The Alternating Knot 10₆₉

Visit 10₆₉'s page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 10₆₉'s page at Knotilus!

Acknowledgement

KnotPlot

PD Presentation: X₁₄₂₅ X_7,12,8,13 X_3,11,4,10 X_11,3,12,2 X_13,17,14,16 X_5,15,6,14 X_15,7,16,6 X_17,20,18,1 X_9,19,10,18 X_19,9,20,8

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

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

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 2

Alexander Polynomial: t^-3 - 7t^-2 + 21t^-1 - 29 + 21t - 7t² + t³

Conway Polynomial: 1 + 2z² - z⁴ + z⁶

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

Determinant and Signature: {87, 2}

Jones Polynomial: - q^-2 + 4q^-1 - 7 + 11q - 14q² + 15q³ - 13q⁴ + 11q⁵ - 7q⁶ + 3q⁷ - q⁸

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

A2 (sl(3)) Invariant: - q^-6 + 2q^-4 - q^-2 + 4q² - 3q⁴ + 2q⁶ - q⁸ + 2q¹² - 2q¹⁴ + 3q¹⁶ - q¹⁸ - q²⁰ + 2q²² - q²⁴ - q²⁶

HOMFLY-PT Polynomial: - a^-8 + 2a^-6 + 3a^-6z² - 2a^-4 - 5a^-4z² - 3a^-4z⁴ + 2a^-2 + 5a^-2z² + 3a^-2z⁴ + a^-2z⁶ - z² - z⁴

Kauffman Polynomial: a^-9z - 2a^-9z³ + a^-9z⁵ - a^-8 + 3a^-8z² - 5a^-8z⁴ + 3a^-8z⁶ - 2a^-7z + 5a^-7z³ - 8a^-7z⁵ + 5a^-7z⁷ - 2a^-6 + 7a^-6z² - 9a^-6z⁴ + 4a^-6z⁸ - 6a^-5z + 23a^-5z³ - 30a^-5z⁵ + 13a^-5z⁷ + a^-5z⁹ - 2a^-4 + 12a^-4z² - 14a^-4z⁴ - 4a^-4z⁶ + 8a^-4z⁸ - 4a^-3z + 22a^-3z³ - 32a^-3z⁵ + 14a^-3z⁷ + a^-3z⁹ - 2a^-2 + 11a^-2z² - 17a^-2z⁴ + 3a^-2z⁶ + 4a^-2z⁸ - a^-1z + 5a^-1z³ - 10a^-1z⁵ + 6a^-1z⁷ + 3z² - 7z⁴ + 4z⁶ - az³ + az⁵

V₂ and V₃, the type 2 and 3 Vassiliev invariants: {2, 4}

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

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

j = 17 1

j = 15 2

j = 13 5 1

j = 11 6 2

j = 9 7 5

j = 7 8 6

j = 5 6 7

j = 3 5 8

j = 1 3 7

j = -1 1 4

j = -3 3

j = -5 1

n Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q^-7 - 4q^-6 + 2q^-5 + 13q^-4 - 24q^-3 + 51q^-1 - 61 - 19q + 117q² - 95q³ - 59q⁴ + 180q⁵ - 104q⁶ - 101q⁷ + 202q⁸ - 83q⁹ - 117q¹⁰ + 172q¹¹ - 43q¹² - 100q¹³ + 107q¹⁴ - 9q¹⁵ - 60q¹⁶ + 43q¹⁷ + 4q¹⁸ - 21q¹⁹ + 9q²⁰ + 2q²¹ - 3q²² + q²³

3 - q^-15 + 4q^-14 - 2q^-13 - 8q^-12 + 23q^-10 + 6q^-9 - 53q^-8 - 16q^-7 + 90q^-6 + 55q^-5 - 153q^-4 - 114q^-3 + 214q^-2 + 224q^-1 - 284 - 362q + 321q² + 552q³ - 341q⁴ - 744q⁵ + 306q⁶ + 945q⁷ - 241q⁸ - 1115q⁹ + 143q¹⁰ + 1244q¹¹ - 30q¹² - 1309q¹³ - 100q¹⁴ + 1330q¹⁵ + 207q¹⁶ - 1270q¹⁷ - 324q¹⁸ + 1175q¹⁹ + 397q²⁰ - 1010q²¹ - 464q²² + 832q²³ + 477q²⁴ - 625q²⁵ - 461q²⁶ + 434q²⁷ + 402q²⁸ - 265q²⁹ - 320q³⁰ + 135q³¹ + 235q³² - 61q³³ - 143q³⁴ + 11q³⁵ + 83q³⁶ + 2q³⁷ - 39q³⁸ - 5q³⁹ + 17q⁴⁰ + q⁴¹ - 4q⁴² - 2q⁴³ + 3q⁴⁴ - q⁴⁵

