The Alternating Knot 10₁₉

Visit 10₁₉'s page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 10₁₉'s page at Knotilus!

Acknowledgement

KnotPlot

PD Presentation: X₁₆₂₇ X_3,12,4,13 X_15,1,16,20 X_7,17,8,16 X_19,9,20,8 X_9,19,10,18 X_17,11,18,10 X_5,14,6,15 X_11,2,12,3 X_13,4,14,5

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

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

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

Reversible 2 3 2 / NotAvailable 1

Alexander Polynomial: 2t^-3 - 7t^-2 + 11t^-1 - 11 + 11t - 7t² + 2t³

Conway Polynomial: 1 + z² + 5z⁴ + 2z⁶

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

Determinant and Signature: {51, -2}

Jones Polynomial: - q^-6 + 3q^-5 - 5q^-4 + 7q^-3 - 8q^-2 + 8q^-1 - 7 + 6q - 3q² + 2q³ - q⁴

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

A2 (sl(3)) Invariant: - q^-18 + q^-16 + q^-10 - 2q^-8 + q^-6 - q^-4 + q^-2 + 2 + 2q⁴ - q¹²

HOMFLY-PT Polynomial: - a^-2 - 3a^-2z² - a^-2z⁴ + 3 + 5z² + 4z⁴ + z⁶ - a² + a²z² + 3a²z⁴ + a²z⁶ - 2a⁴z² - a⁴z⁴

Kauffman Polynomial: - 2a^-3z + 7a^-3z³ - 5a^-3z⁵ + a^-3z⁷ + a^-2 - 9a^-2z² + 16a^-2z⁴ - 10a^-2z⁶ + 2a^-2z⁸ - 4a^-1z + 13a^-1z³ - 8a^-1z⁵ - a^-1z⁷ + a^-1z⁹ + 3 - 13z² + 23z⁴ - 19z⁶ + 5z⁸ - 2az + 11az³ - 15az⁵ + 3az⁷ + az⁹ + a² - 4a²z⁴ - 3a²z⁶ + 3a²z⁸ + a³z - 7a³z⁵ + 5a³z⁷ + 3a⁴z² - 8a⁴z⁴ + 6a⁴z⁶ + a⁵z - 4a⁵z³ + 5a⁵z⁵ - a⁶z² + 3a⁶z⁴ + a⁷z³

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=-2 is the signature of 10₁₉. 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 = 9 1

j = 7 1

j = 5 2 1

j = 3 4 1

j = 1 3 2

j = -1 5 4

j = -3 4 4

j = -5 3 4

j = -7 2 4

j = -9 1 3

j = -11 2

j = -13 1

n Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q^-17 - 3q^-16 + 2q^-15 + 5q^-14 - 13q^-13 + 9q^-12 + 11q^-11 - 29q^-10 + 19q^-9 + 18q^-8 - 44q^-7 + 25q^-6 + 26q^-5 - 52q^-4 + 21q^-3 + 33q^-2 - 49q^-1 + 11 + 34q - 36q² + q³ + 27q⁴ - 20q⁵ - 4q⁶ + 16q⁷ - 7q⁸ - 5q⁹ + 6q¹⁰ - q¹¹ - 2q¹² + q¹³

3 - q^-33 + 3q^-32 - 2q^-31 - 2q^-30 + q^-29 + 6q^-28 - 4q^-27 - 10q^-26 + 10q^-25 + 11q^-24 - 15q^-23 - 15q^-22 + 26q^-21 + 14q^-20 - 36q^-19 - 15q^-18 + 52q^-17 + 13q^-16 - 66q^-15 - 16q^-14 + 82q^-13 + 20q^-12 - 91q^-11 - 32q^-10 + 96q^-9 + 45q^-8 - 93q^-7 - 60q^-6 + 86q^-5 + 72q^-4 - 69q^-3 - 87q^-2 + 59q^-1 + 87 - 36q - 96q² + 26q³ + 87q⁴ - 3q⁵ - 87q⁶ - 5q⁷ + 70q⁸ + 24q⁹ - 61q¹⁰ - 28q¹¹ + 41q¹² + 34q¹³ - 26q¹⁴ - 31q¹⁵ + 12q¹⁶ + 24q¹⁷ - q¹⁸ - 18q¹⁹ - 2q²⁰ + 9q²¹ + 4q²² - 5q²³ - 2q²⁴ + q²⁵ + 2q²⁶ - q²⁷

