The Non Alternating Knot 10₁₂₉

Visit 10₁₂₉'s page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 10₁₂₉'s page at Knotilus!

Acknowledgement

KnotPlot

PD Presentation: X₁₄₂₅ X₃₈₄₉ X_5,14,6,15 X_20,16,1,15 X_16,10,17,9 X_10,20,11,19 X_18,12,19,11 X_12,18,13,17 X_13,6,14,7 X₇₂₈₃

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

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

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 1 2 3 / NotAvailable 1

Alexander Polynomial: 2t^-2 - 6t^-1 + 9 - 6t + 2t²

Conway Polynomial: 1 + 2z² + 2z⁴

Other knots with the same Alexander/Conway Polynomial: {8₈, K11n39, K11n45, K11n50, K11n132, ...}

Determinant and Signature: {25, 0}

Jones Polynomial: - q^-5 + 2q^-4 - 3q^-3 + 4q^-2 - 4q^-1 + 5 - 3q + 2q² - q³

Other knots (up to mirrors) with the same Jones Polynomial: {8₈, ...}

A2 (sl(3)) Invariant: - q^-16 - q^-10 + q^-8 + q^-4 + 2q^-2 + 1 + 2q² - q⁴ - q¹⁰

HOMFLY-PT Polynomial: - a^-2 - a^-2z² + 2 + 2z² + z⁴ + a² + 2a²z² + a²z⁴ - a⁴ - a⁴z²

Kauffman Polynomial: - 2a^-3z + a^-3z³ + a^-2 - 3a^-2z² + 2a^-2z⁴ - 5a^-1z + 9a^-1z³ - 4a^-1z⁵ + a^-1z⁷ + 2 - 4z² + 8z⁴ - 4z⁶ + z⁸ - 5az + 15az³ - 11az⁵ + 3az⁷ - a² + 2a²z² - 2a²z⁶ + a²z⁸ - a³z + 4a³z³ - 6a³z⁵ + 2a³z⁷ - a⁴ + 3a⁴z² - 6a⁴z⁴ + 2a⁴z⁶ + a⁵z - 3a⁵z³ + a⁵z⁵

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

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 10₁₂₉. 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

j = 7 1

j = 5 1

j = 3 2 1

j = 1 3 1

j = -1 2 3

j = -3 2 2

j = -5 1 2

j = -7 1 2

j = -9 1

j = -11 1

n Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q^-15 - 2q^-14 + 5q^-12 - 6q^-11 - 3q^-10 + 11q^-9 - 6q^-8 - 9q^-7 + 15q^-6 - 2q^-5 - 15q^-4 + 16q^-3 + 2q^-2 - 17q^-1 + 14 + 4q - 13q² + 6q³ + 4q⁴ - 6q⁵ + q⁶ + 2q⁷ - q⁸

3 - q^-30 + 2q^-29 - 2q^-27 - 2q^-26 + 5q^-25 + 4q^-24 - 7q^-23 - 8q^-22 + 8q^-21 + 12q^-20 - 4q^-19 - 18q^-18 - q^-17 + 19q^-16 + 9q^-15 - 17q^-14 - 20q^-13 + 14q^-12 + 27q^-11 - 6q^-10 - 36q^-9 + 2q^-8 + 39q^-7 + 8q^-6 - 46q^-5 - 9q^-4 + 43q^-3 + 18q^-2 - 47q^-1 - 15 + 37q + 25q² - 35q³ - 23q⁴ + 23q⁵ + 24q⁶ - 14q⁷ - 21q⁸ + 5q⁹ + 16q¹⁰ + q¹¹ - 10q¹² - 3q¹³ + 4q¹⁴ + 4q¹⁵ - 2q¹⁶ - q¹⁷ - q¹⁸ + q¹⁹

