The Alternating Knot 10₁₁₉

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Acknowledgement

KnotPlot

PD Presentation: X₁₆₂₇ X_7,18,8,19 X₃₉₄₈ X_17,3,18,2 X_5,15,6,14 X_9,17,10,16 X_15,11,16,10 X_11,5,12,4 X_13,20,14,1 X_19,12,20,13

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

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

Minimum Braid Representative:

Length is 11, width is 4
Braid index is 4

A Morse Link Presentation:

3D Invariants:

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

Chiral 1 3 3 / NotAvailable 1

Alexander Polynomial: - 2t^-3 + 10t^-2 - 23t^-1 + 31 - 23t + 10t² - 2t³

Conway Polynomial: 1 - z² - 2z⁴ - 2z⁶

Other knots with the same Alexander/Conway Polynomial: {K11a84, ...}

Determinant and Signature: {101, 0}

Jones Polynomial: q^-4 - 4q^-3 + 9q^-2 - 13q^-1 + 16 - 17q + 16q² - 12q³ + 8q⁴ - 4q⁵ + q⁶

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

A2 (sl(3)) Invariant: q^-12 - 2q^-10 + 2q^-8 + 2q^-6 - 3q^-4 + 3q^-2 - 3 + q² + q⁴ - q⁶ + 4q⁸ - 3q¹⁰ + q¹² + q¹⁴ - 2q¹⁶ + q¹⁸

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

Kauffman Polynomial: a^-6z² - 2a^-6z⁴ + a^-6z⁶ - a^-5z + 7a^-5z³ - 10a^-5z⁵ + 4a^-5z⁷ + 8a^-4z⁴ - 14a^-4z⁶ + 6a^-4z⁸ - 3a^-3z + 19a^-3z³ - 26a^-3z⁵ + 5a^-3z⁷ + 3a^-3z⁹ - a^-2 + a^-2z² + 13a^-2z⁴ - 31a^-2z⁶ + 15a^-2z⁸ - 4a^-1z + 22a^-1z³ - 37a^-1z⁵ + 13a^-1z⁷ + 3a^-1z⁹ - 1 + 6z² - 7z⁴ - 7z⁶ + 9z⁸ - 2az + 9az³ - 17az⁵ + 12az⁷ - a² + 4a²z² - 9a²z⁴ + 9a²z⁶ - a³z³ + 4a³z⁵ + a⁴z⁴

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 10₁₁₉. 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 = 13 1

j = 11 3

j = 9 5 1

j = 7 7 3

j = 5 9 5

j = 3 8 7

j = 1 8 9

j = -1 6 9

j = -3 3 7

j = -5 1 6

j = -7 3

j = -9 1

n Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q^-12 - 4q^-11 + 5q^-10 + 7q^-9 - 30q^-8 + 27q^-7 + 34q^-6 - 99q^-5 + 56q^-4 + 97q^-3 - 185q^-2 + 59q^-1 + 173 - 232q + 26q² + 216q³ - 214q⁴ - 21q⁵ + 206q⁶ - 144q⁷ - 54q⁸ + 146q⁹ - 61q¹⁰ - 54q¹¹ + 67q¹² - 9q¹³ - 26q¹⁴ + 15q¹⁵ + 2q¹⁶ - 4q¹⁷ + q¹⁸

3 q^-24 - 4q^-23 + 5q^-22 + 3q^-21 - 10q^-20 - 9q^-19 + 28q^-18 + 23q^-17 - 74q^-16 - 45q^-15 + 151q^-14 + 100q^-13 - 261q^-12 - 227q^-11 + 414q^-10 + 417q^-9 - 547q^-8 - 701q^-7 + 645q^-6 + 1028q^-5 - 653q^-4 - 1370q^-3 + 584q^-2 + 1654q^-1 - 427 - 1864q + 229q² + 1962q³ + 5q⁴ - 1971q⁵ - 232q⁶ + 1881q⁷ + 458q⁸ - 1721q⁹ - 648q¹⁰ + 1476q¹¹ + 816q¹² - 1196q¹³ - 901q¹⁴ + 861q¹⁵ + 920q¹⁶ - 534q¹⁷ - 843q¹⁸ + 243q¹⁹ + 690q²⁰ - 27q²¹ - 495q²² - 95q²³ + 304q²⁴ + 128q²⁵ - 152q²⁶ - 105q²⁷ + 55q²⁸ + 67q²⁹ - 15q³⁰ - 29q³¹ + q³² + 9q³³ + 2q³⁴ - 4q³⁵ + q³⁶

