The Alternating Knot 10₈₉

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Acknowledgement

KnotPlot

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

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

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

Minimum Braid Representative:

Length is 12, 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

Reversible 2 3 3 / NotAvailable 1

Alexander Polynomial: t^-3 - 8t^-2 + 24t^-1 - 33 + 24t - 8t² + t³

Conway Polynomial: 1 + z² - 2z⁴ + z⁶

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

Determinant and Signature: {99, -2}

Jones Polynomial: - q^-8 + 3q^-7 - 7q^-6 + 12q^-5 - 15q^-4 + 17q^-3 - 16q^-2 + 13q^-1 - 9 + 5q - q²

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

A2 (sl(3)) Invariant: - q^-26 - q^-24 + 2q^-22 - q^-20 - q^-18 + 4q^-16 - 2q^-14 + 2q^-12 - 2q^-8 + 2q^-6 - 4q^-4 + 4q^-2 - q² + 3q⁴ - q⁶

HOMFLY-PT Polynomial: 1 - z⁴ + 2a²z² + 2a²z⁴ + a²z⁶ - a⁴ - 4a⁴z² - 3a⁴z⁴ + 2a⁶ + 3a⁶z² - a⁸

Kauffman Polynomial: a^-1z⁵ + 1 - 6z⁴ + 5z⁶ + 5az³ - 15az⁵ + 9az⁷ + 3a²z² - 9a²z⁴ - 4a²z⁶ + 7a²z⁸ - 2a³z + 19a³z³ - 35a³z⁵ + 15a³z⁷ + 2a³z⁹ - a⁴ + 6a⁴z² - 2a⁴z⁴ - 15a⁴z⁶ + 12a⁴z⁸ - 4a⁵z + 20a⁵z³ - 27a⁵z⁵ + 11a⁵z⁷ + 2a⁵z⁹ - 2a⁶ + 6a⁶z² - 4a⁶z⁴ - 3a⁶z⁶ + 5a⁶z⁸ - a⁷z + 4a⁷z³ - 7a⁷z⁵ + 5a⁷z⁷ - a⁸ + 3a⁸z² - 5a⁸z⁴ + 3a⁸z⁶ + a⁹z - 2a⁹z³ + a⁹z⁵

V₂ and V₃, the type 2 and 3 Vassiliev invariants: {1, -3}

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

t^rq^j r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3

j = 5 1

j = 3 4

j = 1 5 1

j = -1 8 4

j = -3 9 6

j = -5 8 7

j = -7 7 9

j = -9 5 8

j = -11 2 7

j = -13 1 5

j = -15 2

j = -17 1

n Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q^-23 - 3q^-22 + 2q^-21 + 8q^-20 - 21q^-19 + 9q^-18 + 40q^-17 - 71q^-16 + 7q^-15 + 116q^-14 - 136q^-13 - 26q^-12 + 207q^-11 - 171q^-10 - 77q^-9 + 258q^-8 - 155q^-7 - 114q^-6 + 241q^-5 - 99q^-4 - 119q^-3 + 167q^-2 - 35q^-1 - 86 + 76q + q² - 36q³ + 17q⁴ + 4q⁵ - 5q⁶ + q⁷

3 - q^-45 + 3q^-44 - 2q^-43 - 3q^-42 + q^-41 + 14q^-40 - 10q^-39 - 31q^-38 + 20q^-37 + 75q^-36 - 35q^-35 - 153q^-34 + 28q^-33 + 292q^-32 + 6q^-31 - 467q^-30 - 114q^-29 + 673q^-28 + 307q^-27 - 881q^-26 - 562q^-25 + 1026q^-24 + 885q^-23 - 1117q^-22 - 1204q^-21 + 1114q^-20 + 1507q^-19 - 1053q^-18 - 1730q^-17 + 910q^-16 + 1896q^-15 - 741q^-14 - 1957q^-13 + 524q^-12 + 1939q^-11 - 294q^-10 - 1827q^-9 + 57q^-8 + 1633q^-7 + 159q^-6 - 1368q^-5 - 325q^-4 + 1062q^-3 + 411q^-2 - 736q^-1 - 432 + 461q + 368q² - 236q³ - 277q⁴ + 101q⁵ + 167q⁶ - 18q⁷ - 97q⁸ + 3q⁹ + 36q¹⁰ + 6q¹¹ - 12q¹² - 4q¹³ + 5q¹⁴ - q¹⁵