4 q^-26 - 4q^-25 + 2q^-24 + 8q^-23 - 5q^-22 + q^-21 - 29q^-20 + 12q^-19 + 54q^-18 - 9q^-17 - 146q^-15 + 7q^-14 + 213q^-13 + 65q^-12 + 25q^-11 - 503q^-10 - 150q^-9 + 528q^-8 + 433q^-7 + 285q^-6 - 1227q^-5 - 814q^-4 + 785q^-3 + 1320q^-2 + 1260q^-1 - 2106 - 2313q + 384q² + 2524q³ + 3364q⁴ - 2450q⁵ - 4413q⁶ - 1184q⁷ + 3313q⁸ + 6290q⁹ - 1678q¹⁰ - 6261q¹¹ - 3655q¹² + 3088q¹³ + 9050q¹⁴ + 19q¹⁵ - 7108q¹⁶ - 6116q¹⁷ + 1940q¹⁸ + 10743q¹⁹ + 1953q²⁰ - 6805q²¹ - 7816q²² + 349q²³ + 11061q²⁴ + 3583q²⁵ - 5573q²⁶ - 8457q²⁷ - 1326q²⁸ + 10016q²⁹ + 4668q³⁰ - 3616q³¹ - 7961q³² - 2842q³³ + 7772q³⁴ + 4975q³⁵ - 1312q³⁶ - 6333q³⁷ - 3754q³⁸ + 4814q³⁹ + 4270q⁴⁰ + 614q⁴¹ - 3958q⁴² - 3607q⁴³ + 2072q⁴⁴ + 2751q⁴⁵ + 1462q⁴⁶ - 1703q⁴⁷ - 2512q⁴⁸ + 396q⁴⁹ + 1184q⁵⁰ + 1233q⁵¹ - 354q⁵² - 1218q⁵³ - 126q⁵⁴ + 249q⁵⁵ + 606q⁵⁶ + 67q⁵⁷ - 393q⁵⁸ - 95q⁵⁹ - 28q⁶⁰ + 184q⁶¹ + 63q⁶² - 87q⁶³ - 14q⁶⁴ - 29q⁶⁵ + 34q⁶⁶ + 16q⁶⁷ - 16q⁶⁸ + 3q⁶⁹ - 6q⁷⁰ + 4q⁷¹ + 2q⁷² - 3q⁷³ + q⁷⁴

5 - q^-40 + 4q^-39 - 2q^-38 - 8q^-37 + 5q^-36 + 4q^-35 + 5q^-34 + 11q^-33 - 13q^-32 - 45q^-31 - 5q^-30 + 46q^-29 + 64q^-28 + 48q^-27 - 66q^-26 - 185q^-25 - 133q^-24 + 133q^-23 + 370q^-22 + 300q^-21 - 134q^-20 - 672q^-19 - 742q^-18 + 55q^-17 + 1186q^-16 + 1451q^-15 + 277q^-14 - 1681q^-13 - 2741q^-12 - 1209q^-11 + 2265q^-10 + 4554q^-9 + 2915q^-8 - 2282q^-7 - 6966q^-6 - 5948q^-5 + 1561q^-4 + 9605q^-3 + 10277q^-2 + 810q^-1 - 12053 - 16044q - 4986q² + 13425q³ + 22657q⁴ + 11564q⁵ - 13195q⁶ - 29561q⁷ - 19945q⁸ + 10586q⁹ + 35679q¹⁰ + 29961q¹¹ - 5663q¹² - 40415q¹³ - 40314q¹⁴ - 1405q¹⁵ + 42957q¹⁶ + 50365q¹⁷ + 9905q¹⁸ - 43384q¹⁹ - 59028q²⁰ - 19019q²¹ + 41739q²² + 65885q²³ + 27940q²⁴ - 38518q²⁵ - 70632q²⁶ - 36114q²⁷ + 34167q²⁸ + 73458q²⁹ + 43038q³⁰ - 29103q³¹ - 74269q³² - 48866q³³ + 23481q³⁴ + 73624q³⁵ + 53330q³⁶ - 17556q³⁷ - 71073q³⁸ - 56786q³⁹ + 11026q⁴⁰ + 67230q⁴¹ + 58882q⁴² - 4259q⁴³ - 61387q⁴⁴ - 59730q⁴⁵ - 2934q⁴⁶ + 54146q⁴⁷ + 58825q⁴⁸ + 9830q⁴⁹ - 45110q⁵⁰ - 56120q⁵¹ - 16172q⁵² + 35214q⁵³ + 51260q⁵⁴ + 21041q⁵⁵ - 24687q⁵⁶ - 44606q⁵⁷ - 24044q⁵⁸ + 14800q⁵⁹ + 36446q⁶⁰ + 24674q⁶¹ - 6160q⁶² - 27634q⁶³ - 23159q⁶⁴ - 339q⁶⁵ + 19118q⁶⁶ + 19803q⁶⁷ + 4407q⁶⁸ - 11638q⁶⁹ - 15435q⁷⁰ - 6302q⁷¹ + 5989q⁷² + 10922q⁷³ + 6237q⁷⁴ - 2134q⁷⁵ - 6859q⁷⁶ - 5244q⁷⁷ + 33q⁷⁸ + 3875q⁷⁹ + 3682q⁸⁰ + 831q⁸¹ - 1819q⁸² - 2331q⁸³ - 927q⁸⁴ + 709q⁸⁵ + 1263q⁸⁶ + 712q⁸⁷ - 173q⁸⁸ - 629q⁸⁹ - 416q⁹⁰ - 4q⁹¹ + 240q⁹² + 237q⁹³ + 43q⁹⁴ - 111q⁹⁵ - 91q⁹⁶ - 14q⁹⁷ + 11q⁹⁸ + 43q⁹⁹ + 23q¹⁰⁰ - 22q¹⁰¹ - 11q¹⁰² + 5q¹⁰³ - 4q¹⁰⁴ + 2q¹⁰⁵ + 6q¹⁰⁶ - 4q¹⁰⁷ - 2q¹⁰⁸ + 3q¹⁰⁹ - q¹¹⁰