4 q^-40 - 4q^-39 + 5q^-38 + 3q^-37 - 14q^-36 + 11q^-35 - 8q^-34 + 27q^-33 - 2q^-32 - 91q^-31 + 63q^-30 + 38q^-29 + 124q^-28 - 84q^-27 - 454q^-26 + 151q^-25 + 403q^-24 + 666q^-23 - 309q^-22 - 1752q^-21 - 242q^-20 + 1345q^-19 + 2698q^-18 + 81q^-17 - 4508q^-16 - 2602q^-15 + 1895q^-14 + 6895q^-13 + 2925q^-12 - 7376q^-11 - 7596q^-10 - 153q^-9 + 11442q^-8 + 8713q^-7 - 7653q^-6 - 12982q^-5 - 5243q^-4 + 13309q^-3 + 14854q^-2 - 4667q^-1 - 15659 - 10923q + 11734q² + 18417q³ - 359q⁴ - 14952q⁵ - 14791q⁶ + 8212q⁷ + 18858q⁸ + 3581q⁹ - 12057q¹⁰ - 16512q¹¹ + 3954q¹² + 17045q¹³ + 6885q¹⁴ - 7818q¹⁵ - 16431q¹⁶ - 726q¹⁷ + 13279q¹⁸ + 9273q¹⁹ - 2500q²⁰ - 14134q²¹ - 4976q²² + 7637q²³ + 9461q²⁴ + 2669q²⁵ - 9280q²⁶ - 6835q²⁷ + 1610q²⁸ + 6610q²⁹ + 5323q³⁰ - 3503q³¹ - 5254q³² - 2044q³³ + 2383q³⁴ + 4410q³⁵ + 229q³⁶ - 2063q³⁷ - 2244q³⁸ - 284q³⁹ + 1908q⁴⁰ + 923q⁴¹ - 60q⁴² - 912q⁴³ - 653q⁴⁴ + 343q⁴⁵ + 344q⁴⁶ + 256q⁴⁷ - 123q⁴⁸ - 234q⁴⁹ - 2q⁵⁰ + 24q⁵¹ + 78q⁵² + 9q⁵³ - 35q⁵⁴ - 2q⁵⁵ - 5q⁵⁶ + 9q⁵⁷ + 2q⁵⁸ - 4q⁵⁹ + q⁶⁰

5 q^-60 - 4q^-59 + 5q^-58 + 3q^-57 - 14q^-56 + 7q^-55 + 12q^-54 - 9q^-53 + 2q^-52 - 9q^-51 - 33q^-50 + 34q^-49 + 96q^-48 - q^-47 - 133q^-46 - 198q^-45 - 45q^-44 + 361q^-43 + 615q^-42 + 176q^-41 - 928q^-40 - 1566q^-39 - 611q^-38 + 1790q^-37 + 3599q^-36 + 2103q^-35 - 2904q^-34 - 7396q^-33 - 5625q^-32 + 3546q^-31 + 13217q^-30 + 12681q^-29 - 2018q^-28 - 20758q^-27 - 24745q^-26 - 3794q^-25 + 28465q^-24 + 41826q^-23 + 16513q^-22 - 33162q^-21 - 63145q^-20 - 37406q^-19 + 31853q^-18 + 85046q^-17 + 65902q^-16 - 21353q^-15 - 103670q^-14 - 99257q^-13 + 1431q^-12 + 114703q^-11 + 132782q^-10 + 26501q^-9 - 115969q^-8 - 161747q^-7 - 58453q^-6 + 107326q^-5 + 182525q^-4 + 90096q^-3 - 91167q^-2 - 193539q^-1 - 117398 + 70496q + 195494q² + 138440q³ - 48880q⁴ - 190371q⁵ - 152563q⁶ + 28179q⁷ + 180507q⁸ + 161160q⁹ - 9396q¹⁰ - 167906q¹¹ - 165525q¹² - 8048q¹³ + 153331q¹⁴ + 167338q¹⁵ + 24919q¹⁶ - 136672q¹⁷ - 166996q¹⁸ - 42344q¹⁹ + 117290q²⁰ + 164281q²¹ + 60130q²² - 94208q²³ - 157705q²⁴ - 77858q²⁵ + 67383q²⁶ + 146068q²⁷ + 92962q²⁸ - 37512q²⁹ - 127723q³⁰ - 103332q³¹ + 6837q³² + 103113q³³ + 105906q³⁴ + 21200q³⁵ - 73282q³⁶ - 99477q³⁷ - 43024q³⁸ + 41841q³⁹ + 84185q⁴⁰ + 55504q⁴¹ - 12860q⁴² - 62501q⁴³ - 57548q⁴⁴ - 9576q⁴⁵ + 38553q⁴⁶ + 50391q⁴⁷ + 22875q⁴⁸ - 16878q⁴⁹ - 37290q⁵⁰ - 26863q⁵¹ + 908q⁵² + 22560q⁵³ + 23717q⁵⁴ + 7863q⁵⁵ - 10010q⁵⁶ - 16735q⁵⁷ - 10271q⁵⁸ + 1607q⁵⁹ + 9376q⁶⁰ + 8706q⁶¹ + 2301q⁶² - 3818q⁶³ - 5523q⁶⁴ - 3068q⁶⁵ + 615q⁶⁶ + 2741q⁶⁷ + 2323q⁶⁸ + 508q⁶⁹ - 964q⁷⁰ - 1233q⁷¹ - 629q⁷² + 142q⁷³ + 535q⁷⁴ + 386q⁷⁵ + 43q⁷⁶ - 157q⁷⁷ - 149q⁷⁸ - 74q⁷⁹ + 29q⁸⁰ + 71q⁸¹ + 20q⁸² - 11q⁸³ - 8q⁸⁴ - 8q⁸⁵ - 5q⁸⁶ + 9q⁸⁷ + 2q⁸⁸ - 4q⁸⁹ + q⁹⁰

