10 150

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["10 150"];

In[4]:=

PD[K]

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

Out[4]=

X4251 X8493 X5,12,6,13 X9,17,10,16 X17,1,18,20 X13,19,14,18 X19,15,20,14 X15,11,16,10 X11,6,12,7 X2837

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

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

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[(21,2)(3,2-)]

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[{3, 9}, {2, 4}, {1, 3}, {14, 10}, {9, 13}, {11, 14}, {5, 8}, {10, 7}, {8, 6}, {7, 12}, {4, 11}, {12, 5}, {13, 2}, {6, 1}]

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Further quantum knot invariants for 10_150.

A1 Invariants.

Weight	Invariant
1	q−q−1 + 2q−3 + q−7−q−11 + q−13−2q−15 + q−17
2	q6−q4−2q2 + 4 + 2q−2−5q−4 + 2q−6 + 5q−8−3q−10−2q−12 + 5q−14−4q−18 + 2q−20 + 2q−22−4q−24−q−26 + 4q−28−q−30−5q−32 + 4q−34 + 3q−36−5q−38 + q−40 + 2q−42−q−44
3	q15−q13−2q11 + 5q7 + 4q5−6q3−9q + 2q−1 + 13q−3 + 7q−5−12q−7−12q−9 + 5q−11 + 19q−13 + 5q−15−17q−17−12q−19 + 13q−21 + 18q−23−7q−25−21q−27 + q−29 + 20q−31 + 2q−33−18q−35−5q−37 + 18q−39 + 5q−41−15q−43−9q−45 + 11q−47 + 10q−49−5q−51−15q−53−3q−55 + 17q−57 + 14q−59−13q−61−21q−63 + 7q−65 + 25q−67−21q−71−6q−73 + 11q−75 + 9q−77−6q−79−5q−81 + q−83 + 2q−85 + q−87−q−89

A2 Invariants.

Weight	Invariant
1,0	1 + q−4 + q−6 + 2q−10−q−12 + q−14−q−16−q−18−q−22 + q−24
1,1	q4−2q2 + 8−16q−2 + 27q−4−36q−6 + 48q−8−46q−10 + 44q−12−24q−14 + 6q−16 + 22q−18−48q−20 + 64q−22−80q−24 + 82q−26−81q−28 + 66q−30−48q−32 + 28q−34−22q−38 + 42q−40−52q−42 + 51q−44−46q−46 + 30q−48−18q−50 + 7q−52 + 2q−54−4q−56 + 4q−58−2q−66 + q−68
2,0	q4−1 + 2q−4 + 2q−6 + 4q−12 + 2q−14−q−16−q−22−q−24−q−26 + q−28−q−30 + 2q−32 + q−34−2q−36−q−38 + q−40−q−42−3q−44 + q−46 + q−48 + 2q−50−2q−52 + q−56 + q−60−q−62

A3 Invariants.

Weight	Invariant
0,1,0	1−q−2 + 2q−4 + 2q−6−q−8 + 4q−10 + q−12−q−14 + 4q−16−q−18−q−20 + 2q−22−3q−26−q−28−2q−30−q−32−2q−34 + 4q−38−2q−40 + 2q−42 + 3q−44−3q−46 + 2q−50−2q−52−q−54 + q−56
1,0,0	q−1 + 2q−5 + 2q−9 + q−13−q−21−2q−25 + q−27−q−29 + q−31

A4 Invariants.

Weight	Invariant
0,1,0,0	q−2 + q−6 + 3q−8 + q−10 + 2q−12 + 4q−14 + q−16 + 3q−20 + 2q−22−2q−24 + q−26 + 4q−28 + q−30−6q−32 + q−34−9q−38−5q−40 + q−42−4q−44−2q−46 + 6q−48 + 4q−50 + q−52 + 3q−54 + 5q−56−2q−58−4q−60 + q−62−4q−66 + 2q−70
1,0,0,0	q−2 + 2q−6 + q−8 + q−10 + 2q−12 + q−16−q−18 + q−20−q−22−q−26−q−30−q−32 + q−34−q−36 + q−38

B2 Invariants.

Weight	Invariant
0,1	1−q−2 + 4q−4−4q−6 + 5q−8−4q−10 + 5q−12−3q−14 + 2q−16 + q−18−3q−20 + 6q−22−8q−24 + 9q−26−9q−28 + 8q−30−7q−32 + 4q−34−2q−36 + 2q−40−4q−42 + 5q−44−5q−46 + 4q−48−4q−50 + 2q−52−q−54 + q−56
1,0	q2−q−2−q−4 + 3q−6 + 3q−8−2q−10−3q−12 + 2q−14 + 5q−16 + 2q−18−5q−20−2q−22 + 5q−24 + 5q−26−2q−28−4q−30 + 4q−34 + q−36−3q−38−2q−40 + 2q−42 + q−44−3q−46−4q−48 + 4q−52−q−54−5q−56−q−58 + 5q−60 + 3q−62−3q−64−3q−66 + 3q−68 + 5q−70−q−72−4q−74−q−76 + 2q−78 + 3q−80−q−82−2q−84−q−86 + q−90

D4 Invariants.