4 q^-54 - 3q^-53 + 2q^-52 + 2q^-51 - 4q^-50 + 6q^-49 - 11q^-48 + 9q^-47 + 5q^-46 - 15q^-45 + 20q^-44 - 29q^-43 + 21q^-42 + 9q^-41 - 34q^-40 + 51q^-39 - 51q^-38 + 25q^-37 - 2q^-36 - 52q^-35 + 120q^-34 - 65q^-33 + 4q^-32 - 51q^-31 - 77q^-30 + 237q^-29 - 38q^-28 - 29q^-27 - 155q^-26 - 137q^-25 + 370q^-24 + 40q^-23 - 22q^-22 - 265q^-21 - 246q^-20 + 437q^-19 + 122q^-18 + 52q^-17 - 301q^-16 - 354q^-15 + 401q^-14 + 136q^-13 + 154q^-12 - 238q^-11 - 402q^-10 + 302q^-9 + 78q^-8 + 228q^-7 - 129q^-6 - 382q^-5 + 193q^-4 - 6q^-3 + 267q^-2 - 20q^-1 - 331 + 90q - 83q² + 276q³ + 77q⁴ - 254q⁵ + 4q⁶ - 160q⁷ + 241q⁸ + 157q⁹ - 141q¹⁰ - 38q¹¹ - 223q¹² + 148q¹³ + 179q¹⁴ - 15q¹⁵ - 9q¹⁶ - 233q¹⁷ + 29q¹⁸ + 123q¹⁹ + 62q²⁰ + 56q²¹ - 168q²² - 44q²³ + 32q²⁴ + 57q²⁵ + 89q²⁶ - 73q²⁷ - 43q²⁸ - 22q²⁹ + 13q³⁰ + 66q³¹ - 12q³² - 13q³³ - 23q³⁴ - 10q³⁵ + 27q³⁶ + 2q³⁷ + 2q³⁸ - 7q³⁹ - 8q⁴⁰ + 6q⁴¹ + q⁴² + 2q⁴³ - q⁴⁴ - 2q⁴⁵ + q⁴⁶

5 - q^-80 + 3q^-79 - 2q^-78 - 2q^-77 + 4q^-76 - 3q^-75 - q^-74 + 6q^-73 - 4q^-72 - 6q^-71 + 8q^-70 - q^-69 - q^-68 + 8q^-67 - 6q^-66 - 12q^-65 - 2q^-64 + 6q^-63 + 16q^-62 + 17q^-61 + 2q^-60 - 39q^-59 - 53q^-58 - 2q^-57 + 71q^-56 + 97q^-55 + 36q^-54 - 112q^-53 - 205q^-52 - 79q^-51 + 186q^-50 + 326q^-49 + 164q^-48 - 240q^-47 - 523q^-46 - 300q^-45 + 313q^-44 + 736q^-43 + 491q^-42 - 330q^-41 - 987q^-40 - 740q^-39 + 306q^-38 + 1211q^-37 + 1037q^-36 - 205q^-35 - 1402q^-34 - 1329q^-33 + 40q^-32 + 1489q^-31 + 1602q^-30 + 178q^-29 - 1496q^-28 - 1792q^-27 - 393q^-26 + 1388q^-25 + 1880q^-24 + 608q^-23 - 1234q^-22 - 1869q^-21 - 745q^-20 + 1030q^-19 + 1770q^-18 + 834q^-17 - 832q^-16 - 1622q^-15 - 857q^-14 + 639q^-13 + 1453q^-12 + 861q^-11 - 491q^-10 - 1269q^-9 - 831q^-8 + 311q^-7 + 1109q^-6 + 833q^-5 - 192q^-4 - 921q^-3 - 791q^-2 - 10q^-1 + 752 + 790q + 133q² - 528q³ - 715q⁴ - 329q⁵ + 323q⁶ + 645q⁷ + 410q⁸ - 71q⁹ - 484q¹⁰ - 518q¹¹ - 126q¹² + 319q¹³ + 473q¹⁴ + 323q¹⁵ - 90q¹⁶ - 436q¹⁷ - 409q¹⁸ - 92q¹⁹ + 257q²⁰ + 445q²¹ + 277q²² - 115q²³ - 377q²⁴ - 343q²⁵ - 84q²⁶ + 251q²⁷ + 378q²⁸ + 199q²⁹ - 107q³⁰ - 298q³¹ - 270q³² - 43q³³ + 202q³⁴ + 266q³⁵ + 129q³⁶ - 80q³⁷ - 203q³⁸ - 173q³⁹ - 17q⁴⁰ + 128q⁴¹ + 154q⁴² + 71q⁴³ - 44q⁴⁴ - 115q⁴⁵ - 87q⁴⁶ - 3q⁴⁷ + 60q⁴⁸ + 70q⁴⁹ + 37q⁵⁰ - 27q⁵¹ - 49q⁵² - 28q⁵³ - 2q⁵⁴ + 21q⁵⁵ + 29q⁵⁶ + 7q⁵⁷ - 13q⁵⁸ - 10q⁵⁹ - 7q⁶⁰ - 2q⁶¹ + 9q⁶² + 6q⁶³ - 2q⁶⁴ - 2q⁶⁵ - q⁶⁶ - 2q⁶⁷ + q⁶⁸ + 2q⁶⁹ - q⁷⁰