6 q^-84 - 4q^-83 + 5q^-82 + 3q^-81 - 14q^-80 + 7q^-79 + 8q^-78 + 11q^-77 - 34q^-76 - 5q^-75 + 49q^-74 - 52q^-73 + 42q^-72 + 55q^-71 - 16q^-70 - 199q^-69 - 120q^-68 + 266q^-67 + 109q^-66 + 369q^-65 + 221q^-64 - 502q^-63 - 1411q^-62 - 1037q^-61 + 1067q^-60 + 2009q^-59 + 3263q^-58 + 1772q^-57 - 2932q^-56 - 8255q^-55 - 8165q^-54 + 734q^-53 + 10333q^-52 + 19492q^-51 + 15681q^-50 - 4874q^-49 - 32343q^-48 - 43985q^-47 - 21110q^-46 + 22346q^-45 + 72557q^-44 + 83394q^-43 + 29488q^-42 - 70701q^-41 - 149712q^-40 - 133588q^-39 - 20098q^-38 + 157234q^-37 + 267735q^-36 + 205459q^-35 - 34549q^-34 - 312460q^-33 - 421230q^-32 - 263289q^-31 + 145088q^-30 + 537974q^-29 + 622264q^-28 + 263383q^-27 - 355208q^-26 - 821489q^-25 - 807055q^-24 - 186409q^-23 + 662338q^-22 + 1169119q^-21 + 905971q^-20 - 30361q^-19 - 1046864q^-18 - 1479924q^-17 - 886523q^-16 + 382318q^-15 + 1507905q^-14 + 1658528q^-13 + 667373q^-12 - 843657q^-11 - 1908538q^-10 - 1666401q^-9 - 260131q^-8 + 1404434q^-7 + 2136498q^-6 + 1414673q^-5 - 296676q^-4 - 1893519q^-3 - 2157307q^-2 - 931323q^-1 + 976970 + 2187926q + 1883026q² + 273280q³ - 1577469q⁴ - 2260711q⁵ - 1359164q⁶ + 515939q⁷ + 1970649q⁸ + 2026626q⁹ + 657080q¹⁰ - 1212056q¹¹ - 2140097q¹² - 1543644q¹³ + 170021q¹⁴ + 1695002q¹⁵ + 2005998q¹⁶ + 889871q¹⁷ - 894103q¹⁸ - 1964344q¹⁹ - 1634951q²⁰ - 119797q²¹ + 1414470q²² + 1949534q²³ + 1105672q²⁴ - 549293q²⁵ - 1748009q²⁶ - 1721154q²⁷ - 468324q²⁸ + 1041271q²⁹ + 1832666q³⁰ + 1351700q³¹ - 80578q³² - 1385111q³³ - 1734843q³⁴ - 882073q³⁵ + 489665q³⁶ + 1523101q³⁷ + 1513405q³⁸ + 480296q³⁹ - 793359q⁴⁰ - 1510284q⁴¹ - 1196663q⁴² - 174355q⁴³ + 936713q⁴⁴ + 1389200q⁴⁵ + 920165q⁴⁶ - 71166q⁴⁷ - 962377q⁴⁸ - 1177502q⁴⁹ - 692790q⁵⁰ + 205544q⁵¹ + 900348q⁵² + 980938q⁵³ + 483253q⁵⁴ - 261150q⁵⁵ - 763179q⁵⁶ - 797916q⁵⁷ - 338753q⁵⁸ + 258986q⁵⁹ + 632205q⁶⁰ + 607679q⁶¹ + 237928q⁶² - 207811q⁶³ - 502014q⁶⁴ - 455762q⁶⁵ - 169952q⁶⁶ + 165312q⁶⁷ + 362644q⁶⁸ + 331550q⁶⁹ + 132952q⁷⁰ - 121546q⁷¹ - 253500q⁷² - 232947q⁷³ - 92693q⁷⁴ + 70659q⁷⁵ + 167622q⁷⁶ + 162419q⁷⁷ + 63176q⁷⁸ - 39317q⁷⁹ - 104108q⁸⁰ - 100827q⁸¹ - 48250q⁸² + 18940q⁸³ + 63137q⁸⁴ + 59060q⁸⁵ + 31458q⁸⁶ - 7345q⁸⁷ - 31743q⁸⁸ - 35574q⁸⁹ - 19152q⁹⁰ + 3147q⁹¹ + 14755q⁹² + 18227q⁹³ + 10641q⁹⁴ + 740q⁹⁵ - 7628q⁹⁶ - 8642q⁹⁷ - 4766q⁹⁸ - 1066q⁹⁹ + 2879q¹⁰⁰ + 3644q¹⁰¹ + 2687q¹⁰² + 156q¹⁰³ - 1074q¹⁰⁴ - 1170q¹⁰⁵ - 1077q¹⁰⁶ - 193q¹⁰⁷ + 324q¹⁰⁸ + 597q¹⁰⁹ + 212q¹¹⁰ + 32q¹¹¹ - 37q¹¹² - 173q¹¹³ - 87q¹¹⁴ - 19q¹¹⁵ + 76q¹¹⁶ + 13q¹¹⁷ + 16q¹¹⁹ - 14q¹²⁰ - 8q¹²¹ - 5q¹²² + 9q¹²³ + 2q¹²⁴ - 4q¹²⁵ + q¹²⁶