4 q^-74 - 3q^-73 + 2q^-72 + 3q^-71 - 6q^-70 + 6q^-69 - 13q^-68 + 16q^-67 + 20q^-66 - 45q^-65 - 40q^-63 + 106q^-62 + 130q^-61 - 181q^-60 - 161q^-59 - 218q^-58 + 419q^-57 + 695q^-56 - 268q^-55 - 774q^-54 - 1186q^-53 + 751q^-52 + 2326q^-51 + 594q^-50 - 1584q^-49 - 3859q^-48 - 128q^-47 + 4756q^-46 + 3577q^-45 - 1046q^-44 - 7924q^-43 - 3589q^-42 + 6165q^-41 + 8297q^-40 + 2290q^-39 - 11306q^-38 - 9075q^-37 + 4913q^-36 + 12568q^-35 + 7749q^-34 - 12208q^-33 - 14273q^-32 + 1434q^-31 + 14582q^-30 + 13103q^-29 - 10699q^-28 - 17371q^-27 - 2636q^-26 + 14189q^-25 + 16767q^-24 - 7773q^-23 - 18027q^-22 - 6245q^-21 + 11911q^-20 + 18345q^-19 - 3972q^-18 - 16384q^-17 - 9072q^-16 + 7990q^-15 + 17662q^-14 + 285q^-13 - 12452q^-12 - 10439q^-11 + 2985q^-10 + 14355q^-9 + 3737q^-8 - 6894q^-7 - 9359q^-6 - 1392q^-5 + 9022q^-4 + 4781q^-3 - 1766q^-2 - 6045q^-1 - 3238 + 3825q + 3370q² + 888q³ - 2490q⁴ - 2544q⁵ + 809q⁶ + 1336q⁷ + 1097q⁸ - 484q⁹ - 1109q¹⁰ - 53q¹¹ + 224q¹² + 480q¹³ + 34q¹⁴ - 278q¹⁵ - 56q¹⁶ - 21q¹⁷ + 104q¹⁸ + 33q¹⁹ - 40q²⁰ - 6q²¹ - 11q²² + 12q²³ + 4q²⁴ - 5q²⁵ + q²⁶

5 - q^-110 + 3q^-109 - 2q^-108 - 3q^-107 + 6q^-106 - q^-105 - 7q^-104 + 7q^-103 - 5q^-102 - 10q^-101 + 32q^-100 + 18q^-99 - 38q^-98 - 40q^-97 - 49q^-96 + 12q^-95 + 183q^-94 + 202q^-93 - 67q^-92 - 404q^-91 - 522q^-90 - 107q^-89 + 827q^-88 + 1359q^-87 + 601q^-86 - 1305q^-85 - 2871q^-84 - 2082q^-83 + 1500q^-82 + 5264q^-81 + 5144q^-80 - 497q^-79 - 8233q^-78 - 10599q^-77 - 2801q^-76 + 10815q^-75 + 18409q^-74 + 10043q^-73 - 11294q^-72 - 28159q^-71 - 21982q^-70 + 7668q^-69 + 37774q^-68 + 38711q^-67 + 2139q^-66 - 44958q^-65 - 58903q^-64 - 18807q^-63 + 47105q^-62 + 79782q^-61 + 41861q^-60 - 42121q^-59 - 98701q^-58 - 69237q^-57 + 30011q^-56 + 112623q^-55 + 97755q^-54 - 11282q^-53 - 120244q^-52 - 124675q^-51 - 11230q^-50 + 121050q^-49 + 147300q^-48 + 35395q^-47 - 116191q^-46 - 164552q^-45 - 58440q^-44 + 106998q^-43 + 176022q^-42 + 79292q^-41 - 95395q^-40 - 182393q^-39 - 96779q^-38 + 82072q^-37 + 184355q^-36 + 111567q^-35 - 67920q^-34 - 182731q^-33 - 123496q^-32 + 52498q^-31 + 177419q^-30 + 133463q^-29 - 35656q^-28 - 168406q^-27 - 140939q^-26 + 17071q^-25 + 154852q^-24 + 145474q^-23 + 2968q^-22 - 136469q^-21 - 145828q^-20 - 23266q^-19 + 113285q^-18 + 140724q^-17 + 41963q^-16 - 86346q^-15 - 129342q^-14 - 56827q^-13 + 57841q^-12 + 111902q^-11 + 65579q^-10 - 30559q^-9 - 89671q^-8 - 67235q^-7 + 7408q^-6 + 65515q^-5 + 61828q^-4 + 9165q^-3 - 42095q^-2 - 51260q^-1 - 18483 + 22598q + 37988q² + 21019q³ - 8346q⁴ - 25002q⁵ - 18809q⁶ + 18q⁷ + 14066q⁸ + 14187q⁹ + 3800q¹⁰ - 6649q¹¹ - 9204q¹² - 4223q¹³ + 2143q¹⁴ + 5070q¹⁵ + 3485q¹⁶ - 222q¹⁷ - 2458q¹⁸ - 2084q¹⁹ - 429q²⁰ + 895q²¹ + 1152q²² + 449q²³ - 319q²⁴ - 490q²⁵ - 254q²⁶ + 54q²⁷ + 171q²⁸ + 140q²⁹ + 7q³⁰ - 80q³¹ - 40q³² + 4q³³ + 10q³⁴ + 11q³⁵ + 11q³⁶ - 12q³⁷ - 4q³⁸ + 5q³⁹ - q⁴⁰