6 q^-111 - 3q^-110 + 2q^-109 + 2q^-108 - 4q^-107 + 3q^-106 - 2q^-105 + 6q^-104 - 11q^-103 + 5q^-102 + 13q^-101 - 21q^-100 + 7q^-99 - 3q^-98 + 13q^-97 - 22q^-96 + 19q^-95 + 38q^-94 - 58q^-93 + 2q^-92 - 13q^-91 + 17q^-90 - 35q^-89 + 74q^-88 + 109q^-87 - 121q^-86 - 39q^-85 - 85q^-84 - 18q^-83 - 26q^-82 + 240q^-81 + 308q^-80 - 195q^-79 - 189q^-78 - 338q^-77 - 179q^-76 + 50q^-75 + 662q^-74 + 783q^-73 - 238q^-72 - 579q^-71 - 979q^-70 - 606q^-69 + 229q^-68 + 1541q^-67 + 1767q^-66 - 122q^-65 - 1294q^-64 - 2258q^-63 - 1556q^-62 + 382q^-61 + 2950q^-60 + 3521q^-59 + 486q^-58 - 2112q^-57 - 4198q^-56 - 3341q^-55 + 41q^-54 + 4498q^-53 + 5998q^-52 + 2001q^-51 - 2361q^-50 - 6223q^-49 - 5863q^-48 - 1318q^-47 + 5297q^-46 + 8435q^-45 + 4292q^-44 - 1423q^-43 - 7289q^-42 - 8205q^-41 - 3485q^-40 + 4702q^-39 + 9672q^-38 + 6336q^-37 + 403q^-36 - 6809q^-35 - 9210q^-34 - 5375q^-33 + 3131q^-32 + 9258q^-31 + 7094q^-30 + 2024q^-29 - 5347q^-28 - 8648q^-27 - 6122q^-26 + 1644q^-25 + 7895q^-24 + 6594q^-23 + 2784q^-22 - 3911q^-21 - 7356q^-20 - 5953q^-19 + 726q^-18 + 6509q^-17 + 5699q^-16 + 3018q^-15 - 2826q^-14 - 6126q^-13 - 5644q^-12 + 6q^-11 + 5280q^-10 + 4984q^-9 + 3341q^-8 - 1720q^-7 - 4974q^-6 - 5506q^-5 - 959q^-4 + 3832q^-3 + 4267q^-2 + 3821q^-1 - 282 - 3486q - 5182q² - 2060q³ + 1959q⁴ + 3078q⁵ + 3967q⁶ + 1250q⁷ - 1507q⁸ - 4167q⁹ - 2711q¹⁰ + 4q¹¹ + 1290q¹² + 3249q¹³ + 2207q¹⁴ + 536q¹⁵ - 2379q¹⁶ - 2345q¹⁷ - 1286q¹⁸ - 599q¹⁹ + 1642q²⁰ + 2006q²¹ + 1814q²² - 422q²³ - 984q²⁴ - 1314q²⁵ - 1697q²⁶ - 137q²⁷ + 757q²⁸ + 1742q²⁹ + 729q³⁰ + 540q³¹ - 284q³² - 1463q³³ - 1061q³⁴ - 589q³⁵ + 651q³⁶ + 582q³⁷ + 1190q³⁸ + 775q³⁹ - 368q⁴⁰ - 761q⁴¹ - 1029q⁴² - 351q⁴³ - 273q⁴⁴ + 740q⁴⁵ + 961q⁴⁶ + 474q⁴⁷ + 53q⁴⁸ - 519q⁴⁹ - 495q⁵⁰ - 778q⁵¹ - 20q⁵² + 403q⁵³ + 494q⁵⁴ + 437q⁵⁵ + 110q⁵⁶ - 63q⁵⁷ - 585q⁵⁸ - 306q⁵⁹ - 104q⁶⁰ + 107q⁶¹ + 257q⁶² + 263q⁶³ + 234q⁶⁴ - 176q⁶⁵ - 154q⁶⁶ - 179q⁶⁷ - 100q⁶⁸ - 5q⁶⁹ + 108q⁷⁰ + 198q⁷¹ + 12q⁷² + 9q⁷³ - 62q⁷⁴ - 69q⁷⁵ - 71q⁷⁶ - 7q⁷⁷ + 75q⁷⁸ + 16q⁷⁹ + 34q⁸⁰ + 2q⁸¹ - 10q⁸² - 35q⁸³ - 20q⁸⁴ + 17q⁸⁵ - q⁸⁶ + 12q⁸⁷ + 6q⁸⁸ + 5q⁸⁹ - 9q⁹⁰ - 8q⁹¹ + 4q⁹² - 2q⁹³ + 2q⁹⁴ + q⁹⁵ + 2q⁹⁶ - q⁹⁷ - 2q⁹⁸ + q⁹⁹