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, 119]]

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

In[3]:=
GaussCode[Knot[10, 119]]

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

In[4]:=
DTCode[Knot[10, 119]]

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

In[5]:=
br = BR[Knot[10, 119]]

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

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

Out[6]=
{4, 11}

In[7]:=
BraidIndex[Knot[10, 119]]

Out[7]=
4

In[8]:=
Show[DrawMorseLink[Knot[10, 119]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[10, 119]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

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

In[10]:=
alex = Alexander[Knot[10, 119]][t]

Out[10]=
2 10 23 2 3 31 - -- + -- - -- - 23 t + 10 t - 2 t 3 2 t t t

In[11]:=
Conway[Knot[10, 119]][z]

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

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

Out[12]=
{Knot[10, 119], Knot[11, Alternating, 84]}

In[13]:=
{KnotDet[Knot[10, 119]], KnotSignature[Knot[10, 119]]}

Out[13]=
{101, 0}

In[14]:=
Jones[Knot[10, 119]][q]

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

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

Out[15]=
{Knot[10, 119]}

In[16]:=
A2Invariant[Knot[10, 119]][q]

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

In[17]:=
HOMFLYPT[Knot[10, 119]][a, z]

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

In[18]:=
Kauffman[Knot[10, 119]][a, z]

Out[18]=
2 2 3 -2 2 z 3 z 4 z 2 z z 2 2 7 z -1 - a - a - -- - --- - --- - 2 a z + 6 z + -- + -- + 4 a z + ---- + 5 3 a 6 2 5 a a a a a 3 3 4 4 4 19 z 22 z 3 3 3 4 2 z 8 z 13 z 2 4 > ----- + ----- + 9 a z - a z - 7 z - ---- + ---- + ----- - 9 a z + 3 a 6 4 2 a a a a 5 5 5 6 6 4 4 10 z 26 z 37 z 5 3 5 6 z 14 z > a z - ----- - ----- - ----- - 17 a z + 4 a z - 7 z + -- - ----- - 5 3 a 6 4 a a a a 6 7 7 7 8 8 31 z 2 6 4 z 5 z 13 z 7 8 6 z 15 z > ----- + 9 a z + ---- + ---- + ----- + 12 a z + 9 z + ---- + ----- + 2 5 3 a 4 2 a a a a a 9 9 3 z 3 z > ---- + ---- 3 a a

In[19]:=
{Vassiliev[2][Knot[10, 119]], Vassiliev[3][Knot[10, 119]]}

Out[19]=
{-1, 0}

In[20]:=
Kh[Knot[10, 119]][q, t]

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

In[21]:=
ColouredJones[Knot[10, 119], 2][q]