4 q^-50 - 2q^-49 + 2q^-47 - q^-46 + 3q^-45 - 7q^-44 + 7q^-42 - q^-41 + 9q^-40 - 18q^-39 - 7q^-38 + 12q^-37 + 2q^-36 + 27q^-35 - 22q^-34 - 19q^-33 - 11q^-31 + 51q^-30 - q^-29 - 10q^-28 - 17q^-27 - 56q^-26 + 44q^-25 + 28q^-24 + 40q^-23 - 3q^-22 - 111q^-21 - 6q^-20 + 31q^-19 + 102q^-18 + 46q^-17 - 140q^-16 - 70q^-15 + 5q^-14 + 147q^-13 + 101q^-12 - 145q^-11 - 117q^-10 - 23q^-9 + 168q^-8 + 138q^-7 - 139q^-6 - 143q^-5 - 42q^-4 + 170q^-3 + 156q^-2 - 121q^-1 - 150 - 59q + 148q² + 162q³ - 77q⁴ - 139q⁵ - 79q⁶ + 97q⁷ + 146q⁸ - 17q⁹ - 93q¹⁰ - 86q¹¹ + 27q¹² + 99q¹³ + 27q¹⁴ - 31q¹⁵ - 62q¹⁶ - 18q¹⁷ + 40q¹⁸ + 28q¹⁹ + 7q²⁰ - 22q²¹ - 21q²² + 5q²³ + 9q²⁴ + 9q²⁵ - q²⁶ - 7q²⁷ - q²⁸ + 2q³⁰ + q³¹ - q³²

5 - q^-75 + 2q^-74 - 2q^-72 + q^-71 - q^-69 + 3q^-68 + q^-67 - 6q^-66 - 2q^-65 + 3q^-64 + 3q^-63 + 7q^-62 + 3q^-61 - 9q^-60 - 16q^-59 - 4q^-58 + 8q^-57 + 16q^-56 + 20q^-55 - 20q^-53 - 27q^-52 - 16q^-51 + 5q^-50 + 30q^-49 + 35q^-48 + 22q^-47 - 10q^-46 - 48q^-45 - 56q^-44 - 29q^-43 + 27q^-42 + 90q^-41 + 92q^-40 + 20q^-39 - 97q^-38 - 158q^-37 - 101q^-36 + 63q^-35 + 214q^-34 + 206q^-33 + 3q^-32 - 242q^-31 - 306q^-30 - 107q^-29 + 229q^-28 + 407q^-27 + 221q^-26 - 195q^-25 - 471q^-24 - 338q^-23 + 131q^-22 + 524q^-21 + 440q^-20 - 76q^-19 - 539q^-18 - 523q^-17 + 8q^-16 + 563q^-15 + 579q^-14 + 29q^-13 - 550q^-12 - 623q^-11 - 77q^-10 + 563q^-9 + 647q^-8 + 89q^-7 - 537q^-6 - 662q^-5 - 136q^-4 + 542q^-3 + 671q^-2 + 141q^-1 - 489 - 669q - 201q² + 461q³ + 658q⁴ + 226q⁵ - 376q⁶ - 617q⁷ - 287q⁸ + 290q⁹ + 557q¹⁰ + 317q¹¹ - 174q¹² - 465q¹³ - 334q¹⁴ + 60q¹⁵ + 352q¹⁶ + 317q¹⁷ + 38q¹⁸ - 229q¹⁹ - 273q²⁰ - 97q²¹ + 114q²² + 202q²³ + 121q²⁴ - 28q²⁵ - 125q²⁶ - 112q²⁷ - 20q²⁸ + 61q²⁹ + 76q³⁰ + 39q³¹ - 13q³² - 48q³³ - 33q³⁴ - q³⁵ + 15q³⁶ + 20q³⁷ + 11q³⁸ - 6q³⁹ - 10q⁴⁰ - 3q⁴¹ - q⁴² + 2q⁴³ + 3q⁴⁴ - q⁴⁶