Weight	Invariant
1,0,0,0	q−2−q−4 + 3q−6−2q−8 + 6q−10−2q−12 + 5q−14−2q−16 + 6q−18−2q−20 + q−22−q−26 + 3q−28−4q−30 + 5q−32−7q−34 + 6q−36−8q−38 + 5q−40−8q−42 + 4q−44−6q−46 + 3q−48−q−50 + q−52 + 2q−54−q−56 + 4q−58−3q−60 + 5q−62−4q−64 + 2q−66−3q−68 + 3q−70−2q−72−q−76 + q−78

G2 Invariants.

Weight

Invariant

1,0

q−2−q−4 + 4q−6−5q−8 + 5q−10−q−12−5q−14 + 14q−16−15q−18 + 14q−20−5q−22−7q−24 + 17q−26−19q−28 + 15q−30−q−32−9q−34 + 15q−36−12q−38 + 2q−40 + 11q−42−18q−44 + 16q−46−7q−48−4q−50 + 16q−52−21q−54 + 21q−56−13q−58 + 2q−60 + 8q−62−17q−64 + 19q−66−16q−68 + 7q−70 + 3q−72−12q−74 + 14q−76−11q−78−3q−80 + 12q−82−17q−84 + 12q−86−2q−88−12q−90 + 20q−92−18q−94 + 11q−96−10q−100 + 14q−102−11q−104 + 6q−106 + 2q−108−3q−110 + 4q−112−3q−114 + q−118 + q−120−q−122−q−126−q−132 + q−134

.

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["10 150"];

In[4]:=

Alexander[K][t]

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

Out[4]=

−t3 + 4t2−6t + 7−6t−1 + 4t−2−t−3

In[5]:=

Conway[K][z]

Out[5]=

−z6−2z4 + z2 + 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]=

{ 29, 4 }

In[8]:=

Jones[K][q]

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

Out[8]=

q8−3q7 + 4q6−5q5 + 5q4−4q3 + 4q2−2q + 1

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

−z6a−4 + z4a−2−4z4a−4 + z4a−6 + 3z2a−2−4z2a−4 + 2z2a−6 + 2a−2−a−4

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

z8a−4 + z8a−6 + 2z7a−3 + 4z7a−5 + 2z7a−7 + z6a−2 + z6a−8−7z5a−3−12z5a−5−5z5a−7−4z4a−2−9z4a−4−5z4a−6 + 5z3a−3 + 8z3a−5 + 6z3a−7 + 3z3a−9 + 5z2a−2 + 8z2a−4 + 3z2a−6 + z2a−8 + z2a−10−2za−5−3za−7−za−9−2a−2−a−4

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["10 150"];