6 q^-57 - 4q^-56 + 2q^-55 + 8q^-54 - 5q^-53 - 4q^-52 - 10q^-51 + 13q^-50 - 10q^-49 + 4q^-48 + 59q^-47 - 25q^-46 - 40q^-45 - 74q^-44 + 32q^-43 - 3q^-42 + 56q^-41 + 268q^-40 - 30q^-39 - 197q^-38 - 407q^-37 - 55q^-36 - 35q^-35 + 354q^-34 + 1102q^-33 + 305q^-32 - 516q^-31 - 1647q^-30 - 1015q^-29 - 702q^-28 + 1091q^-27 + 3823q^-26 + 2616q^-25 - 53q^-24 - 4545q^-23 - 5119q^-22 - 4943q^-21 + 788q^-20 + 9900q^-19 + 11202q^-18 + 6080q^-17 - 7012q^-16 - 15023q^-15 - 19992q^-14 - 8288q^-13 + 16168q^-12 + 30777q^-11 + 29120q^-10 + 2555q^-9 - 26252q^-8 - 53117q^-7 - 42476q^-6 + 5971q^-5 + 55332q^-4 + 79144q^-3 + 46574q^-2 - 16455q^-1 - 96180 - 114531q - 49406q² + 56360q³ + 145060q⁴ + 139926q⁵ + 48543q⁶ - 114346q⁷ - 209517q⁸ - 165324q⁹ - 4633q¹⁰ + 185928q¹¹ + 263001q¹² + 182376q¹³ - 66294q¹⁴ - 280859q¹⁵ - 316308q¹⁶ - 138309q¹⁷ + 159174q¹⁸ + 364485q¹⁹ + 353221q²⁰ + 54973q²¹ - 285746q²² - 447067q²³ - 308130q²⁴ + 61232q²⁵ + 402191q²⁶ + 503619q²⁷ + 210606q²⁸ - 222600q²⁹ - 516598q³⁰ - 458164q³¹ - 69685q³² + 375256q³³ + 595214q³⁴ + 349536q³⁵ - 125675q³⁶ - 523805q³⁷ - 555571q³⁸ - 190207q³⁹ + 312203q⁴⁰ + 627244q⁴¹ + 446200q⁴² - 28635q⁴³ - 491012q⁴⁴ - 601618q⁴⁵ - 282948q⁴⁶ + 236940q⁴⁷ + 616966q⁴⁸ + 504456q⁴⁹ + 59356q⁵⁰ - 433462q⁵¹ - 609988q⁵² - 354261q⁵³ + 151582q⁵⁴ + 572713q⁵⁵ + 534670q⁵⁶ + 146413q⁵⁷ - 347730q⁵⁸ - 582966q⁵⁹ - 410722q⁶⁰ + 47351q⁶¹ + 486687q⁶² + 532645q⁶³ + 234221q⁶⁴ - 225474q⁶⁵ - 507943q⁶⁶ - 440451q⁶⁷ - 71583q⁶⁸ + 351254q⁶⁹ + 479632q⁷⁰ + 301775q⁷¹ - 77438q⁷² - 376572q⁷³ - 417104q⁷⁴ - 174563q⁷⁵ + 183183q⁷⁶ + 365081q⁷⁷ + 314426q⁷⁸ + 56438q⁷⁹ - 209585q⁸⁰ - 327261q⁸¹ - 219573q⁸² + 29782q⁸³ + 212139q⁸⁴ + 256318q⁸⁵ + 128868q⁸⁶ - 58377q⁸⁷ - 196007q⁸⁸ - 190230q⁸⁹ - 59584q⁹⁰ + 73981q⁹¹ + 154418q⁹² + 125127q⁹³ + 29385q⁹⁴ - 76895q⁹⁵ - 115855q⁹⁶ - 73901q⁹⁷ - 5489q⁹⁸ + 61052q⁹⁹ + 76748q¹⁰⁰ + 48021q¹⁰¹ - 9077q¹⁰² - 46371q¹⁰³ - 46128q¹⁰⁴ - 25339q¹⁰⁵ + 9643q¹⁰⁶ + 29753q¹⁰⁷ + 30667q¹⁰⁸ + 9969q¹⁰⁹ - 9338q¹¹⁰ - 17035q¹¹¹ - 16430q¹¹² - 4632q¹¹³ + 5902q¹¹⁴ + 11702q¹¹⁵ + 7049q¹¹⁶ + 999q¹¹⁷ - 3119q¹¹⁸ - 5840q¹¹⁹ - 3582q¹²⁰ - 345q¹²¹ + 2830q¹²² + 2222q¹²³ + 1187q¹²⁴ + 185q¹²⁵ - 1254q¹²⁶ - 1153q¹²⁷ - 561q¹²⁸ + 480q¹²⁹ + 360q¹³⁰ + 308q¹³¹ + 263q¹³² - 159q¹³³ - 228q¹³⁴ - 172q¹³⁵ + 95q¹³⁶ + 16q¹³⁷ + 25q¹³⁸ + 77q¹³⁹ - 13q¹⁴⁰ - 30q¹⁴¹ - 37q¹⁴² + 28q¹⁴³ - q¹⁴⁴ - 10q¹⁴⁵ + 15q¹⁴⁶ - q¹⁴⁷ - 2q¹⁴⁸ - 6q¹⁴⁹ + 4q¹⁵⁰ + 2q¹⁵¹ - 3q¹⁵² + q¹⁵³