6 q^-105 - 2q^-104 + 2q^-102 - q^-101 - 2q^-99 + 5q^-98 - 4q^-97 - 2q^-96 + 8q^-95 - 2q^-94 - 2q^-93 - 9q^-92 + 8q^-91 - 7q^-90 - 2q^-89 + 21q^-88 + 4q^-87 + q^-86 - 24q^-85 + 7q^-84 - 24q^-83 - 15q^-82 + 34q^-81 + 20q^-80 + 29q^-79 - 22q^-78 + 21q^-77 - 43q^-76 - 57q^-75 + 2q^-73 + 48q^-72 + 5q^-71 + 101q^-70 + 12q^-69 - 49q^-68 - 55q^-67 - 96q^-66 - 52q^-65 - 77q^-64 + 157q^-63 + 162q^-62 + 136q^-61 + 67q^-60 - 100q^-59 - 231q^-58 - 385q^-57 - 69q^-56 + 138q^-55 + 365q^-54 + 449q^-53 + 275q^-52 - 135q^-51 - 682q^-50 - 588q^-49 - 337q^-48 + 222q^-47 + 762q^-46 + 948q^-45 + 470q^-44 - 535q^-43 - 1003q^-42 - 1093q^-41 - 447q^-40 + 609q^-39 + 1484q^-38 + 1328q^-37 + 112q^-36 - 976q^-35 - 1696q^-34 - 1316q^-33 + 31q^-32 + 1615q^-31 + 2018q^-30 + 895q^-29 - 607q^-28 - 1941q^-27 - 1998q^-26 - 608q^-25 + 1467q^-24 + 2374q^-23 + 1474q^-22 - 213q^-21 - 1941q^-20 - 2367q^-19 - 1047q^-18 + 1283q^-17 + 2495q^-16 + 1776q^-15 + 33q^-14 - 1875q^-13 - 2515q^-12 - 1270q^-11 + 1160q^-10 + 2518q^-9 + 1912q^-8 + 166q^-7 - 1804q^-6 - 2568q^-5 - 1404q^-4 + 1033q^-3 + 2484q^-2 + 2011q^-1 + 328 - 1653q - 2560q² - 1567q³ + 753q⁴ + 2300q⁵ + 2093q⁶ + 627q⁷ - 1262q⁸ - 2370q⁹ - 1750q¹⁰ + 223q¹¹ + 1790q¹² + 1994q¹³ + 1016q¹⁴ - 567q¹⁵ - 1816q¹⁶ - 1742q¹⁷ - 417q¹⁸ + 943q¹⁹ + 1506q²⁰ + 1199q²¹ + 201q²² - 937q²³ - 1321q²⁴ - 775q²⁵ + 91q²⁶ + 714q²⁷ + 927q²⁸ + 597q²⁹ - 126q³⁰ - 618q³¹ - 625q³² - 320q³³ + 45q³⁴ + 388q³⁵ + 471q³⁶ + 217q³⁷ - 74q³⁸ - 233q³⁹ - 245q⁴⁰ - 179q⁴¹ + 14q⁴² + 163q⁴³ + 153q⁴⁴ + 80q⁴⁵ + 2q⁴⁶ - 53q⁴⁷ - 101q⁴⁸ - 57q⁴⁹ + 6q⁵⁰ + 31q⁵¹ + 33q⁵² + 26q⁵³ + 15q⁵⁴ - 19q⁵⁵ - 17q⁵⁶ - 7q⁵⁷ - 2q⁵⁸ + 3q⁶⁰ + 8q⁶¹ - q⁶² - q⁶³ - q⁶⁶ - q⁶⁷ + q⁶⁸

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, 129]]

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

In[3]:=
GaussCode[Knot[10, 129]]

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

In[4]:=
DTCode[Knot[10, 129]]

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

In[5]:=
br = BR[Knot[10, 129]]

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

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

Out[6]=
{4, 11}

In[7]:=
BraidIndex[Knot[10, 129]]

Out[7]=
4

In[8]:=
Show[DrawMorseLink[Knot[10, 129]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[10, 129]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

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

In[10]:=
alex = Alexander[Knot[10, 129]][t]

Out[10]=
2 6 2 9 + -- - - - 6 t + 2 t 2 t t

In[11]:=
Conway[Knot[10, 129]][z]

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

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

Out[12]=
{Knot[8, 8], Knot[10, 129], Knot[11, NonAlternating, 39], > Knot[11, NonAlternating, 45], Knot[11, NonAlternating, 50], > Knot[11, NonAlternating, 132]}

In[13]:=
{KnotDet[Knot[10, 129]], KnotSignature[Knot[10, 129]]}

Out[13]=
{25, 0}

In[14]:=
Jones[Knot[10, 129]][q]

Out[14]=
-5 2 3 4 4 2 3 5 - q + -- - -- + -- - - - 3 q + 2 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[8, 8], Knot[10, 129]}

In[16]:=
A2Invariant[Knot[10, 129]][q]

Out[16]=
-16 -10 -8 -4 2 2 4 10 1 - q - q + q + q + -- + 2 q - q - q 2 q

In[17]:=
HOMFLYPT[Knot[10, 129]][a, z]

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

In[18]:=
Kauffman[Knot[10, 129]][a, z]

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

In[19]:=
{Vassiliev[2][Knot[10, 129]], Vassiliev[3][Knot[10, 129]]}

Out[19]=
{2, -1}

In[20]:=
Kh[Knot[10, 129]][q, t]

Out[20]=
3 1 1 1 2 1 2 2 2 2 - + 3 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 7 3 > q t + 2 q t + q t + q t + q t

In[21]:=
ColouredJones[Knot[10, 129], 2][q]