6 q^-153 - 3q^-152 + 2q^-151 + 3q^-150 - 6q^-149 + q^-148 + 2q^-147 + 13q^-146 - 18q^-145 - 5q^-144 + 23q^-143 - 35q^-142 - 2q^-141 + 29q^-140 + 84q^-139 - 39q^-138 - 80q^-137 + 6q^-136 - 183q^-135 - 39q^-134 + 222q^-133 + 538q^-132 + 163q^-131 - 299q^-130 - 474q^-129 - 1264q^-128 - 723q^-127 + 757q^-126 + 2778q^-125 + 2561q^-124 + 633q^-123 - 1947q^-122 - 6486q^-121 - 6551q^-120 - 1409q^-119 + 8319q^-118 + 13858q^-117 + 11931q^-116 + 1825q^-115 - 18510q^-114 - 30296q^-113 - 23995q^-112 + 5490q^-111 + 37906q^-110 + 55570q^-109 + 41676q^-108 - 16054q^-107 - 76580q^-106 - 101709q^-105 - 54925q^-104 + 38741q^-103 + 135206q^-102 + 167164q^-101 + 74382q^-100 - 89280q^-99 - 234071q^-98 - 237747q^-97 - 85271q^-96 + 169279q^-95 + 368560q^-94 + 330955q^-93 + 61894q^-92 - 309994q^-91 - 518360q^-90 - 419792q^-89 + 1901q^-88 + 503434q^-87 + 708363q^-86 + 461572q^-85 - 154155q^-84 - 724847q^-83 - 893256q^-82 - 441190q^-81 + 385642q^-80 + 1008702q^-79 + 1009106q^-78 + 290363q^-77 - 669820q^-76 - 1287386q^-75 - 1031393q^-74 - 20239q^-73 + 1049771q^-72 + 1474204q^-71 + 873701q^-70 - 339098q^-69 - 1430264q^-68 - 1536799q^-67 - 553067q^-66 + 832731q^-65 + 1700374q^-64 + 1374897q^-63 + 107102q^-62 - 1332607q^-61 - 1821200q^-60 - 1014325q^-59 + 504094q^-58 + 1704725q^-57 + 1683461q^-56 + 502100q^-55 - 1120225q^-54 - 1907246q^-53 - 1322990q^-52 + 192914q^-51 + 1591422q^-50 + 1828892q^-49 + 795906q^-48 - 885422q^-47 - 1879548q^-46 - 1516518q^-45 - 80945q^-44 + 1420734q^-43 + 1879600q^-42 + 1033995q^-41 - 622190q^-40 - 1770803q^-39 - 1649075q^-38 - 369544q^-37 + 1164970q^-36 + 1843110q^-35 + 1256130q^-34 - 274757q^-33 - 1533185q^-32 - 1702203q^-31 - 695670q^-30 + 766439q^-29 + 1649785q^-28 + 1414442q^-27 + 159076q^-26 - 1106663q^-25 - 1580502q^-24 - 976920q^-23 + 245365q^-22 + 1234571q^-21 + 1383990q^-20 + 559209q^-19 - 533358q^-18 - 1211702q^-17 - 1056845q^-16 - 241986q^-15 + 658679q^-14 + 1085958q^-13 + 741503q^-12 + 85q^-11 - 674173q^-10 - 853697q^-9 - 493832q^-8 + 130178q^-7 + 615513q^-6 + 629512q^-5 + 287272q^-4 - 188657q^-3 - 478721q^-2 - 449328q^-1 - 153211 + 197545q + 350261q² + 290494q³ + 65923q⁴ - 151498q⁵ - 249195q⁶ - 179840q⁷ - 14670q⁸ + 110029q⁹ + 157541q¹⁰ + 101533q¹¹ + 4575q¹² - 79513q¹³ - 95331q¹⁴ - 50689q¹⁵ + 2707q¹⁶ + 47384q¹⁷ + 52006q¹⁸ + 29398q¹⁹ - 7712q²⁰ - 27520q²¹ - 25068q²² - 13711q²³ + 4612q²⁴ + 13763q²⁵ + 14196q²⁶ + 4221q²⁷ - 3275q²⁸ - 5872q²⁹ - 6117q³⁰ - 1862q³¹ + 1467q³² + 3589q³³ + 1823q³⁴ + 358q³⁵ - 492q³⁶ - 1315q³⁷ - 786q³⁸ - 162q³⁹ + 592q⁴⁰ + 285q⁴¹ + 159q⁴² + 70q⁴³ - 166q⁴⁴ - 130q⁴⁵ - 76q⁴⁶ + 94q⁴⁷ + 16q⁴⁸ + 3q⁴⁹ + 26q⁵⁰ - 15q⁵¹ - 11q⁵² - 11q⁵³ + 12q⁵⁴ + 4q⁵⁵ - 5q⁵⁶ + q⁵⁷