Out[21]=
-12 4 5 7 30 27 34 99 56 97 185 59 173 + q - --- + --- + -- - -- + -- + -- - -- + -- + -- - --- + -- - 232 q + 11 10 9 8 7 6 5 4 3 2 q q q q q q q q q q q 2 3 4 5 6 7 8 9 > 26 q + 216 q - 214 q - 21 q + 206 q - 144 q - 54 q + 146 q - 10 11 12 13 14 15 16 17 18 > 61 q - 54 q + 67 q - 9 q - 26 q + 15 q + 2 q - 4 q + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 10₁₁₉

10₁₁₈
10₁₂₀

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-12 - 4q^-11 + 5q^-10 + 7q^-9 - 30q^-8 + 27q^-7 + 34q^-6 - 99q^-5 + 56q^-4 + 97q^-3 - 185q^-2 + 59q^-1 + 173 - 232q + 26q² + 216q³ - 214q⁴ - 21q⁵ + 206q⁶ - 144q⁷ - 54q⁸ + 146q⁹ - 61q¹⁰ - 54q¹¹ + 67q¹² - 9q¹³ - 26q¹⁴ + 15q¹⁵ + 2q¹⁶ - 4q¹⁷ + q¹⁸
3	q^-24 - 4q^-23 + 5q^-22 + 3q^-21 - 10q^-20 - 9q^-19 + 28q^-18 + 23q^-17 - 74q^-16 - 45q^-15 + 151q^-14 + 100q^-13 - 261q^-12 - 227q^-11 + 414q^-10 + 417q^-9 - 547q^-8 - 701q^-7 + 645q^-6 + 1028q^-5 - 653q^-4 - 1370q^-3 + 584q^-2 + 1654q^-1 - 427 - 1864q + 229q² + 1962q³ + 5q⁴ - 1971q⁵ - 232q⁶ + 1881q⁷ + 458q⁸ - 1721q⁹ - 648q¹⁰ + 1476q¹¹ + 816q¹² - 1196q¹³ - 901q¹⁴ + 861q¹⁵ + 920q¹⁶ - 534q¹⁷ - 843q¹⁸ + 243q¹⁹ + 690q²⁰ - 27q²¹ - 495q²² - 95q²³ + 304q²⁴ + 128q²⁵ - 152q²⁶ - 105q²⁷ + 55q²⁸ + 67q²⁹ - 15q³⁰ - 29q³¹ + q³² + 9q³³ + 2q³⁴ - 4q³⁵ + q³⁶
4	q^-40 - 4q^-39 + 5q^-38 + 3q^-37 - 14q^-36 + 11q^-35 - 8q^-34 + 27q^-33 - 2q^-32 - 91q^-31 + 63q^-30 + 38q^-29 + 124q^-28 - 84q^-27 - 454q^-26 + 151q^-25 + 403q^-24 + 666q^-23 - 309q^-22 - 1752q^-21 - 242q^-20 + 1345q^-19 + 2698q^-18 + 81q^-17 - 4508q^-16 - 2602q^-15 + 1895q^-14 + 6895q^-13 + 2925q^-12 - 7376q^-11 - 7596q^-10 - 153q^-9 + 11442q^-8 + 8713q^-7 - 7653q^-6 - 12982q^-5 - 5243q^-4 + 13309q^-3 + 14854q^-2 - 4667q^-1 - 15659 - 10923q + 11734q² + 18417q³ - 359q⁴ - 14952q⁵ - 14791q⁶ + 8212q⁷ + 18858q⁸ + 3581q⁹ - 12057q¹⁰ - 16512q¹¹ + 3954q¹² + 17045q¹³ + 6885q¹⁴ - 7818q¹⁵ - 16431q¹⁶ - 726q¹⁷ + 13279q¹⁸ + 9273q¹⁹ - 2500q²⁰ - 14134q²¹ - 4976q²² + 7637q²³ + 9461q²⁴ + 2669q²⁵ - 9280q²⁶ - 6835q²⁷ + 1610q²⁸ + 6610q²⁹ + 5323q³⁰ - 3503q³¹ - 5254q³² - 2044q³³ + 2383q³⁴ + 4410q³⁵ + 229q³⁶ - 2063q³⁷ - 2244q³⁸ - 284q³⁹ + 1908q⁴⁰ + 923q⁴¹ - 60q⁴² - 912q⁴³ - 653q⁴⁴ + 343q⁴⁵ + 344q⁴⁶ + 256q⁴⁷ - 123q⁴⁸ - 234q⁴⁹ - 2q⁵⁰ + 24q⁵¹ + 78q⁵² + 9q⁵³ - 35q⁵⁴ - 2q⁵⁵ - 5q⁵⁶ + 9q⁵⁷ + 2q⁵⁸ - 4q⁵⁹ + q⁶⁰