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, 69]]

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

In[3]:=
GaussCode[Knot[10, 69]]

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

In[4]:=
DTCode[Knot[10, 69]]

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

In[5]:=
br = BR[Knot[10, 69]]

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

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

Out[6]=
{5, 12}

In[7]:=
BraidIndex[Knot[10, 69]]

Out[7]=
5

In[8]:=
Show[DrawMorseLink[Knot[10, 69]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[10, 69]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

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

In[10]:=
alex = Alexander[Knot[10, 69]][t]

Out[10]=
-3 7 21 2 3 -29 + t - -- + -- + 21 t - 7 t + t 2 t t

In[11]:=
Conway[Knot[10, 69]][z]

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

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

Out[12]=
{Knot[10, 69]}

In[13]:=
{KnotDet[Knot[10, 69]], KnotSignature[Knot[10, 69]]}

Out[13]=
{87, 2}

In[14]:=
Jones[Knot[10, 69]][q]

Out[14]=
-2 4 2 3 4 5 6 7 8 -7 - q + - + 11 q - 14 q + 15 q - 13 q + 11 q - 7 q + 3 q - q q

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

Out[15]=
{Knot[10, 69]}

In[16]:=
A2Invariant[Knot[10, 69]][q]

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

In[17]:=
HOMFLYPT[Knot[10, 69]][a, z]

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

In[18]:=
Kauffman[Knot[10, 69]][a, z]

Out[18]=
2 2 2 -8 2 2 2 z 2 z 6 z 4 z z 2 3 z 7 z 12 z -a - -- - -- - -- + -- - --- - --- - --- - - + 3 z + ---- + ---- + ----- + 6 4 2 9 7 5 3 a 8 6 4 a a a a a a a a a a 2 3 3 3 3 3 4 4 11 z 2 z 5 z 23 z 22 z 5 z 3 4 5 z 9 z > ----- - ---- + ---- + ----- + ----- + ---- - a z - 7 z - ---- - ---- - 2 9 7 5 3 a 8 6 a a a a a a a 4 4 5 5 5 5 5 6 14 z 17 z z 8 z 30 z 32 z 10 z 5 6 3 z > ----- - ----- + -- - ---- - ----- - ----- - ----- + a z + 4 z + ---- - 4 2 9 7 5 3 a 8 a a a a a a a 6 6 7 7 7 7 8 8 8 9 9 4 z 3 z 5 z 13 z 14 z 6 z 4 z 8 z 4 z z z > ---- + ---- + ---- + ----- + ----- + ---- + ---- + ---- + ---- + -- + -- 4 2 7 5 3 a 6 4 2 5 3 a a a a a a a a a a

In[19]:=
{Vassiliev[2][Knot[10, 69]], Vassiliev[3][Knot[10, 69]]}

Out[19]=
{2, 4}

In[20]:=
Kh[Knot[10, 69]][q, t]

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

In[21]:=
ColouredJones[Knot[10, 69], 2][q]