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, 89]]

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

In[3]:=
GaussCode[Knot[10, 89]]

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

In[4]:=
DTCode[Knot[10, 89]]

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

In[5]:=
br = BR[Knot[10, 89]]

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

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

Out[6]=
{5, 12}

In[7]:=
BraidIndex[Knot[10, 89]]

Out[7]=
5

In[8]:=
Show[DrawMorseLink[Knot[10, 89]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[10, 89]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

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

In[10]:=
alex = Alexander[Knot[10, 89]][t]

Out[10]=
-3 8 24 2 3 -33 + t - -- + -- + 24 t - 8 t + t 2 t t

In[11]:=
Conway[Knot[10, 89]][z]

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

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

Out[12]=
{Knot[10, 89]}

In[13]:=
{KnotDet[Knot[10, 89]], KnotSignature[Knot[10, 89]]}

Out[13]=
{99, -2}

In[14]:=
Jones[Knot[10, 89]][q]

Out[14]=
-8 3 7 12 15 17 16 13 2 -9 - q + -- - -- + -- - -- + -- - -- + -- + 5 q - q 7 6 5 4 3 2 q q q q q q q

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

Out[15]=
{Knot[10, 89]}

In[16]:=
A2Invariant[Knot[10, 89]][q]

Out[16]=
-26 -24 2 -20 -18 4 2 2 2 2 4 4 2 -q - q + --- - q - q + --- - --- + --- - -- + -- - -- + -- - q + 22 16 14 12 8 6 4 2 q q q q q q q q 4 6 > 3 q - q

In[17]:=
HOMFLYPT[Knot[10, 89]][a, z]

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

In[18]:=
Kauffman[Knot[10, 89]][a, z]

Out[18]=
4 6 8 3 5 7 9 2 2 4 2 1 - a - 2 a - a - 2 a z - 4 a z - a z + a z + 3 a z + 6 a z + 6 2 8 2 3 3 3 5 3 7 3 9 3 > 6 a z + 3 a z + 5 a z + 19 a z + 20 a z + 4 a z - 2 a z - 5 4 2 4 4 4 6 4 8 4 z 5 3 5 > 6 z - 9 a z - 2 a z - 4 a z - 5 a z + -- - 15 a z - 35 a z - a 5 5 7 5 9 5 6 2 6 4 6 6 6 > 27 a z - 7 a z + a z + 5 z - 4 a z - 15 a z - 3 a z + 8 6 7 3 7 5 7 7 7 2 8 4 8 > 3 a z + 9 a z + 15 a z + 11 a z + 5 a z + 7 a z + 12 a z + 6 8 3 9 5 9 > 5 a z + 2 a z + 2 a z

In[19]:=
{Vassiliev[2][Knot[10, 89]], Vassiliev[3][Knot[10, 89]]}

Out[19]=
{1, -3}

In[20]:=
Kh[Knot[10, 89]][q, t]

Out[20]=
6 8 1 2 1 5 2 7 5 8 -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 3 q 17 7 15 6 13 6 13 5 11 5 11 4 9 4 9 3 q q t q t q t q t q t q t q t q t 7 9 8 7 9 4 t 2 3 2 5 3 > ----- + ----- + ----- + ---- + ---- + --- + 5 q t + q t + 4 q t + q t 7 3 7 2 5 2 5 3 q q t q t q t q t q t