5	q^-60 - 4q^-59 + 5q^-58 + 3q^-57 - 14q^-56 + 7q^-55 + 12q^-54 - 9q^-53 + 2q^-52 - 9q^-51 - 33q^-50 + 34q^-49 + 96q^-48 - q^-47 - 133q^-46 - 198q^-45 - 45q^-44 + 361q^-43 + 615q^-42 + 176q^-41 - 928q^-40 - 1566q^-39 - 611q^-38 + 1790q^-37 + 3599q^-36 + 2103q^-35 - 2904q^-34 - 7396q^-33 - 5625q^-32 + 3546q^-31 + 13217q^-30 + 12681q^-29 - 2018q^-28 - 20758q^-27 - 24745q^-26 - 3794q^-25 + 28465q^-24 + 41826q^-23 + 16513q^-22 - 33162q^-21 - 63145q^-20 - 37406q^-19 + 31853q^-18 + 85046q^-17 + 65902q^-16 - 21353q^-15 - 103670q^-14 - 99257q^-13 + 1431q^-12 + 114703q^-11 + 132782q^-10 + 26501q^-9 - 115969q^-8 - 161747q^-7 - 58453q^-6 + 107326q^-5 + 182525q^-4 + 90096q^-3 - 91167q^-2 - 193539q^-1 - 117398 + 70496q + 195494q² + 138440q³ - 48880q⁴ - 190371q⁵ - 152563q⁶ + 28179q⁷ + 180507q⁸ + 161160q⁹ - 9396q¹⁰ - 167906q¹¹ - 165525q¹² - 8048q¹³ + 153331q¹⁴ + 167338q¹⁵ + 24919q¹⁶ - 136672q¹⁷ - 166996q¹⁸ - 42344q¹⁹ + 117290q²⁰ + 164281q²¹ + 60130q²² - 94208q²³ - 157705q²⁴ - 77858q²⁵ + 67383q²⁶ + 146068q²⁷ + 92962q²⁸ - 37512q²⁹ - 127723q³⁰ - 103332q³¹ + 6837q³² + 103113q³³ + 105906q³⁴ + 21200q³⁵ - 73282q³⁶ - 99477q³⁷ - 43024q³⁸ + 41841q³⁹ + 84185q⁴⁰ + 55504q⁴¹ - 12860q⁴² - 62501q⁴³ - 57548q⁴⁴ - 9576q⁴⁵ + 38553q⁴⁶ + 50391q⁴⁷ + 22875q⁴⁸ - 16878q⁴⁹ - 37290q⁵⁰ - 26863q⁵¹ + 908q⁵² + 22560q⁵³ + 23717q⁵⁴ + 7863q⁵⁵ - 10010q⁵⁶ - 16735q⁵⁷ - 10271q⁵⁸ + 1607q⁵⁹ + 9376q⁶⁰ + 8706q⁶¹ + 2301q⁶² - 3818q⁶³ - 5523q⁶⁴ - 3068q⁶⁵ + 615q⁶⁶ + 2741q⁶⁷ + 2323q⁶⁸ + 508q⁶⁹ - 964q⁷⁰ - 1233q⁷¹ - 629q⁷² + 142q⁷³ + 535q⁷⁴ + 386q⁷⁵ + 43q⁷⁶ - 157q⁷⁷ - 149q⁷⁸ - 74q⁷⁹ + 29q⁸⁰ + 71q⁸¹ + 20q⁸² - 11q⁸³ - 8q⁸⁴ - 8q⁸⁵ - 5q⁸⁶ + 9q⁸⁷ + 2q⁸⁸ - 4q⁸⁹ + q⁹⁰
6	q^-84 - 4q^-83 + 5q^-82 + 3q^-81 - 14q^-80 + 7q^-79 + 8q^-78 + 11q^-77 - 34q^-76 - 5q^-75 + 49q^-74 - 52q^-73 + 42q^-72 + 55q^-71 - 16q^-70 - 199q^-69 - 120q^-68 + 266q^-67 + 109q^-66 + 369q^-65 + 221q^-64 - 502q^-63 - 1411q^-62 - 1037q^-61 + 1067q^-60 + 2009q^-59 + 3263q^-58 + 1772q^-57 - 2932q^-56 - 8255q^-55 - 8165q^-54 + 734q^-53 + 10333q^-52 + 19492q^-51 + 