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, 19]]

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

In[3]:=
GaussCode[Knot[10, 19]]

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

In[4]:=
DTCode[Knot[10, 19]]

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

In[5]:=
br = BR[Knot[10, 19]]

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

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

Out[6]=
{4, 11}

In[7]:=
BraidIndex[Knot[10, 19]]

Out[7]=
4

In[8]:=
Show[DrawMorseLink[Knot[10, 19]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[10, 19]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

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

In[10]:=
alex = Alexander[Knot[10, 19]][t]

Out[10]=
2 7 11 2 3 -11 + -- - -- + -- + 11 t - 7 t + 2 t 3 2 t t t

In[11]:=
Conway[Knot[10, 19]][z]

Out[11]=
2 4 6 1 + z + 5 z + 2 z

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

Out[12]=
{Knot[10, 19]}

In[13]:=
{KnotDet[Knot[10, 19]], KnotSignature[Knot[10, 19]]}

Out[13]=
{51, -2}

In[14]:=
Jones[Knot[10, 19]][q]

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

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

Out[15]=
{Knot[10, 19]}

In[16]:=
A2Invariant[Knot[10, 19]][q]

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

In[17]:=
HOMFLYPT[Knot[10, 19]][a, z]

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

In[18]:=
Kauffman[Knot[10, 19]][a, z]

Out[18]=
2 -2 2 2 z 4 z 3 5 2 9 z 4 2 3 + a + a - --- - --- - 2 a z + a z + a z - 13 z - ---- + 3 a z - 3 a 2 a a 3 3 4 6 2 7 z 13 z 3 5 3 7 3 4 16 z > a z + ---- + ----- + 11 a z - 4 a z + a z + 23 z + ----- - 3 a 2 a a 5 5 2 4 4 4 6 4 5 z 8 z 5 3 5 5 5 > 4 a z - 8 a z + 3 a z - ---- - ---- - 15 a z - 7 a z + 5 a z - 3 a a 6 7 7 6 10 z 2 6 4 6 z z 7 3 7 8 > 19 z - ----- - 3 a z + 6 a z + -- - -- + 3 a z + 5 a z + 5 z + 2 3 a a a 8 9 2 z 2 8 z 9 > ---- + 3 a z + -- + a z 2 a a

