10 129

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of May 31, 2006, 14:15:20.091. Read more at http://katlas.org/wiki/KnotTheory.

In[3]:=

K = Knot["10 129"];

In[4]:=

PD[K]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

X1425 X3849 X5,14,6,15 X20,16,1,15 X16,10,17,9 X10,20,11,19 X18,12,19,11 X12,18,13,17 X13,6,14,7 X7283

In[5]:=

GaussCode[K]

Out[5]=

-1, 10, -2, 1, -3, 9, -10, 2, 5, -6, 7, -8, -9, 3, 4, -5, 8, -7, 6, -4

In[6]:=

DTCode[K]

Out[6]=

4 8 14 2 -16 -18 6 -20 -12 -10

(The path below may be different on your system)

In[7]:=

AppendTo[$Path, "C:/bin/LinKnot/"];

In[8]:=

ConwayNotation[K]

Out[8]=

[32,21,2-]

In[9]:=

br = BR[K]

KnotTheory::credits: The minimum braids representing the knots with up to 10 crossings were provided by Thomas Gittings. See arXiv:math.GT/0401051.

Out[9]=

BR(4,{1,1,1,−2,−1,−1,3,−2,−1,3,−2})

In[10]:=

{First[br], Crossings[br], BraidIndex[K]}

KnotTheory::credits: The braid index data known to KnotTheory` is taken from Charles Livingston's http://www.indiana.edu/~knotinfo/.

KnotTheory::loading: Loading precomputed data in IndianaData`.

Out[10]=

{ 4, 11, 4 }

In[11]:=

Show[BraidPlot[br]]

Out[11]=

-Graphics-

In[12]:=

Show[DrawMorseLink[K]]

KnotTheory::credits: "MorseLink was added to KnotTheory` by Siddarth Sankaran at the University of Toronto in the summer of 2005."

KnotTheory::credits: "DrawMorseLink was written by Siddarth Sankaran at the University of Toronto in the summer of 2005."

Out[12]=

-Graphics-

In[13]:=

ap = ArcPresentation[K]

Out[13]=

ArcPresentation[{12, 2}, {1, 10}, {8, 11}, {10, 12}, {9, 3}, {2, 8}, {5, 1}, {4, 6}, {7, 5}, {6, 9}, {3, 7}, {11, 4}]

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Further quantum knot invariants for 10_129.

A1 Invariants.

Weight	Invariant
1	−q11 + q9−q7 + q5 + q + 2q−1−q−3 + q−5−q−7
2	q32−q30−q28 + 3q26−q24−4q22 + 2q20 + 2q18−4q16 + 4q12−2q10−q8 + 3q6 + q4−q2 + 1 + 5q−2−3q−4−3q−6 + 4q−8−q−10−3q−12 + 2q−14 + q−16−q−18
3	−q63 + q61 + q59−q57−2q55 + q53 + 5q51−6q47−3q45 + 5q43 + 8q41−2q39−11q37−4q35 + 9q33 + 10q31−9q29−14q27 + 4q25 + 15q23−q21−13q19−q17 + 13q15 + 3q13−8q11−4q9 + 6q7 + 5q5−q3−7q + 12q−3 + 4q−5−10q−7−11q−9 + 10q−11 + 12q−13−6q−15−14q−17 + q−19 + 12q−21 + 4q−23−8q−25−5q−27 + 3q−29 + 5q−31−3q−35−q−37 + q−41

A2 Invariants.

Weight	Invariant
1,0	−q16−q10 + q8 + q4 + 2q2 + 1 + 2q−2−q−4−q−10
1,1	q44−2q42 + 4q40−8q38 + 15q36−18q34 + 24q32−32q30 + 29q28−24q26 + 14q24−4q22−19q20 + 32q18−48q16 + 54q14−54q12 + 58q10−38q8 + 36q6−13q4 + 2q2 + 14−26q−2 + 30q−4−38q−6 + 32q−8−22q−10 + 15q−12−8q−14 + 2q−16 + 4q−18−3q−20−2q−24 + q−28
2,0	q42−q38 + 2q34 + 2q32−2q30−3q28−q26−q24−3q22−3q20 + q18 + 2q16 + q14 + q12 + 3q10 + 2q8 + 2q6 + 4q4 + q2 + 2 + q−2 + q−4−4q−6−3q−8 + q−10 + q−12−2q−14−q−16 + 2q−18 + q−20−q−24

A3 Invariants.