Out[21]=
-15 2 5 6 3 11 6 9 15 2 15 16 2 14 + q - --- + --- - --- - --- + -- - -- - -- + -- - -- - -- + -- + -- - 14 12 11 10 9 8 7 6 5 4 3 2 q q q q q q q q q q q q 17 2 3 4 5 6 7 8 > -- + 4 q - 13 q + 6 q + 4 q - 6 q + q + 2 q - q q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 10₁₂₉

10₁₂₈
10₁₃₀

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-15 - 2q^-14 + 5q^-12 - 6q^-11 - 3q^-10 + 11q^-9 - 6q^-8 - 9q^-7 + 15q^-6 - 2q^-5 - 15q^-4 + 16q^-3 + 2q^-2 - 17q^-1 + 14 + 4q - 13q² + 6q³ + 4q⁴ - 6q⁵ + q⁶ + 2q⁷ - q⁸
3	- q^-30 + 2q^-29 - 2q^-27 - 2q^-26 + 5q^-25 + 4q^-24 - 7q^-23 - 8q^-22 + 8q^-21 + 12q^-20 - 4q^-19 - 18q^-18 - q^-17 + 19q^-16 + 9q^-15 - 17q^-14 - 20q^-13 + 14q^-12 + 27q^-11 - 6q^-10 - 36q^-9 + 2q^-8 + 39q^-7 + 8q^-6 - 46q^-5 - 9q^-4 + 43q^-3 + 18q^-2 - 47q^-1 - 15 + 37q + 25q² - 35q³ - 23q⁴ + 23q⁵ + 24q⁶ - 14q⁷ - 21q⁸ + 5q⁹ + 16q¹⁰ + q¹¹ - 10q¹² - 3q¹³ + 4q¹⁴ + 4q¹⁵ - 2q¹⁶ - q¹⁷ - q¹⁸ + q¹⁹
4	q^-50 - 2q^-49 + 2q^-47 - q^-46 + 3q^-45 - 7q^-44 + 7q^-42 - q^-41 + 9q^-40 - 18q^-39 - 7q^-38 + 12q^-37 + 2q^-36 + 27q^-35 - 22q^-34 - 19q^-33 - 11q^-31 + 51q^-30 - q^-29 - 10q^-28 - 17q^-27 - 56q^-26 + 44q^-25 + 28q^-24 + 40q^-23 - 3q^-22 - 111q^-21 - 6q^-20 + 31q^-19 + 102q^-18 + 46q^-17 - 140q^-16 - 70q^-15 + 5q^-14 + 147q^-13 + 101q^-12 - 145q^-11 - 117q^-10 - 23q^-9 + 168q^-8 + 138q^-7 - 139q^-6 - 143q^-5 - 42q^-4 + 170q^-3 + 156q^-2 - 121q^-1 - 150 - 59q + 148q² + 162q³ - 77q⁴ - 139q⁵ - 79q⁶ + 97q⁷ + 146q⁸ - 17q⁹ - 93q¹⁰ - 86q¹¹ + 27q¹² + 99q¹³ + 27q¹⁴ - 31q¹⁵ - 62q¹⁶ - 18q¹⁷ + 40q¹⁸ + 28q¹⁹ + 7q²⁰ - 22q²¹ - 21q²² + 5q²³ + 9q²⁴ + 9q²⁵ - q²⁶ - 7q²⁷ - q²⁸ + 2q³⁰ + q³¹ - q³²
5	- q^-75 + 2q^-74 - 2q^-72 + q^-71 - q^-69 + 3q^-68 + q^-67 - 6q^-66 - 2q^-65 + 3q^-64 + 3q^-63 + 7q^-62 + 3q^-61 - 9q^-60 - 16q^-59 - 4q^-58 + 8q^-57 + 16q^-56 + 20q^-55 - 20q^-53 - 27q^-52 - 16q^-51 + 5q^-50 + 30q^-49 + 35q^-48 + 22q^-47 - 10q^-46 - 48q^-45 - 56q^-44 - 29q^-43 + 27q^-42 + 90q^-41 + 92q^-40 + 20q^-39 - 97q^-38 - 158q^-37 - 101q^-36 + 63q^-35 + 214q^-34 + 206q^-33 + 3q^-32 - 242q^-31 - 306q^-30 - 107q^-29 + 229q^-28 + 407q^-27 + 221q^-26 - 195q^-25 - 471q^-24 - 338q^-23 + 131q^-22 + 524q^-21 + 440q^-20 - 76q^-19 - 539q^-18 - 523q^-17 + 8q^-16 + 563q^-15 + 579q^-14 + 29q^-13 - 550q^-12 - 623q^-11 - 77q^-10 + 563q^-9 + 647q^-8 + 89q^-7 - 537q^-6 - 662q^-5 - 136q^-4 + 542q^-3 + 671q^-2 + 141q^-1 - 489 - 669q - 201q² + 461q³ + 658q⁴ + 226q⁵ - 376q⁶ - 617q⁷ - 287q⁸ + 290q⁹ + 557q¹⁰ + 317q¹¹ - 174q¹² - 465q¹³ - 334q¹⁴ + 60q¹⁵ + 352q¹⁶ + 317q¹⁷ + 38q¹⁸ - 229q¹⁹ - 273q²⁰ - 97q²¹ + 114q²² + 202q²³ + 121q²⁴ - 28q²⁵ - 125q²⁶ - 112q²⁷ - 20q²⁸ + 61q²⁹ + 76q³⁰ + 39q³¹ - 13q³² - 48q³³ - 33q³⁴ - q³⁵ + 15q³⁶ + 20q³⁷ + 11q³⁸ - 6q³⁹ - 10q⁴⁰ - 3q⁴¹ - q⁴² + 2q⁴³ + 3q⁴⁴ - q⁴⁶
6	q^-105 - 2q^-104 + 2q^-102 - q^-101 - 2q^-99 + 5q^-98 - 4q^-97 - 2q^-96 + 8q^-95 - 2q^-94 - 2q^-93 - 9q^-92 + 8q^-91 - 7q^-90 - 2q^-89 + 21q^-88 + 4q^-87 + q^-86 - 24q^-85 + 7q^-84 - 24q^-83 - 15q^-82 + 34q^-81 + 20q^-80 + 29q^-79 - 22q^-78 + 21q^-77 - 43q^-76 - 57q^-75 + 2q^-73 + 48q^-72 + 5q^-71 + 101q^-70 + 12q^-69 - 49q^-68 - 55q^-67 - 96q^-66 - 52q^-65 - 77q^-64 + 157q^-63 + 162q^-62 + 136q^-61 + 67q^-60 - 100q^-59 - 231q^-58 - 385q^-57 - 69q^-56 + 138q^-55 + 365q^-54 + 449q^-53 + 275q^-52 - 135q^-51 - 682q^-50 - 588q^-49 - 337q^-48 + 222q^-47 + 762q^-46 + 948q^-45 + 470q^-44 - 535q^-43 - 1003q^-42 - 1093q^-41 - 447q^-40 + 609q^-39 + 1484q^-38 + 1328q^-37 + 112q^-36 - 976q^-35 - 1696q^-34 - 1316q^-33 + 31q^-32 + 1615q^-31 + 2018q^-30 + 895q^-29 - 607q^-28 - 1941q^-27 - 1998q^-26 - 608q^-25 + 1467q^-24 + 2374q^-23 + 1474q^-22 - 213q^-21 - 1941q^-20 - 2367q^-19 - 1047q^-18 + 1283q^-17 + 2495q^-16 + 1776q^-15 + 33q^-14 - 1875q^-13 - 2515q^-12 - 1270q^-11 + 1160q^-10 + 2518q^-9 + 1912q^-8 + 166q^-7 - 1804q^-6 - 2568q^-5 - 1404q^-4 + 1033q^-3 + 2484q^-2 + 2011q^-1 + 328 - 1653q - 2560q² - 1567q³ + 753q⁴ + 2300q⁵ + 2093q⁶ + 627q⁷ - 1262q⁸ - 2370q⁹ - 1750q¹⁰ + 223q¹¹ + 1790q¹² + 1994q¹³ + 1016q¹⁴ - 567q¹⁵ - 1816q¹⁶ - 1742q¹⁷ - 417q¹⁸ + 943q¹⁹ + 1506q²⁰ + 1199q²¹ + 201q²² - 937q²³ - 1321q²⁴ - 775q²⁵ + 91q²⁶ + 714q²⁷ + 927q²⁸ + 597q²⁹ - 126q³⁰ - 618q³¹ - 625q³² - 320q³³ + 45q³⁴ + 388q³⁵ + 471q³⁶ + 217q³⁷ - 74q³⁸ - 233q³⁹ - 245q⁴⁰ - 179q⁴¹ + 14q⁴² + 163q⁴³ + 153q⁴⁴ + 80q⁴⁵ + 2q⁴⁶ - 53q⁴⁷ - 101q⁴⁸ - 57q⁴⁹ + 6q⁵⁰ + 31q⁵¹ + 33q⁵² + 26q⁵³ + 15q⁵⁴ - 19q⁵⁵ - 17q⁵⁶ - 7q⁵⁷ - 2q⁵⁸ + 3q⁶⁰ + 8q⁶¹ - q⁶² - q⁶³ - q⁶⁶ - q⁶⁷ + q⁶⁸