Out[21]=
-7 4 2 13 24 51 2 3 4 5 -61 + q - -- + -- + -- - -- + -- - 19 q + 117 q - 95 q - 59 q + 180 q - 6 5 4 3 q q q q q 6 7 8 9 10 11 12 13 > 104 q - 101 q + 202 q - 83 q - 117 q + 172 q - 43 q - 100 q + 14 15 16 17 18 19 20 21 > 107 q - 9 q - 60 q + 43 q + 4 q - 21 q + 9 q + 2 q - 22 23 > 3 q + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 10₆₉

10₆₈
10₇₀

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-7 - 4q^-6 + 2q^-5 + 13q^-4 - 24q^-3 + 51q^-1 - 61 - 19q + 117q² - 95q³ - 59q⁴ + 180q⁵ - 104q⁶ - 101q⁷ + 202q⁸ - 83q⁹ - 117q¹⁰ + 172q¹¹ - 43q¹² - 100q¹³ + 107q¹⁴ - 9q¹⁵ - 60q¹⁶ + 43q¹⁷ + 4q¹⁸ - 21q¹⁹ + 9q²⁰ + 2q²¹ - 3q²² + q²³
3	- q^-15 + 4q^-14 - 2q^-13 - 8q^-12 + 23q^-10 + 6q^-9 - 53q^-8 - 16q^-7 + 90q^-6 + 55q^-5 - 153q^-4 - 114q^-3 + 214q^-2 + 224q^-1 - 284 - 362q + 321q² + 552q³ - 341q⁴ - 744q⁵ + 306q⁶ + 945q⁷ - 241q⁸ - 1115q⁹ + 143q¹⁰ + 1244q¹¹ - 30q¹² - 1309q¹³ - 100q¹⁴ + 1330q¹⁵ + 207q¹⁶ - 1270q¹⁷ - 324q¹⁸ + 1175q¹⁹ + 397q²⁰ - 1010q²¹ - 464q²² + 832q²³ + 477q²⁴ - 625q²⁵ - 461q²⁶ + 434q²⁷ + 402q²⁸ - 265q²⁹ - 320q³⁰ + 135q³¹ + 235q³² - 61q³³ - 143q³⁴ + 11q³⁵ + 83q³⁶ + 2q³⁷ - 39q³⁸ - 5q³⁹ + 17q⁴⁰ + q⁴¹ - 4q⁴² - 2q⁴³ + 3q⁴⁴ - q⁴⁵
4	q^-26 - 4q^-25 + 2q^-24 + 8q^-23 - 5q^-22 + q^-21 - 29q^-20 + 12q^-19 + 54q^-18 - 9q^-17 - 146q^-15 + 7q^-14 + 213q^-13 + 65q^-12 + 25q^-11 - 503q^-10 - 150q^-9 + 528q^-8 + 433q^-7 + 285q^-6 - 1227q^-5 - 814q^-4 + 785q^-3 + 1320q^-2 + 1260q^-1 - 2106 - 2313q + 384q² + 2524q³ + 3364q⁴ - 2450q⁵ - 4413q⁶ - 1184q⁷ + 3313q⁸ + 6290q⁹ - 1678q¹⁰ - 6261q¹¹ - 3655q¹² + 3088q¹³ + 9050q¹⁴ + 19q¹⁵ - 7108q¹⁶ - 6116q¹⁷ + 1940q¹⁸ + 10743q¹⁹ + 1953q²⁰ - 6805q²¹ - 7816q²² + 349q²³ + 11061q²⁴ + 3583q²⁵ - 5573q²⁶ - 8457q²⁷ - 1326q²⁸ + 10016q²⁹ + 4668q³⁰ - 3616q³¹ - 7961q³² - 2842q³³ + 7772q³⁴ + 4975q³⁵ - 1312q³⁶ - 6333q³⁷ - 3754q³⁸ + 4814q³⁹ + 4270q⁴⁰ + 614q⁴¹ - 3958q⁴² - 3607q⁴³ + 2072q⁴⁴ + 2751q⁴⁵ + 1462q⁴⁶ - 1703q⁴⁷ - 2512q⁴⁸ + 396q⁴⁹ + 1184q⁵⁰ + 1233q⁵¹ - 354q⁵² - 1218q⁵³ - 126q⁵⁴ + 249q⁵⁵ + 606q⁵⁶ + 67q⁵⁷ - 393q⁵⁸ - 95q⁵⁹ - 28q⁶⁰ + 184q⁶¹ + 63q⁶² - 87q⁶³ - 14q⁶⁴ - 29q⁶⁵ + 34q⁶⁶ + 16q⁶⁷ - 16q⁶⁸ + 3q⁶⁹ - 6q⁷⁰ + 4q⁷¹ + 2q⁷² - 3q⁷³ + q⁷⁴