Weight	Invariant
0,1,0	q34−q32 + q28−2q26 + q24−4q20 + q18−3q14 + q12 + q10 + q6 + 3q4 + 4q2 + 4 + 2q−2 + 5q−4−2q−6−3q−8 + q−10−4q−12−3q−14 + q−16 + q−22
1,0,0	−q21−q17−q13 + q11 + q7 + q5 + 2q3 + 2q + q−1 + 2q−3−q−5−q−9−q−13

A4 Invariants.

Weight	Invariant
0,1,0,0	q44 + q38−2q34−q32 + q30−2q28−4q26 + 2q22−3q20−3q18 + 2q16−q14−4q12 + 3q8 + 3q6 + 5q4 + 12q2 + 9 + 5q−2 + 5q−4 + 3q−6−6q−8−7q−10−3q−12−4q−14−5q−16−2q−18 + 2q−20 + q−22 + q−26 + q−28
1,0,0,0	−q26−q22−q20−q16 + q14 + q10 + q8 + q6 + 2q4 + 2q2 + 2 + q−2 + 2q−4−q−6−q−10−q−12−q−16

B2 Invariants.

Weight	Invariant
0,1	−q34 + q32−2q30 + 3q28−4q26 + 3q24−4q22 + 2q20−q18 + 3q14−3q12 + 7q10−6q8 + 7q6−5q4 + 6q2−4 + 2q−2 + q−4−2q−6 + 3q−8−3q−10 + 4q−12−3q−14 + 3q−16−2q−18−q−22
1,0	q56−q52−q50 + q48 + 2q46−q44−3q42 + 3q38 + 2q36−4q34−4q32 + q30 + 3q28−4q24−q22 + 2q20 + 2q18−2q16−q14 + 2q12 + 3q10−q6 + q4 + 5q2 + 3−q−2−q−4 + 4q−6 + 3q−8−2q−10−4q−12 + 3q−16−4q−20−3q−22 + 2q−26−q−30 + q−36

D4 Invariants.

Weight	Invariant
1,0,0,0	q46−q44 + q42−2q40 + 3q38−3q36 + 2q34−4q32 + 2q30−3q28−q24−q22 + q20−3q18 + 4q16−4q14 + 5q12−5q10 + 6q8−2q6 + 8q4 + 7 + 2q−2 + 4q−4 + 3q−6−2q−8−5q−12 + q−14−5q−16−4q−20 + 2q−22−q−24 + q−26 + q−30

G2 Invariants.

Weight	Invariant
1,0	q80−q78 + 2q76−3q74 + 2q72−q70−3q68 + 6q66−8q64 + 8q62−7q60−q58 + 6q56−11q54 + 12q52−7q50 + 6q46−9q44 + 7q42−q40−7q38 + 11q36−10q34 + 4q32 + 6q30−13q28 + 16q26−12q24 + 5q22 + 2q20−10q18 + 14q16−12q14 + 9q12−3q8 + 10q6−8q4 + 6q2 + 2−4q−2 + 10q−4−6q−6 + q−8 + 10q−10−12q−12 + 13q−14−7q−16−4q−18 + 8q−20−11q−22 + 9q−24−5q−26−q−28 + 3q−30−5q−32 + 2q−34−q−36−q−38−q−44 + q−52

.

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["10 129"];

In[4]:=

Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

2t2−6t + 9−6t−1 + 2t−2

In[5]:=

Conway[K][z]

Out[5]=

2z4 + 2z2 + 1

In[6]:=

Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]=

{1}

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 25, 0 }

In[8]:=

Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=

−q3 + 2q2−3q + 5−4q−1 + 4q−2−3q−3 + 2q−4−q−5

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

−z2a4−a4 + z4a2 + 2z2a2 + a2 + z4 + 2z2 + 2−z2a−2−a−2

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

a2z8 + z8 + 2a3z7 + 3az7 + z7a−1 + 2a4z6−2a2z6−4z6 + a5z5−6a3z5−11az5−4z5a−1−6a4z4 + 2z4a−2 + 8z4−3a5z3 + 4a3z3 + 15az3 + 9z3a−1 + z3a−3 + 3a4z2 + 2a2z2−3z2a−2−4z2 + a5z−a3z−5az−5za−1−2za−3−a4−a2 + a−2 + 2

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of May 31, 2006, 14:15:20.091. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["10 129"];

In[4]:=

{A = Alexander[K][t], J = Jones[K][q]}

KnotTheory::loading: Loading precomputed data in PD4Knots`.

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[4]=

{ 2t2−6t + 9−6t−1 + 2t−2, −q3 + 2q2−3q + 5−4q−1 + 4q−2−3q−3 + 2q−4−q−5 }

In[5]:=

DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]

KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.

KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.

Out[5]=

{8_8, K11n39, K11n45, K11n50, K11n132,}

In[6]:=

DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]=

{8_8,}

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