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

Out[19]=
{1, 0}

In[20]:=
Kh[Knot[10, 19]][q, t]

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

In[21]:=
ColouredJones[Knot[10, 19], 2][q]

Out[21]=
-17 3 2 5 13 9 11 29 19 18 44 25 26 11 + q - --- + --- + --- - --- + --- + --- - --- + -- + -- - -- + -- + -- - 16 15 14 13 12 11 10 9 8 7 6 5 q q q q q q q q q q q q 52 21 33 49 2 3 4 5 6 7 > -- + -- + -- - -- + 34 q - 36 q + q + 27 q - 20 q - 4 q + 16 q - 4 3 2 q q q q 8 9 10 11 12 13 > 7 q - 5 q + 6 q - q - 2 q + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 10₁₉

10₁₈
10₂₀

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-17 - 3q^-16 + 2q^-15 + 5q^-14 - 13q^-13 + 9q^-12 + 11q^-11 - 29q^-10 + 19q^-9 + 18q^-8 - 44q^-7 + 25q^-6 + 26q^-5 - 52q^-4 + 21q^-3 + 33q^-2 - 49q^-1 + 11 + 34q - 36q² + q³ + 27q⁴ - 20q⁵ - 4q⁶ + 16q⁷ - 7q⁸ - 5q⁹ + 6q¹⁰ - q¹¹ - 2q¹² + q¹³
3	- q^-33 + 3q^-32 - 2q^-31 - 2q^-30 + q^-29 + 6q^-28 - 4q^-27 - 10q^-26 + 10q^-25 + 11q^-24 - 15q^-23 - 15q^-22 + 26q^-21 + 14q^-20 - 36q^-19 - 15q^-18 + 52q^-17 + 13q^-16 - 66q^-15 - 16q^-14 + 82q^-13 + 20q^-12 - 91q^-11 - 32q^-10 + 96q^-9 + 45q^-8 - 93q^-7 - 60q^-6 + 86q^-5 + 72q^-4 - 69q^-3 - 87q^-2 + 59q^-1 + 87 - 36q - 96q² + 26q³ + 87q⁴ - 3q⁵ - 87q⁶ - 5q⁷ + 70q⁸ + 24q⁹ - 61q¹⁰ - 28q¹¹ + 41q¹² + 34q¹³ - 26q¹⁴ - 31q¹⁵ + 12q¹⁶ + 24q¹⁷ - q¹⁸ - 18q¹⁹ - 2q²⁰ + 9q²¹ + 4q²² - 5q²³ - 2q²⁴ + q²⁵ + 2q²⁶ - q²⁷
4	q^-54 - 3q^-53 + 2q^-52 + 2q^-51 - 4q^-50 + 6q^-49 - 11q^-48 + 9q^-47 + 5q^-46 - 15q^-45 + 20q^-44 - 29q^-43 + 21q^-42 + 9q^-41 - 34q^-40 + 51q^-39 - 51q^-38 + 25q^-37 - 2q^-36 - 52q^-35 + 120q^-34 - 65q^-33 + 4q^-32 - 51q^-31 - 77q^-30 + 237q^-29 - 38q^-28 - 29q^-27 - 155q^-26 - 137q^-25 + 370q^-24 + 40q^-23 - 22q^-22 - 265q^-21 - 246q^-20 + 437q^-19 + 122q^-18 + 52q^-17 - 301q^-16 - 354q^-15 + 401q^-14 + 136q^-13 + 154q^-12 - 238q^-11 - 402q^-10 + 302q^-9 + 78q^-8 + 228q^-7 - 129q^-6 - 382q^-5 + 193q^-4 - 6q^-3 + 267q^-2 - 