In[21]:=
ColouredJones[Knot[10, 89], 2][q]

Out[21]=
-23 3 2 8 21 9 40 71 7 116 136 26 -86 + q - --- + --- + --- - --- + --- + --- - --- + --- + --- - --- - --- + 22 21 20 19 18 17 16 15 14 13 12 q q q q q q q q q q q 207 171 77 258 155 114 241 99 119 167 35 2 > --- - --- - -- + --- - --- - --- + --- - -- - --- + --- - -- + 76 q + q - 11 10 9 8 7 6 5 4 3 2 q q q q q q q q q q q 3 4 5 6 7 > 36 q + 17 q + 4 q - 5 q + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 10₈₉

10₈₈
10₉₀

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-23 - 3q^-22 + 2q^-21 + 8q^-20 - 21q^-19 + 9q^-18 + 40q^-17 - 71q^-16 + 7q^-15 + 116q^-14 - 136q^-13 - 26q^-12 + 207q^-11 - 171q^-10 - 77q^-9 + 258q^-8 - 155q^-7 - 114q^-6 + 241q^-5 - 99q^-4 - 119q^-3 + 167q^-2 - 35q^-1 - 86 + 76q + q² - 36q³ + 17q⁴ + 4q⁵ - 5q⁶ + q⁷
3	- q^-45 + 3q^-44 - 2q^-43 - 3q^-42 + q^-41 + 14q^-40 - 10q^-39 - 31q^-38 + 20q^-37 + 75q^-36 - 35q^-35 - 153q^-34 + 28q^-33 + 292q^-32 + 6q^-31 - 467q^-30 - 114q^-29 + 673q^-28 + 307q^-27 - 881q^-26 - 562q^-25 + 1026q^-24 + 885q^-23 - 1117q^-22 - 1204q^-21 + 1114q^-20 + 1507q^-19 - 1053q^-18 - 1730q^-17 + 910q^-16 + 1896q^-15 - 741q^-14 - 1957q^-13 + 524q^-12 + 1939q^-11 - 294q^-10 - 1827q^-9 + 57q^-8 + 1633q^-7 + 159q^-6 - 1368q^-5 - 325q^-4 + 1062q^-3 + 411q^-2 - 736q^-1 - 432 + 461q + 368q² - 236q³ - 277q⁴ + 101q⁵ + 167q⁶ - 18q⁷ - 97q⁸ + 3q⁹ + 36q¹⁰ + 6q¹¹ - 12q¹² - 4q¹³ + 5q¹⁴ - q¹⁵
4	q^-74 - 3q^-73 + 2q^-72 + 3q^-71 - 6q^-70 + 6q^-69 - 13q^-68 + 16q^-67 + 20q^-66 - 45q^-65 - 40q^-63 + 106q^-62 + 130q^-61 - 181q^-60 - 161q^-59 - 218q^-58 + 419q^-57 + 695q^-56 - 268q^-55 - 774q^-54 - 1186q^-53 + 751q^-52 + 2326q^-51 + 594q^-50 - 1584q^-49 - 3859q^-48 - 128q^-47 + 4756q^-46 + 3577q^-45 - 1046q^-44 - 7924q^-43 - 3589q^-42 + 6165q^-41 + 8297q^-40 + 2290q^-39 - 11306q^-38 - 9075q^-37 + 4913q^-36 + 12568q^-35 + 7749q^-34 - 12208q^-33 - 14273q^-32 + 1434q^-31 + 14582q^-30 + 13103q^-29 - 10699q^-28 - 17371q^-27 - 2636q^-26 + 14189q^-25 + 16767q^-24 - 7773q^-23 - 18027q^-22 - 6245q^-21 + 11911q^-20 + 18345q^-19 - 3972q^-18 - 16384q^-17 - 9072q^-16 + 7990q^-15 + 17662q^-14 + 285q^-13 - 12452q^-12 - 10439q^-11 + 2985q^-10 + 14355q^-9 + 3737q^-8 - 6894q^-7 - 9359q^-6 - 1392q^-5 + 9022q^-4 + 4781q^-3 - 1766q^-2 - 6045q^-1 - 3238 + 3825q + 3370q² + 888q³ - 2490q⁴ - 2544q⁵ + 809q⁶ + 1336q⁷ + 1097q⁸ - 484q⁹ - 1109q¹⁰ - 53q¹¹ + 224q¹² + 480q¹³ + 34q¹⁴ - 278q¹⁵ - 56q¹⁶ - 21q¹⁷ + 104q¹⁸ + 33q¹⁹ - 40q²⁰ - 6q²¹ - 11q²² + 12q²³ + 4q²⁴ - 5q²⁵ + q²⁶