15681q^-50 - 4874q^-49 - 32343q^-48 - 43985q^-47 - 21110q^-46 + 22346q^-45 + 72557q^-44 + 83394q^-43 + 29488q^-42 - 70701q^-41 - 149712q^-40 - 133588q^-39 - 20098q^-38 + 157234q^-37 + 267735q^-36 + 205459q^-35 - 34549q^-34 - 312460q^-33 - 421230q^-32 - 263289q^-31 + 145088q^-30 + 537974q^-29 + 622264q^-28 + 263383q^-27 - 355208q^-26 - 821489q^-25 - 807055q^-24 - 186409q^-23 + 662338q^-22 + 1169119q^-21 + 905971q^-20 - 30361q^-19 - 1046864q^-18 - 1479924q^-17 - 886523q^-16 + 382318q^-15 + 1507905q^-14 + 1658528q^-13 + 667373q^-12 - 843657q^-11 - 1908538q^-10 - 1666401q^-9 - 260131q^-8 + 1404434q^-7 + 2136498q^-6 + 1414673q^-5 - 296676q^-4 - 1893519q^-3 - 2157307q^-2 - 931323q^-1 + 976970 + 2187926q + 1883026q² + 273280q³ - 1577469q⁴ - 2260711q⁵ - 1359164q⁶ + 515939q⁷ + 1970649q⁸ + 2026626q⁹ + 657080q¹⁰ - 1212056q¹¹ - 2140097q¹² - 1543644q¹³ + 170021q¹⁴ + 1695002q¹⁵ + 2005998q¹⁶ + 889871q¹⁷ - 894103q¹⁸ - 1964344q¹⁹ - 1634951q²⁰ - 119797q²¹ + 1414470q²² + 1949534q²³ + 1105672q²⁴ - 549293q²⁵ - 1748009q²⁶ - 1721154q²⁷ - 468324q²⁸ + 1041271q²⁹ + 1832666q³⁰ + 1351700q³¹ - 80578q³² - 1385111q³³ - 1734843q³⁴ - 882073q³⁵ + 489665q³⁶ + 1523101q³⁷ + 1513405q³⁸ + 480296q³⁹ - 793359q⁴⁰ - 1510284q⁴¹ - 1196663q⁴² - 174355q⁴³ + 936713q⁴⁴ + 1389200q⁴⁵ + 920165q⁴⁶ - 71166q⁴⁷ - 962377q⁴⁸ - 1177502q⁴⁹ - 692790q⁵⁰ + 205544q⁵¹ + 900348q⁵² + 980938q⁵³ + 483253q⁵⁴ - 261150q⁵⁵ - 763179q⁵⁶ - 797916q⁵⁷ - 338753q⁵⁸ + 258986q⁵⁹ + 632205q⁶⁰ + 607679q⁶¹ + 237928q⁶² - 207811q⁶³ - 502014q⁶⁴ - 455762q⁶⁵ - 169952q⁶⁶ + 165312q⁶⁷ + 362644q⁶⁸ + 331550q⁶⁹ + 132952q⁷⁰ - 121546q⁷¹ - 253500q⁷² - 232947q⁷³ - 92693q⁷⁴ + 70659q⁷⁵ + 167622q⁷⁶ + 162419q⁷⁷ + 63176q⁷⁸ - 39317q⁷⁹ - 104108q⁸⁰ - 100827q⁸¹ - 48250q⁸² + 18940q⁸³ + 63137q⁸⁴ + 59060q⁸⁵ + 31458q⁸⁶ - 7345q⁸⁷ - 31743q⁸⁸ - 35574q⁸⁹ - 19152q⁹⁰ + 3147q⁹¹ + 14755q⁹² + 18227q⁹³ + 10641q⁹⁴ + 740q⁹⁵ - 7628q⁹⁶ - 8642q⁹⁷ - 4766q⁹⁸ - 1066q⁹⁹ + 2879q¹⁰⁰ + 3644q¹⁰¹ + 2687q¹⁰² + 156q¹⁰³ - 1074q¹⁰⁴ - 1170q¹⁰⁵ - 1077q¹⁰⁶ - 193q¹⁰⁷ + 324q¹⁰⁸ + 597q¹⁰⁹ + 212q¹¹⁰ + 32q¹¹¹ - 37q¹¹² - 173q¹¹³ - 87q¹¹⁴ - 19q¹¹⁵ + 76q¹¹⁶ + 13q¹¹⁷ + 16q¹¹⁹ - 14q¹²⁰ - 8q¹²¹ - 5q¹²² + 9q¹²³ + 2q¹²⁴ - 4q¹²⁵ + q¹²⁶