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	PD[Knot[10, 129]]
Out[2]=	PD[X[1, 4, 2, 5], X[3, 8, 4, 9], X[5, 14, 6, 15], X[20, 16, 1, 15], > X[16, 10, 17, 9], X[10, 20, 11, 19], X[18, 12, 19, 11], X[12, 18, 13, 17], > X[13, 6, 14, 7], X[7, 2, 8, 3]]
In[3]:=	GaussCode[Knot[10, 129]]
Out[3]=	GaussCode[-1, 10, -2, 1, -3, 9, -10, 2, 5, -6, 7, -8, -9, 3, 4, -5, 8, -7, 6, > -4]
In[4]:=	DTCode[Knot[10, 129]]
Out[4]=	DTCode[4, 8, 14, 2, -16, -18, 6, -20, -12, -10]
In[5]:=	br = BR[Knot[10, 129]]
Out[5]=	BR[4, {1, 1, 1, -2, -1, -1, 3, -2, -1, 3, -2}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 11}
In[7]:=	BraidIndex[Knot[10, 129]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[10, 129]]]

Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 129]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 1, 2, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 129]][t]
Out[10]=	2 6 2 9 + -- - - - 6 t + 2 t 2 t t
In[11]:=	Conway[Knot[10, 129]][z]
Out[11]=	2 4 1 + 2 z + 2 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[8, 8], Knot[10, 129], Knot[11, NonAlternating, 39], > Knot[11, NonAlternating, 45], Knot[11, NonAlternating, 50], > Knot[11, NonAlternating, 132]}
In[13]:=	{KnotDet[Knot[10, 129]], KnotSignature[Knot[10, 129]]}
Out[13]=	{25, 0}
In[14]:=	Jones[Knot[10, 129]][q]
Out[14]=	-5 2 3 4 4 2 3 5 - q + -- - -- + -- - - - 3 q + 2 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[8, 8], Knot[10, 129]}
In[16]:=	A2Invariant[Knot[10, 129]][q]
Out[16]=	-16 -10 -8 -4 2 2 4 10 1 - q - q + q + q + -- + 2 q - q - q 2 q
In[17]:=	HOMFLYPT[Knot[10, 129]][a, z]
Out[17]=	2 -2 2 4 2 z 2 2 4 2 4 2 4 2 - a + a - a + 2 z - -- + 2 a z - a z + z + a z 2 a
In[18]:=	Kauffman[Knot[10, 129]][a, z]
Out[18]=	2 -2 2 4 2 z 5 z 3 5 2 3 z 2 2 2 + a - a - a - --- - --- - 5 a z - a z + a z - 4 z - ---- + 2 a z + 3 a 2 a a 3 3 4 4 2 z 9 z 3 3 3 5 3 4 2 z 4 4 > 3 a z + -- + ---- + 15 a z + 4 a z - 3 a z + 8 z + ---- - 6 a z - 3 a 2 a a 5 7 4 z 5 3 5 5 5 6 2 6 4 6 z 7 > ---- - 11 a z - 6 a z + a z - 4 z - 2 a z + 2 a z + -- + 3 a z + a a 3 7 8 2 8 > 2 a z + z + a z
In[19]:=	{Vassiliev[2][Knot[10, 129]], Vassiliev[3][Knot[10, 129]]}
Out[19]=	{2, -1}
In[20]:=	Kh[Knot[10, 129]][q, t]
Out[20]=	3 1 1 1 2 1 2 2 2 2 - + 3 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 7 3 > q t + 2 q t + q t + q t + q t
In[21]:=	ColouredJones[Knot[10, 129], 2][q]
Out[21]=	-15 2 5 6 3 11 6 9 15 2 15 16 2 14 + q - --- + --- - --- - --- + -- - -- - -- + -- - -- - -- + -- + -- - 14 12 11 10 9 8 7 6 5 4 3 2 q q q q q q q q q q q q 17 2 3 4 5 6 7 8 > -- + 4 q - 13 q + 6 q + 4 q - 6 q + q + 2 q - q q

The Non Alternating Knot 10129

The Non Alternating Knot 10₁₂₉