5	- q^-40 + 4q^-39 - 2q^-38 - 8q^-37 + 5q^-36 + 4q^-35 + 5q^-34 + 11q^-33 - 13q^-32 - 45q^-31 - 5q^-30 + 46q^-29 + 64q^-28 + 48q^-27 - 66q^-26 - 185q^-25 - 133q^-24 + 133q^-23 + 370q^-22 + 300q^-21 - 134q^-20 - 672q^-19 - 742q^-18 + 55q^-17 + 1186q^-16 + 1451q^-15 + 277q^-14 - 1681q^-13 - 2741q^-12 - 1209q^-11 + 2265q^-10 + 4554q^-9 + 2915q^-8 - 2282q^-7 - 6966q^-6 - 5948q^-5 + 1561q^-4 + 9605q^-3 + 10277q^-2 + 810q^-1 - 12053 - 16044q - 4986q² + 13425q³ + 22657q⁴ + 11564q⁵ - 13195q⁶ - 29561q⁷ - 19945q⁸ + 10586q⁹ + 35679q¹⁰ + 29961q¹¹ - 5663q¹² - 40415q¹³ - 40314q¹⁴ - 1405q¹⁵ + 42957q¹⁶ + 50365q¹⁷ + 9905q¹⁸ - 43384q¹⁹ - 59028q²⁰ - 19019q²¹ + 41739q²² + 65885q²³ + 27940q²⁴ - 38518q²⁵ - 70632q²⁶ - 36114q²⁷ + 34167q²⁸ + 73458q²⁹ + 43038q³⁰ - 29103q³¹ - 74269q³² - 48866q³³ + 23481q³⁴ + 73624q³⁵ + 53330q³⁶ - 17556q³⁷ - 71073q³⁸ - 56786q³⁹ + 11026q⁴⁰ + 67230q⁴¹ + 58882q⁴² - 4259q⁴³ - 61387q⁴⁴ - 59730q⁴⁵ - 2934q⁴⁶ + 54146q⁴⁷ + 58825q⁴⁸ + 9830q⁴⁹ - 45110q⁵⁰ - 56120q⁵¹ - 16172q⁵² + 35214q⁵³ + 51260q⁵⁴ + 21041q⁵⁵ - 24687q⁵⁶ - 44606q⁵⁷ - 24044q⁵⁸ + 14800q⁵⁹ + 36446q⁶⁰ + 24674q⁶¹ - 6160q⁶² - 27634q⁶³ - 23159q⁶⁴ - 339q⁶⁵ + 19118q⁶⁶ + 19803q⁶⁷ + 4407q⁶⁸ - 11638q⁶⁹ - 15435q⁷⁰ - 6302q⁷¹ + 5989q⁷² + 10922q⁷³ + 6237q⁷⁴ - 2134q⁷⁵ - 6859q⁷⁶ - 5244q⁷⁷ + 33q⁷⁸ + 3875q⁷⁹ + 3682q⁸⁰ + 831q⁸¹ - 1819q⁸² - 2331q⁸³ - 927q⁸⁴ + 709q⁸⁵ + 1263q⁸⁶ + 712q⁸⁷ - 173q⁸⁸ - 629q⁸⁹ - 416q⁹⁰ - 4q⁹¹ + 240q⁹² + 237q⁹³ + 43q⁹⁴ - 111q⁹⁵ - 91q⁹⁶ - 14q⁹⁷ + 11q⁹⁸ + 43q⁹⁹ + 23q¹⁰⁰ - 22q¹⁰¹ - 11q¹⁰² + 5q¹⁰³ - 4q¹⁰⁴ + 2q¹⁰⁵ + 6q¹⁰⁶ - 4q¹⁰⁷ - 2q¹⁰⁸ + 3q¹⁰⁹ - q¹¹⁰
6	q^-57 - 4q^-56 + 2q^-55 + 8q^-54 - 5q^-53 - 4q^-52 - 10q^-51 + 13q^-50 - 10q^-49 + 4q^-48 + 59q^-47 - 25q^-46 - 40q^-45 - 74q^-44 + 32q^-43 - 3q^-42 + 56q^-41 + 268q^-40 - 30q^-39 - 197q^-38 - 407q^-37 - 55q^-36 - 35q^-35 + 354q^-34 + 1102q^-33 + 305q^-32 - 516q^-31 - 1647q^-30 - 1015q^-29 - 702q^-28 + 1091q^-27 + 3823q^-26 + 2616q^-25 - 