q¹¹¹ + 2823q¹¹² + 1493q¹¹³ + 135q¹¹⁴ - 608q¹¹⁵ - 1273q¹¹⁶ - 744q¹¹⁷ + 28q¹¹⁸ + 206q¹¹⁹ + 345q¹²⁰ + 153q¹²¹ + 72q¹²² + 67q¹²³ - 168q¹²⁴ - 121q¹²⁵ + 13q¹²⁶ + 30q¹²⁷ + 42q¹²⁸ - 16q¹²⁹ - 8q¹³⁰ + 33q¹³¹ - 15q¹³² - 17q¹³³ + 3q¹³⁴ + 4q¹³⁵ + 9q¹³⁶ - 7q¹³⁷ - 3q¹³⁸ + 4q¹³⁹ - q¹⁴⁰

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	PD[Knot[10, 88]]
Out[2]=	PD[X[4, 2, 5, 1], X[20, 14, 1, 13], X[8, 3, 9, 4], X[2, 9, 3, 10], > X[14, 7, 15, 8], X[18, 15, 19, 16], X[12, 6, 13, 5], X[10, 18, 11, 17], > X[16, 12, 17, 11], X[6, 19, 7, 20]]
In[3]:=	GaussCode[Knot[10, 88]]
Out[3]=	GaussCode[1, -4, 3, -1, 7, -10, 5, -3, 4, -8, 9, -7, 2, -5, 6, -9, 8, -6, 10, > -2]
In[4]:=	DTCode[Knot[10, 88]]
Out[4]=	DTCode[4, 8, 12, 14, 2, 16, 20, 18, 10, 6]
In[5]:=	br = BR[Knot[10, 88]]
Out[5]=	BR[5, {-1, 2, -1, -3, 2, -3, 2, 4, -3, 4}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{5, 10}
In[7]:=	BraidIndex[Knot[10, 88]]
Out[7]=	5
In[8]:=	Show[DrawMorseLink[Knot[10, 88]]]

Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 88]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{NegativeAmphicheiral, 1, 3, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 88]][t]
Out[10]=	-3 8 24 2 3 35 - t + -- - -- - 24 t + 8 t - t 2 t t
In[11]:=	Conway[Knot[10, 88]][z]
Out[11]=	2 4 6 1 - z + 2 z - z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 88]}
In[13]:=	{KnotDet[Knot[10, 88]], KnotSignature[Knot[10, 88]]}
Out[13]=	{101, 0}
In[14]:=	Jones[Knot[10, 88]][q]
Out[14]=	-5 4 8 13 16 2 3 4 5 17 - q + -- - -- + -- - -- - 16 q + 13 q - 8 q + 4 q - q 4 3 2 q q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 88]}
In[16]:=	A2Invariant[Knot[10, 88]][q]
Out[16]=	-16 -14 2 3 3 2 3 2 4 8 10 -3 - q + q + --- - --- + -- - -- + -- + 3 q - 2 q + 3 q - 3 q + 12 10 8 4 2 q q q q q 12 14 16 > 2 q + q - q
In[17]:=	HOMFLYPT[Knot[10, 88]][a, z]
Out[17]=	2 2 4 -2 2 2 z 2 z 2 2 4 2 4 2 z 2 4 6 -1 + a + a - 3 z - -- + ---- + 2 a z - a z - 2 z + ---- + 2 a z - z 4 2 2 a a a
In[18]:=	Kauffman[Knot[10, 88]][a, z]
Out[18]=	2 2 -2 2 z 4 z 3 2 3 z 7 z 2 2 -1 - a - a - -- - --- - 4 a z - a z + 8 z + ---- + ---- + 7 a z + 3 a 4 2 a a a 3 3 3 4 4 2 z 6 z 19 z 3 3 3 5 3 4 6 z > 3 a z - -- + ---- + ----- + 19 a z + 6 a z - a z - 8 z - ---- - 5 3 a 4 a a a 4 5 5 5 10 z 2 4 4 4 z 11 z 32 z 5 3 5 > ----- - 10 a z - 6 a z + -- - ----- - ----- - 32 a z - 11 a z + 2 5 3 a a a a 6 6 7 7 5 5 6 4 z 2 z 2 6 4 6 7 z 14 z 7 > a z - 12 z + ---- - ---- - 2 a z + 4 a z + ---- + ----- + 14 a z + 4 2 3 a a a a 8 9 3 7 8 6 z 2 8 2 z 9 > 7 a z + 12 z + ---- + 6 a z + ---- + 2 a z 2 a a
In[19]:=	{Vassiliev[2][Knot[10, 88]], Vassiliev[3][Knot[10, 88]]}
Out[19]=	{-1, 0}
In[20]:=	Kh[Knot[10, 88]][q, t]
Out[20]=	9 1 3 1 5 3 8 5 8 8 - + 9 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ---- + --- + q 11 5 9 4 7 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t q t q t 3 3 2 5 2 5 3 7 3 7 4 9 4 > 8 q t + 8 q t + 5 q t + 8 q t + 3 q t + 5 q t + q t + 3 q t + 11 5 > q t
In[21]:=	ColouredJones[Knot[10, 88], 2][q]
Out[21]=	-15 4 3 12 29 7 59 84 12 150 138 65 275 + q - --- + --- + --- - --- + --- + -- - -- - -- + --- - --- - -- + 14 13 12 11 10 9 8 7 6 5 4 q q q q q q q q q q q 239 153 123 2 3 4 5 6 > --- - --- - --- - 123 q - 153 q + 239 q - 65 q - 138 q + 150 q - 3 2 q q q 7 8 9 10 11 12 13 14 15 > 12 q - 84 q + 59 q + 7 q - 29 q + 12 q + 3 q - 4 q + q

The Alternating Knot 1088

The Alternating Knot 10₈₈