20q^-1 - 331 + 90q - 83q² + 276q³ + 77q⁴ - 254q⁵ + 4q⁶ - 160q⁷ + 241q⁸ + 157q⁹ - 141q¹⁰ - 38q¹¹ - 223q¹² + 148q¹³ + 179q¹⁴ - 15q¹⁵ - 9q¹⁶ - 233q¹⁷ + 29q¹⁸ + 123q¹⁹ + 62q²⁰ + 56q²¹ - 168q²² - 44q²³ + 32q²⁴ + 57q²⁵ + 89q²⁶ - 73q²⁷ - 43q²⁸ - 22q²⁹ + 13q³⁰ + 66q³¹ - 12q³² - 13q³³ - 23q³⁴ - 10q³⁵ + 27q³⁶ + 2q³⁷ + 2q³⁸ - 7q³⁹ - 8q⁴⁰ + 6q⁴¹ + q⁴² + 2q⁴³ - q⁴⁴ - 2q⁴⁵ + q⁴⁶
5	- q^-80 + 3q^-79 - 2q^-78 - 2q^-77 + 4q^-76 - 3q^-75 - q^-74 + 6q^-73 - 4q^-72 - 6q^-71 + 8q^-70 - q^-69 - q^-68 + 8q^-67 - 6q^-66 - 12q^-65 - 2q^-64 + 6q^-63 + 16q^-62 + 17q^-61 + 2q^-60 - 39q^-59 - 53q^-58 - 2q^-57 + 71q^-56 + 97q^-55 + 36q^-54 - 112q^-53 - 205q^-52 - 79q^-51 + 186q^-50 + 326q^-49 + 164q^-48 - 240q^-47 - 523q^-46 - 300q^-45 + 313q^-44 + 736q^-43 + 491q^-42 - 330q^-41 - 987q^-40 - 740q^-39 + 306q^-38 + 1211q^-37 + 1037q^-36 - 205q^-35 - 1402q^-34 - 1329q^-33 + 40q^-32 + 1489q^-31 + 1602q^-30 + 178q^-29 - 1496q^-28 - 1792q^-27 - 393q^-26 + 1388q^-25 + 1880q^-24 + 608q^-23 - 1234q^-22 - 1869q^-21 - 745q^-20 + 1030q^-19 + 1770q^-18 + 834q^-17 - 832q^-16 - 1622q^-15 - 857q^-14 + 639q^-13 + 1453q^-12 + 861q^-11 - 491q^-10 - 1269q^-9 - 831q^-8 + 311q^-7 + 1109q^-6 + 833q^-5 - 192q^-4 - 921q^-3 - 791q^-2 - 10q^-1 + 752 + 790q + 133q² - 528q³ - 715q⁴ - 329q⁵ + 323q⁶ + 645q⁷ + 410q⁸ - 71q⁹ - 484q¹⁰ - 518q¹¹ - 126q¹² + 319q¹³ + 473q¹⁴ + 323q¹⁵ - 90q¹⁶ - 436q¹⁷ - 409q¹⁸ - 92q¹⁹ + 257q²⁰ + 445q²¹ + 277q²² - 115q²³ - 377q²⁴ - 343q²⁵ - 84q²⁶ + 251q²⁷ + 378q²⁸ + 199q²⁹ - 107q³⁰ - 298q³¹ - 270q³² - 43q³³ + 202q³⁴ + 266q³⁵ + 129q³⁶ - 80q³⁷ - 203q³⁸ - 173q³⁹ - 17q⁴⁰ + 128q⁴¹ + 154q⁴² + 71q⁴³ - 44q⁴⁴ - 115q⁴⁵ - 87q⁴⁶ - 3q⁴⁷ + 60q⁴⁸ + 70q⁴⁹ + 37q⁵⁰ - 27q⁵¹ - 49q⁵² - 28q⁵³ - 2q⁵⁴ + 21q⁵⁵ + 29q⁵⁶ + 7q⁵⁷ - 13q⁵⁸ - 10q⁵⁹ - 7q⁶⁰ - 2q⁶¹ + 9q⁶² + 6q⁶³ - 2q⁶⁴ - 2q⁶⁵ - q⁶⁶ - 2q⁶⁷ + q⁶⁸ + 2q⁶⁹ - q⁷⁰