53q^-24 - 4545q^-23 - 5119q^-22 - 4943q^-21 + 788q^-20 + 9900q^-19 + 11202q^-18 + 6080q^-17 - 7012q^-16 - 15023q^-15 - 19992q^-14 - 8288q^-13 + 16168q^-12 + 30777q^-11 + 29120q^-10 + 2555q^-9 - 26252q^-8 - 53117q^-7 - 42476q^-6 + 5971q^-5 + 55332q^-4 + 79144q^-3 + 46574q^-2 - 16455q^-1 - 96180 - 114531q - 49406q² + 56360q³ + 145060q⁴ + 139926q⁵ + 48543q⁶ - 114346q⁷ - 209517q⁸ - 165324q⁹ - 4633q¹⁰ + 185928q¹¹ + 263001q¹² + 182376q¹³ - 66294q¹⁴ - 280859q¹⁵ - 316308q¹⁶ - 138309q¹⁷ + 159174q¹⁸ + 364485q¹⁹ + 353221q²⁰ + 54973q²¹ - 285746q²² - 447067q²³ - 308130q²⁴ + 61232q²⁵ + 402191q²⁶ + 503619q²⁷ + 210606q²⁸ - 222600q²⁹ - 516598q³⁰ - 458164q³¹ - 69685q³² + 375256q³³ + 595214q³⁴ + 349536q³⁵ - 125675q³⁶ - 523805q³⁷ - 555571q³⁸ - 190207q³⁹ + 312203q⁴⁰ + 627244q⁴¹ + 446200q⁴² - 28635q⁴³ - 491012q⁴⁴ - 601618q⁴⁵ - 282948q⁴⁶ + 236940q⁴⁷ + 616966q⁴⁸ + 504456q⁴⁹ + 59356q⁵⁰ - 433462q⁵¹ - 609988q⁵² - 354261q⁵³ + 151582q⁵⁴ + 572713q⁵⁵ + 534670q⁵⁶ + 146413q⁵⁷ - 347730q⁵⁸ - 582966q⁵⁹ - 410722q⁶⁰ + 47351q⁶¹ + 486687q⁶² + 532645q⁶³ + 234221q⁶⁴ - 225474q⁶⁵ - 507943q⁶⁶ - 440451q⁶⁷ - 71583q⁶⁸ + 351254q⁶⁹ + 479632q⁷⁰ + 301775q⁷¹ - 77438q⁷² - 376572q⁷³ - 417104q⁷⁴ - 174563q⁷⁵ + 183183q⁷⁶ + 365081q⁷⁷ + 314426q⁷⁸ + 56438q⁷⁹ - 209585q⁸⁰ - 327261q⁸¹ - 219573q⁸² + 29782q⁸³ + 212139q⁸⁴ + 256318q⁸⁵ + 128868q⁸⁶ - 58377q⁸⁷ - 196007q⁸⁸ - 190230q⁸⁹ - 59584q⁹⁰ + 73981q⁹¹ + 154418q⁹² + 125127q⁹³ + 29385q⁹⁴ - 76895q⁹⁵ - 115855q⁹⁶ - 73901q⁹⁷ - 5489q⁹⁸ + 61052q⁹⁹ + 76748q¹⁰⁰ + 48021q¹⁰¹ - 9077q¹⁰² - 46371q¹⁰³ - 46128q¹⁰⁴ - 25339q¹⁰⁵ + 9643q¹⁰⁶ + 29753q¹⁰⁷ + 30667q¹⁰⁸ + 9969q¹⁰⁹ - 9338q¹¹⁰ - 17035q¹¹¹ - 16430q¹¹² - 4632q¹¹³ + 5902q¹¹⁴ + 11702q¹¹⁵ + 7049q¹¹⁶ + 999q¹¹⁷ - 3119q¹¹⁸ - 5840q¹¹⁹ - 3582q¹²⁰ - 345q¹²¹ + 2830q¹²² + 2222q¹²³ + 1187q¹²⁴ + 185q¹²⁵ - 1254q¹²⁶ - 1153q¹²⁷ - 561q¹²⁸ + 480q¹²⁹ + 360q¹³⁰ + 308q¹³¹ + 263q¹³² - 159q¹³³ - 228q¹³⁴ - 172q¹³⁵ + 95q¹³⁶ + 16q¹³⁷ + 25q¹³⁸ + 77q¹³⁹ - 13q¹⁴⁰ - 30q¹⁴¹ - 37q¹⁴² + 28q¹⁴³ - q¹⁴⁴ - 10q¹⁴⁵ + 15q¹⁴⁶ - q¹⁴⁷ - 2q¹⁴⁸ - 6q¹⁴⁹ + 4q¹⁵⁰ + 2q¹⁵¹ - 3q¹⁵² + q¹⁵³