5	- q^-110 + 3q^-109 - 2q^-108 - 3q^-107 + 6q^-106 - q^-105 - 7q^-104 + 7q^-103 - 5q^-102 - 10q^-101 + 32q^-100 + 18q^-99 - 38q^-98 - 40q^-97 - 49q^-96 + 12q^-95 + 183q^-94 + 202q^-93 - 67q^-92 - 404q^-91 - 522q^-90 - 107q^-89 + 827q^-88 + 1359q^-87 + 601q^-86 - 1305q^-85 - 2871q^-84 - 2082q^-83 + 1500q^-82 + 5264q^-81 + 5144q^-80 - 497q^-79 - 8233q^-78 - 10599q^-77 - 2801q^-76 + 10815q^-75 + 18409q^-74 + 10043q^-73 - 11294q^-72 - 28159q^-71 - 21982q^-70 + 7668q^-69 + 37774q^-68 + 38711q^-67 + 2139q^-66 - 44958q^-65 - 58903q^-64 - 18807q^-63 + 47105q^-62 + 79782q^-61 + 41861q^-60 - 42121q^-59 - 98701q^-58 - 69237q^-57 + 30011q^-56 + 112623q^-55 + 97755q^-54 - 11282q^-53 - 120244q^-52 - 124675q^-51 - 11230q^-50 + 121050q^-49 + 147300q^-48 + 35395q^-47 - 116191q^-46 - 164552q^-45 - 58440q^-44 + 106998q^-43 + 176022q^-42 + 79292q^-41 - 95395q^-40 - 182393q^-39 - 96779q^-38 + 82072q^-37 + 184355q^-36 + 111567q^-35 - 67920q^-34 - 182731q^-33 - 123496q^-32 + 52498q^-31 + 177419q^-30 + 133463q^-29 - 35656q^-28 - 168406q^-27 - 140939q^-26 + 17071q^-25 + 154852q^-24 + 145474q^-23 + 2968q^-22 - 136469q^-21 - 145828q^-20 - 23266q^-19 + 113285q^-18 + 140724q^-17 + 41963q^-16 - 86346q^-15 - 129342q^-14 - 56827q^-13 + 57841q^-12 + 111902q^-11 + 65579q^-10 - 30559q^-9 - 89671q^-8 - 67235q^-7 + 7408q^-6 + 65515q^-5 + 61828q^-4 + 9165q^-3 - 42095q^-2 - 51260q^-1 - 18483 + 22598q + 37988q² + 21019q³ - 8346q⁴ - 25002q⁵ - 18809q⁶ + 18q⁷ + 14066q⁸ + 14187q⁹ + 3800q¹⁰ - 6649q¹¹ - 9204q¹² - 4223q¹³ + 2143q¹⁴ + 5070q¹⁵ + 3485q¹⁶ - 222q¹⁷ - 2458q¹⁸ - 2084q¹⁹ - 429q²⁰ + 895q²¹ + 1152q²² + 449q²³ - 319q²⁴ - 490q²⁵ - 254q²⁶ + 54q²⁷ + 171q²⁸ + 140q²⁹ + 7q³⁰ - 80q³¹ - 40q³² + 4q³³ + 10q³⁴ + 11q³⁵ + 11q³⁶ - 12q³⁷ - 4q³⁸ + 5q³⁹ - q⁴⁰
6	q^-153 - 3q^-152 + 2q^-151 + 3q^-150 - 6q^-149 + q^-148 + 2q^-147 + 13q^-146 - 18q^-145 - 5q^-144 + 23q^-143 - 35q^-142 - 2q^-141 + 29q^-140 + 84q^-139 - 39q^-138 - 80q^-137 + 6q^-136 - 183q^-135 - 39q^-134 + 222q^-133 + 538q^-132 + 163q^-131 - 299q^-130 - 474q^-129 - 1264q^-128 - 723q^-127 + 757q^-126 + 2778q^-125 + 2561q^-124 + 633q^-123 - 1947q^-122 - 6486q^-121 - 6551q^-120 - 1409q^-119 + 8319q^-118 + 13858q^-117 + 11931q^-116 + 1825q^-115 - 18510q^-114 - 30296q^-113 - 23995q^-112 + 5490q^-111 + 37906q^-110 + 55570q^-109 + 41676q^-108 - 16054q^-107 - 76580q^-106 - 101709q^-105 - 54925q^-104 + 38741q^-103 + 135206q^-102 + 167164q^-101 + 74382q^-100 - 89280q^-99 - 234071q^-98 - 237747q^-97 - 85271q^-96 + 169279q^-95 + 368560q^-94 + 330955q^-93 + 61894q^-92 - 309994q^-91 - 518360q^-90 - 419792q^-89 + 1901q^-88 + 503434q^-87 + 708363q^-86 + 461572q^-85 - 154155q^-84 - 724847q^-83 - 893256q^-82 - 441190q^-81 + 385642q^-80 + 1008702q^-79 + 1009106q^-78 + 290363q^-77 - 669820q^-76 - 1287386q^-75 - 1031393q^-74 - 20239q^-73 + 1049771q^-72 + 1474204q^-71 + 873701q^-70 - 339098q^-69 - 1430264q^-68 - 1536799q^-67 - 553067q^-66 + 832731q^-65 + 1700374q^-64 + 1374897q^-63 + 107102q^-62 - 1332607q^-61 - 1821200q^-60 - 1014325q^-59 + 504094q^-58 + 1704725q^-57 + 1683461q^-56 + 502100q^-55 - 1120225q^-54 - 1907246q^-53 - 1322990q^-52 + 192914q^-51 + 1591422q^-50 + 1828892q^-49 + 795906q^-48 - 885422q^-47 - 1879548q^-46 - 1516518q^-45 - 80945q^-44 + 1420734q^-43 + 1879600q^-42 + 1033995q^-41 - 622190q^-40 - 1770803q^-39 - 1649075q^-38 - 369544q^-37 + 1164970q^-36 + 1843110q^-35 + 1256130q^-34 - 274757q^-33 - 1533185q^-32 - 1702203q^-31 - 695670q^-30 + 766439q^-29 + 1649785q^-28 + 1414442q^-27 + 159076q^-26 - 1106663q^-25 - 1580502q^-24 - 976920q^-23 + 245365q^-22 + 1234571q^-21 + 1383990q^-20 + 559209q^-19 - 533358q^-18 - 1211702q^-17 - 1056845q^-16 - 241986q^-15 + 658679q^-14 + 1085958q^-13 + 741503q^-12 + 85q^-11 - 674173q^-10 - 853697q^-9 - 493832q^-8 + 130178q^-7 + 615513q^-6 + 629512q^-5 + 287272q^-4 - 188657q^-3 - 478721q^-2 - 449328q^-1 - 153211 + 197545q + 350261q² + 290494q³ + 65923q⁴ - 151498q⁵ - 249195q⁶ - 179840q⁷ - 14670q⁸ + 110029q⁹ + 157541q¹⁰ + 101533q¹¹ + 4575q¹² - 79513q¹³ - 95331q¹⁴ - 50689q¹⁵ + 2707q¹⁶ + 47384q¹⁷ + 52006q¹⁸ + 29398q¹⁹ - 7712q²⁰ - 27520q²¹ - 25068q²² - 13711q²³ + 4612q²⁴ + 13763q²⁵ + 14196q²⁶ + 4221q²⁷ - 3275q²⁸ - 5872q²⁹ - 6117q³⁰ - 1862q³¹ + 1467q³² + 3589q³³ + 1823q³⁴ + 358q³⁵ - 492q³⁶ - 1315q³⁷ - 786q³⁸ - 162q³⁹ + 592q⁴⁰ + 285q⁴¹ + 159q⁴² + 70q⁴³ - 166q⁴⁴ - 130q⁴⁵ - 76q⁴⁶ + 94q⁴⁷ + 16q⁴⁸ + 3q⁴⁹ + 26q⁵⁰ - 15q⁵¹ - 11q⁵² - 11q⁵³ + 12q⁵⁴ + 4q⁵⁵ - 5q⁵⁶ + q⁵⁷