6	q^-111 - 3q^-110 + 2q^-109 + 2q^-108 - 4q^-107 + 3q^-106 - 2q^-105 + 6q^-104 - 11q^-103 + 5q^-102 + 13q^-101 - 21q^-100 + 7q^-99 - 3q^-98 + 13q^-97 - 22q^-96 + 19q^-95 + 38q^-94 - 58q^-93 + 2q^-92 - 13q^-91 + 17q^-90 - 35q^-89 + 74q^-88 + 109q^-87 - 121q^-86 - 39q^-85 - 85q^-84 - 18q^-83 - 26q^-82 + 240q^-81 + 308q^-80 - 195q^-79 - 189q^-78 - 338q^-77 - 179q^-76 + 50q^-75 + 662q^-74 + 783q^-73 - 238q^-72 - 579q^-71 - 979q^-70 - 606q^-69 + 229q^-68 + 1541q^-67 + 1767q^-66 - 122q^-65 - 1294q^-64 - 2258q^-63 - 1556q^-62 + 382q^-61 + 2950q^-60 + 3521q^-59 + 486q^-58 - 2112q^-57 - 4198q^-56 - 3341q^-55 + 41q^-54 + 4498q^-53 + 5998q^-52 + 2001q^-51 - 2361q^-50 - 6223q^-49 - 5863q^-48 - 1318q^-47 + 5297q^-46 + 8435q^-45 + 4292q^-44 - 1423q^-43 - 7289q^-42 - 8205q^-41 - 3485q^-40 + 4702q^-39 + 9672q^-38 + 6336q^-37 + 403q^-36 - 6809q^-35 - 9210q^-34 - 5375q^-33 + 3131q^-32 + 9258q^-31 + 7094q^-30 + 2024q^-29 - 5347q^-28 - 8648q^-27 - 6122q^-26 + 1644q^-25 + 7895q^-24 + 6594q^-23 + 2784q^-22 - 3911q^-21 - 7356q^-20 - 5953q^-19 + 726q^-18 + 6509q^-17 + 5699q^-16 + 3018q^-15 - 2826q^-14 - 6126q^-13 - 5644q^-12 + 6q^-11 + 5280q^-10 + 4984q^-9 + 3341q^-8 - 1720q^-7 - 4974q^-6 - 5506q^-5 - 959q^-4 + 3832q^-3 + 4267q^-2 + 3821q^-1 - 282 - 3486q - 5182q² - 2060q³ + 1959q⁴ + 3078q⁵ + 3967q⁶ + 1250q⁷ - 1507q⁸ - 4167q⁹ - 2711q¹⁰ + 4q¹¹ + 1290q¹² + 3249q¹³ + 2207q¹⁴ + 536q¹⁵ - 2379q¹⁶ - 2345q¹⁷ - 1286q¹⁸ - 599q¹⁹ + 1642q²⁰ + 2006q²¹ + 1814q²² - 422q²³ - 984q²⁴ - 1314q²⁵ - 1697q²⁶ - 137q²⁷ + 757q²⁸ + 1742q²⁹ + 729q³⁰ + 540q³¹ - 284q³² - 1463q³³ - 1061q³⁴ - 589q³⁵ + 651q³⁶ + 582q³⁷ + 1190q³⁸ + 775q³⁹ - 368q⁴⁰ - 761q⁴¹ - 1029q⁴² - 351q⁴³ - 273q⁴⁴ + 740q⁴⁵ + 961q⁴⁶ + 474q⁴⁷ + 53q⁴⁸ - 519q⁴⁹ - 495q⁵⁰ - 778q⁵¹ - 20q⁵² + 403q⁵³ + 494q⁵⁴ + 437q⁵⁵ + 110q⁵⁶ - 63q⁵⁷ - 585q⁵⁸ - 306q⁵⁹ - 104q⁶⁰ + 107q⁶¹ + 257q⁶² + 263q⁶³ + 234q⁶⁴ - 176q⁶⁵ - 154q⁶⁶ - 179q⁶⁷ - 100q⁶⁸ - 5q⁶⁹ + 108q⁷⁰ + 198q⁷¹ + 12q⁷² + 9q⁷³ - 62q⁷⁴ - 69q⁷⁵ - 71q⁷⁶ - 7q⁷⁷ + 75q⁷⁸ + 16q⁷⁹ + 34q⁸⁰ + 2q⁸¹ - 10q⁸² - 35q⁸³ - 20q⁸⁴ + 17q⁸⁵ - q⁸⁶ + 12q⁸⁷ + 6q⁸⁸ + 5q⁸⁹ - 9q⁹⁰ - 8q⁹¹ + 4q⁹² - 2q⁹³ + 2q⁹⁴ + q⁹⁵ + 2q⁹⁶ - q⁹⁷ - 2q⁹⁸ + q⁹⁹