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	PD[Knot[10, 119]]
Out[2]=	PD[X[1, 6, 2, 7], X[7, 18, 8, 19], X[3, 9, 4, 8], X[17, 3, 18, 2], > X[5, 15, 6, 14], X[9, 17, 10, 16], X[15, 11, 16, 10], X[11, 5, 12, 4], > X[13, 20, 14, 1], X[19, 12, 20, 13]]
In[3]:=	GaussCode[Knot[10, 119]]
Out[3]=	GaussCode[-1, 4, -3, 8, -5, 1, -2, 3, -6, 7, -8, 10, -9, 5, -7, 6, -4, 2, -10, > 9]
In[4]:=	DTCode[Knot[10, 119]]
Out[4]=	DTCode[6, 8, 14, 18, 16, 4, 20, 10, 2, 12]
In[5]:=	br = BR[Knot[10, 119]]
Out[5]=	BR[4, {-1, -1, 2, -1, -3, 2, -1, 2, 3, 3, 2}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 11}
In[7]:=	BraidIndex[Knot[10, 119]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[10, 119]]]

Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 119]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Chiral, 1, 3, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 119]][t]
Out[10]=	2 10 23 2 3 31 - -- + -- - -- - 23 t + 10 t - 2 t 3 2 t t t
In[11]:=	Conway[Knot[10, 119]][z]
Out[11]=	2 4 6 1 - z - 2 z - 2 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 119], Knot[11, Alternating, 84]}
In[13]:=	{KnotDet[Knot[10, 119]], KnotSignature[Knot[10, 119]]}
Out[13]=	{101, 0}
In[14]:=	Jones[Knot[10, 119]][q]
Out[14]=	-4 4 9 13 2 3 4 5 6 16 + q - -- + -- - -- - 17 q + 16 q - 12 q + 8 q - 4 q + q 3 2 q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 119]}
In[16]:=	A2Invariant[Knot[10, 119]][q]
Out[16]=	-12 2 2 2 3 3 2 4 6 8 10 12 14 -3 + q - --- + -- + -- - -- + -- + q + q - q + 4 q - 3 q + q + q - 10 8 6 4 2 q q q q q 16 18 > 2 q + q
In[17]:=	HOMFLYPT[Knot[10, 119]][a, z]
Out[17]=	2 2 4 4 6 -2 2 2 z z 2 2 4 z 2 z 2 4 6 z -1 + a + a - 2 z + -- - -- + a z - 2 z + -- - ---- + a z - z - -- 4 2 4 2 2 a a a a a
In[18]:=	Kauffman[Knot[10, 119]][a, z]
Out[18]=	2 2 3 -2 2 z 3 z 4 z 2 z z 2 2 7 z -1 - a - a - -- - --- - --- - 2 a z + 6 z + -- + -- + 4 a z + ---- + 5 3 a 6 2 5 a a a a a 3 3 4 4 4 19 z 22 z 3 3 3 4 2 z 8 z 13 z 2 4 > ----- + ----- + 9 a z - a z - 7 z - ---- + ---- + ----- - 9 a z + 3 a 6 4 2 a a a a 5 5 5 6 6 4 4 10 z 26 z 37 z 5 3 5 6 z 14 z > a z - ----- - ----- - ----- - 17 a z + 4 a z - 7 z + -- - ----- - 5 3 a 6 4 a a a a 6 7 7 7 8 8 31 z 2 6 4 z 5 z 13 z 7 8 6 z 15 z > ----- + 9 a z + ---- + ---- + ----- + 12 a z + 9 z + ---- + ----- + 2 5 3 a 4 2 a a a a a 9 9 3 z 3 z > ---- + ---- 3 a a
In[19]:=	{Vassiliev[2][Knot[10, 119]], Vassiliev[3][Knot[10, 119]]}
Out[19]=	{-1, 0}
In[20]:=	Kh[Knot[10, 119]][q, t]
Out[20]=	9 1 3 1 6 3 7 6 3 - + 8 q + ----- + ----- + ----- + ----- + ----- + ---- + --- + 9 q t + 8 q t + q 9 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t 3 2 5 2 5 3 7 3 7 4 9 4 9 5 > 7 q t + 9 q t + 5 q t + 7 q t + 3 q t + 5 q t + q t + 11 5 13 6 > 3 q t + q t
In[21]:=	ColouredJones[Knot[10, 119], 2][q]
Out[21]=	-12 4 5 7 30 27 34 99 56 97 185 59 173 + q - --- + --- + -- - -- + -- + -- - -- + -- + -- - --- + -- - 232 q + 11 10 9 8 7 6 5 4 3 2 q q q q q q q q q q q 2 3 4 5 6 7 8 9 > 26 q + 216 q - 214 q - 21 q + 206 q - 144 q - 54 q + 146 q - 10 11 12 13 14 15 16 17 18 > 61 q - 54 q + 67 q - 9 q - 26 q + 15 q + 2 q - 4 q + q

The Alternating Knot 10119

The Alternating Knot 10₁₁₉