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	PD[Knot[10, 89]]
Out[2]=	PD[X[4, 2, 5, 1], X[12, 8, 13, 7], X[8, 3, 9, 4], X[2, 9, 3, 10], > X[20, 13, 1, 14], X[14, 5, 15, 6], X[6, 19, 7, 20], X[18, 16, 19, 15], > X[16, 11, 17, 12], X[10, 17, 11, 18]]
In[3]:=	GaussCode[Knot[10, 89]]
Out[3]=	GaussCode[1, -4, 3, -1, 6, -7, 2, -3, 4, -10, 9, -2, 5, -6, 8, -9, 10, -8, 7, > -5]
In[4]:=	DTCode[Knot[10, 89]]
Out[4]=	DTCode[4, 8, 14, 12, 2, 16, 20, 18, 10, 6]
In[5]:=	br = BR[Knot[10, 89]]
Out[5]=	BR[5, {-1, 2, -1, 2, 3, -2, -1, -4, -3, 2, -3, -4}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{5, 12}
In[7]:=	BraidIndex[Knot[10, 89]]
Out[7]=	5
In[8]:=	Show[DrawMorseLink[Knot[10, 89]]]

Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 89]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 3, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 89]][t]
Out[10]=	-3 8 24 2 3 -33 + t - -- + -- + 24 t - 8 t + t 2 t t
In[11]:=	Conway[Knot[10, 89]][z]
Out[11]=	2 4 6 1 + z - 2 z + z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 89]}
In[13]:=	{KnotDet[Knot[10, 89]], KnotSignature[Knot[10, 89]]}
Out[13]=	{99, -2}
In[14]:=	Jones[Knot[10, 89]][q]
Out[14]=	-8 3 7 12 15 17 16 13 2 -9 - q + -- - -- + -- - -- + -- - -- + -- + 5 q - q 7 6 5 4 3 2 q q q q q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 89]}
In[16]:=	A2Invariant[Knot[10, 89]][q]
Out[16]=	-26 -24 2 -20 -18 4 2 2 2 2 4 4 2 -q - q + --- - q - q + --- - --- + --- - -- + -- - -- + -- - q + 22 16 14 12 8 6 4 2 q q q q q q q q 4 6 > 3 q - q
In[17]:=	HOMFLYPT[Knot[10, 89]][a, z]
Out[17]=	4 6 8 2 2 4 2 6 2 4 2 4 4 4 1 - a + 2 a - a + 2 a z - 4 a z + 3 a z - z + 2 a z - 3 a z + 2 6 > a z
In[18]:=	Kauffman[Knot[10, 89]][a, z]
Out[18]=	4 6 8 3 5 7 9 2 2 4 2 1 - a - 2 a - a - 2 a z - 4 a z - a z + a z + 3 a z + 6 a z + 6 2 8 2 3 3 3 5 3 7 3 9 3 > 6 a z + 3 a z + 5 a z + 19 a z + 20 a z + 4 a z - 2 a z - 5 4 2 4 4 4 6 4 8 4 z 5 3 5 > 6 z - 9 a z - 2 a z - 4 a z - 5 a z + -- - 15 a z - 35 a z - a 5 5 7 5 9 5 6 2 6 4 6 6 6 > 27 a z - 7 a z + a z + 5 z - 4 a z - 15 a z - 3 a z + 8 6 7 3 7 5 7 7 7 2 8 4 8 > 3 a z + 9 a z + 15 a z + 11 a z + 5 a z + 7 a z + 12 a z + 6 8 3 9 5 9 > 5 a z + 2 a z + 2 a z
In[19]:=	{Vassiliev[2][Knot[10, 89]], Vassiliev[3][Knot[10, 89]]}
Out[19]=	{1, -3}
In[20]:=	Kh[Knot[10, 89]][q, t]
Out[20]=	6 8 1 2 1 5 2 7 5 8 -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 3 q 17 7 15 6 13 6 13 5 11 5 11 4 9 4 9 3 q q t q t q t q t q t q t q t q t 7 9 8 7 9 4 t 2 3 2 5 3 > ----- + ----- + ----- + ---- + ---- + --- + 5 q t + q t + 4 q t + q t 7 3 7 2 5 2 5 3 q q t q t q t q t q t
In[21]:=	ColouredJones[Knot[10, 89], 2][q]
Out[21]=	-23 3 2 8 21 9 40 71 7 116 136 26 -86 + q - --- + --- + --- - --- + --- + --- - --- + --- + --- - --- - --- + 22 21 20 19 18 17 16 15 14 13 12 q q q q q q q q q q q 207 171 77 258 155 114 241 99 119 167 35 2 > --- - --- - -- + --- - --- - --- + --- - -- - --- + --- - -- + 76 q + q - 11 10 9 8 7 6 5 4 3 2 q q q q q q q q q q q 3 4 5 6 7 > 36 q + 17 q + 4 q - 5 q + q

The Alternating Knot 1089

The Alternating Knot 10₈₉