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]=

{ −t3 + 4t2−6t + 7−6t−1 + 4t−2−t−3, q8−3q7 + 4q6−5q5 + 5q4−4q3 + 4q2−2q + 1 }

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]=

{10_127, K11n51,}

In[6]:=

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

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

Out[6]=

{}

n	Jn
2	−q21 + 3q20−q19−7q18 + 11q17−16q15 + 15q14 + 5q13−21q12 + 12q11 + 11q10−21q9 + 6q8 + 15q7−16q6−q5 + 14q4−8q3−4q2 + 7q−1−2q−1 + q−2
3	−q43 + 2q42 + q41−q40−7q39 + q38 + 16q37 + q36−24q35−14q34 + 37q33 + 26q32−42q31−42q30 + 45q29 + 53q28−39q27−62q26 + 33q25 + 63q24−24q23−61q22 + 13q21 + 57q20−4q19−48q18−10q17 + 44q16 + 16q15−30q14−29q13 + 22q12 + 30q11−5q10−34q9−3q8 + 25q7 + 17q6−20q5−17q4 + 8q3 + 17q2−q−11−3q−1 + 6q−2 + 2q−3−q−4−2q−5 + q−6
4	−q70 + 2q69 + 2q68−4q67−3q66−4q65 + 13q64 + 17q63−15q62−29q61−28q60 + 45q59 + 83q58−9q57−94q56−117q55 + 56q54 + 205q53 + 70q52−144q51−259q50−5q49 + 303q48 + 189q47−118q46−355q45−106q44 + 315q43 + 262q42−49q41−362q40−172q39 + 272q38 + 260q37 + 11q36−311q35−198q34 + 212q33 + 224q32 + 61q31−240q30−210q29 + 136q28 + 177q27 + 117q26−149q25−210q24 + 42q23 + 106q22 + 158q21−37q20−167q19−37q18 + 4q17 + 140q16 + 57q15−74q14−49q13−78q12 + 58q11 + 73q10 + 12q9 + 4q8−83q7−15q6 + 23q5 + 27q4 + 43q3−31q2−22q−14 + q−1 + 30q−2−2q−4−9q−5−7q−6 + 7q−7 + q−8 + 2q−9−q−10−2q−11 + q−12
5	q101−2q100−3q99 + 4q98 + 7q97 + 3q96−4q95−24q94−25q93 + 20q92 + 64q91 + 61q90−20q89−130q88−156q87−10q86 + 228q85 + 315q84 + 90q83−313q82−525q81−273q80 + 353q79 + 789q78 + 514q77−319q76−1001q75−824q74 + 187q73 + 1164q72 + 1123q71−3q70−1213q69−1364q68−221q67 + 1185q66 + 1513q65 + 420q64−1087q63−1577q62−568q61 + 968q60 + 1559q59 + 660q58−843q57−1504q56−705q55 + 736q54 + 1422q53 + 724q52−632q51−1333q50−742q49 + 522q48 + 1249q47 + 767q46−404q45−1141q44−806q43 + 245q42 + 1033q41 + 847q40−85q39−871q38−865q37−123q36 + 695q35 + 856q34 + 288q33−456q32−781q31−452q30 + 217q29 + 651q28 + 522q27 + 27q26−448q25−539q24−206q23 + 231q22 + 435q21 + 321q20−12q19−301q18−324q17−125q16 + 108q15 + 256q14 + 204q13 + 21q12−136q11−174q10−113q9 + 19q8 + 116q7 + 118q6 + 53q5−33q4−84q3−72q2−23q + 33 + 58q−1 + 41q−2−23q−4−34q−5−20q−6 + 6q−7 + 20q−8 + 12q−9 + 4q−10−2q−11−11q−12−5q−13 + 3q−14 + 2q−15 + q−16 + 2q−17−q−18−2q−19 + q−20
6	q143−2q142−q141 + 2q140 + q139 + q138 + 8q136−11q135−18q134−3q133 + 6q132 + 25q131 + 39q130 + 54q129−41q128−128q127−125q126−52q125 + 112q124 + 300q123 + 383q122 + 42q121−425q120−700q119−589q118 + 13q117 + 898q116 + 1484q115 + 874q114−491q113−1796q112−2183q111−1097q110 + 1186q109 + 3219q108 + 2984q107 + 652q106−2453q105−4374q104−3558q103 + 113q102 + 4300q101 + 5512q100 + 3071q99−1631q98−5640q97−6138q96−2096q95 + 3792q94 + 6859q93 + 5341q92 + 186q91−5300q90−7366q89−3973q88 + 2405q87 + 6657q86 + 6268q85 + 1624q84−4215q83−7193q82−4671q81 + 1327q80 + 5824q79 + 6082q78 + 2167q77−3342q76−6515q75−4601q74 + 792q73 + 5081q72 + 5612q71 + 2322q70−2717q69−5873q68−4491q67 + 285q66 + 4371q65 + 5253q64 + 2681q63−1888q62−5172q61−4590q60−621q59 + 3317q58 + 4842q57 + 3338q56−560q55−4056q54−4615q53−1874q52 + 1680q51 + 3944q50 + 3873q49 + 1121q48−2270q47−4009q46−2912q45−332q44 + 2237q43 + 3585q42 + 2472q41−63q40−2407q39−2922q38−1902q37 + 64q36 + 2108q35 + 2588q34 + 1596q33−296q32−1601q31−2070q30−1445q29 + 133q28 + 1309q27 + 1735q26 + 1039q25 + 125q24−882q23−1391q22−945q21−202q20 + 646q19 + 863q18 + 857q17 + 320q16−355q15−640q14−657q13−249q12 + 10q11 + 431q10 + 493q9 + 280q8 + 36q7−225q6−245q5−320q4−75q3 + 89q2 + 172q + 176 + 94q−1 + 47q−2−133q−3−102q−4−83q−5−25q−6 + 20q−7 + 54q−8 + 91q−9 + 5q−10 + q−11−27q−12−28q−13−30q−14−7q−15 + 26q−16 + 5q−17 + 13q−18 + 3q−19 + q−20−11q−21−7q−22 + 5q−23−2q−24 + 2q−25 + q−26 + 2q−27−q−28−2q−29 + q−30