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

Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 19]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 3, 2, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 19]][t]
Out[10]=	2 7 11 2 3 -11 + -- - -- + -- + 11 t - 7 t + 2 t 3 2 t t t
In[11]:=	Conway[Knot[10, 19]][z]
Out[11]=	2 4 6 1 + z + 5 z + 2 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 19]}
In[13]:=	{KnotDet[Knot[10, 19]], KnotSignature[Knot[10, 19]]}
Out[13]=	{51, -2}
In[14]:=	Jones[Knot[10, 19]][q]
Out[14]=	-6 3 5 7 8 8 2 3 4 -7 - q + -- - -- + -- - -- + - + 6 q - 3 q + 2 q - q 5 4 3 2 q q q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 19]}
In[16]:=	A2Invariant[Knot[10, 19]][q]
Out[16]=	-18 -16 -10 2 -6 -4 -2 4 12 2 - q + q + q - -- + q - q + q + 2 q - q 8 q
In[17]:=	HOMFLYPT[Knot[10, 19]][a, z]
Out[17]=	2 4 -2 2 2 3 z 2 2 4 2 4 z 2 4 4 4 3 - a - a + 5 z - ---- + a z - 2 a z + 4 z - -- + 3 a z - a z + 2 2 a a 6 2 6 > z + a z
In[18]:=	Kauffman[Knot[10, 19]][a, z]
Out[18]=	2 -2 2 2 z 4 z 3 5 2 9 z 4 2 3 + a + a - --- - --- - 2 a z + a z + a z - 13 z - ---- + 3 a z - 3 a 2 a a 3 3 4 6 2 7 z 13 z 3 5 3 7 3 4 16 z > a z + ---- + ----- + 11 a z - 4 a z + a z + 23 z + ----- - 3 a 2 a a 5 5 2 4 4 4 6 4 5 z 8 z 5 3 5 5 5 > 4 a z - 8 a z + 3 a z - ---- - ---- - 15 a z - 7 a z + 5 a z - 3 a a 6 7 7 6 10 z 2 6 4 6 z z 7 3 7 8 > 19 z - ----- - 3 a z + 6 a z + -- - -- + 3 a z + 5 a z + 5 z + 2 3 a a a 8 9 2 z 2 8 z 9 > ---- + 3 a z + -- + a z 2 a a
In[19]:=	{Vassiliev[2][Knot[10, 19]], Vassiliev[3][Knot[10, 19]]}
Out[19]=	{1, 0}
In[20]:=	Kh[Knot[10, 19]][q, t]
Out[20]=	4 5 1 2 1 3 2 4 3 4 -- + - + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ---- + 3 q 13 5 11 4 9 4 9 3 7 3 7 2 5 2 5 q q t q t q t q t q t q t q t q t 4 4 t 2 3 2 3 3 5 3 5 4 7 4 > ---- + --- + 3 q t + 2 q t + 4 q t + q t + 2 q t + q t + q t + 3 q q t 9 5 > q t
In[21]:=	ColouredJones[Knot[10, 19], 2][q]
Out[21]=	-17 3 2 5 13 9 11 29 19 18 44 25 26 11 + q - --- + --- + --- - --- + --- + --- - --- + -- + -- - -- + -- + -- - 16 15 14 13 12 11 10 9 8 7 6 5 q q q q q q q q q q q q 52 21 33 49 2 3 4 5 6 7 > -- + -- + -- - -- + 34 q - 36 q + q + 27 q - 20 q - 4 q + 16 q - 4 3 2 q q q q 8 9 10 11 12 13 > 7 q - 5 q + 6 q - q - 2 q + q

The Alternating Knot 1019

The Alternating Knot 10₁₉