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	PD[Knot[10, 69]]
Out[2]=	PD[X[1, 4, 2, 5], X[7, 12, 8, 13], X[3, 11, 4, 10], X[11, 3, 12, 2], > X[13, 17, 14, 16], X[5, 15, 6, 14], X[15, 7, 16, 6], X[17, 20, 18, 1], > X[9, 19, 10, 18], X[19, 9, 20, 8]]
In[3]:=	GaussCode[Knot[10, 69]]
Out[3]=	GaussCode[-1, 4, -3, 1, -6, 7, -2, 10, -9, 3, -4, 2, -5, 6, -7, 5, -8, 9, -10, > 8]
In[4]:=	DTCode[Knot[10, 69]]
Out[4]=	DTCode[4, 10, 14, 12, 18, 2, 16, 6, 20, 8]
In[5]:=	br = BR[Knot[10, 69]]
Out[5]=	BR[5, {1, 1, 2, -1, -3, 2, 1, 4, -3, 2, -3, 4}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{5, 12}
In[7]:=	BraidIndex[Knot[10, 69]]
Out[7]=	5
In[8]:=	Show[DrawMorseLink[Knot[10, 69]]]

Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 69]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 3, 3, NotAvailable, 2}
In[10]:=	alex = Alexander[Knot[10, 69]][t]
Out[10]=	-3 7 21 2 3 -29 + t - -- + -- + 21 t - 7 t + t 2 t t
In[11]:=	Conway[Knot[10, 69]][z]
Out[11]=	2 4 6 1 + 2 z - z + z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 69]}
In[13]:=	{KnotDet[Knot[10, 69]], KnotSignature[Knot[10, 69]]}
Out[13]=	{87, 2}
In[14]:=	Jones[Knot[10, 69]][q]
Out[14]=	-2 4 2 3 4 5 6 7 8 -7 - q + - + 11 q - 14 q + 15 q - 13 q + 11 q - 7 q + 3 q - q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 69]}
In[16]:=	A2Invariant[Knot[10, 69]][q]
Out[16]=	-6 2 -2 2 4 6 8 12 14 16 18 20 -q + -- - q + 4 q - 3 q + 2 q - q + 2 q - 2 q + 3 q - q - q + 4 q 22 24 26 > 2 q - q - q
In[17]:=	HOMFLYPT[Knot[10, 69]][a, z]
Out[17]=	2 2 2 4 4 6 -8 2 2 2 2 3 z 5 z 5 z 4 3 z 3 z z -a + -- - -- + -- - z + ---- - ---- + ---- - z - ---- + ---- + -- 6 4 2 6 4 2 4 2 2 a a a a a a a a a
In[18]:=	Kauffman[Knot[10, 69]][a, z]
Out[18]=	2 2 2 -8 2 2 2 z 2 z 6 z 4 z z 2 3 z 7 z 12 z -a - -- - -- - -- + -- - --- - --- - --- - - + 3 z + ---- + ---- + ----- + 6 4 2 9 7 5 3 a 8 6 4 a a a a a a a a a a 2 3 3 3 3 3 4 4 11 z 2 z 5 z 23 z 22 z 5 z 3 4 5 z 9 z > ----- - ---- + ---- + ----- + ----- + ---- - a z - 7 z - ---- - ---- - 2 9 7 5 3 a 8 6 a a a a a a a 4 4 5 5 5 5 5 6 14 z 17 z z 8 z 30 z 32 z 10 z 5 6 3 z > ----- - ----- + -- - ---- - ----- - ----- - ----- + a z + 4 z + ---- - 4 2 9 7 5 3 a 8 a a a a a a a 6 6 7 7 7 7 8 8 8 9 9 4 z 3 z 5 z 13 z 14 z 6 z 4 z 8 z 4 z z z > ---- + ---- + ---- + ----- + ----- + ---- + ---- + ---- + ---- + -- + -- 4 2 7 5 3 a 6 4 2 5 3 a a a a a a a a a a
In[19]:=	{Vassiliev[2][Knot[10, 69]], Vassiliev[3][Knot[10, 69]]}
Out[19]=	{2, 4}
In[20]:=	Kh[Knot[10, 69]][q, t]
Out[20]=	3 1 3 1 4 3 q 3 5 5 2 7 q + 5 q + ----- + ----- + ---- + --- + --- + 8 q t + 6 q t + 7 q t + 5 3 3 2 2 q t t q t q t q t 7 2 7 3 9 3 9 4 11 4 11 5 13 5 > 8 q t + 6 q t + 7 q t + 5 q t + 6 q t + 2 q t + 5 q t + 13 6 15 6 17 7 > q t + 2 q t + q t
In[21]:=	ColouredJones[Knot[10, 69], 2][q]
Out[21]=	-7 4 2 13 24 51 2 3 4 5 -61 + q - -- + -- + -- - -- + -- - 19 q + 117 q - 95 q - 59 q + 180 q - 6 5 4 3 q q q q q 6 7 8 9 10 11 12 13 > 104 q - 101 q + 202 q - 83 q - 117 q + 172 q - 43 q - 100 q + 14 15 16 17 18 19 20 21 > 107 q - 9 q - 60 q + 43 q + 4 q - 21 q + 9 q + 2 q - 22 23 > 3 q + q

The Alternating Knot 1069

The Alternating Knot 10₆₉