10 144

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["10 144"];

In[4]:=

PD[K]

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

Out[4]=

X1425 X3,10,4,11 X18,11,19,12 X5,15,6,14 X17,7,18,6 X7,17,8,16 X15,9,16,8 X20,13,1,14 X12,19,13,20 X9,2,10,3

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

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

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[31,21,21-]

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Further quantum knot invariants for 10_144.

A1 Invariants.

Weight	Invariant
1	q15−2q13 + 2q11−q9 + q7−2q3 + 2q−q−1 + 2q−3
2	q42−2q40−q38 + 6q36−4q34−6q32 + 11q30−q28−10q26 + 9q24 + 2q22−9q20 + q18 + 4q16−q14−6q12 + 6q10 + 8q8−10q6 + 2q4 + 10q2−9−3q−2 + 7q−4−3q−6−3q−8 + 3q−10 + q−12
3	q81−2q79−q77 + 3q75 + 3q73−4q71−9q69 + 8q67 + 16q65−7q63−26q61 + 37q57 + 8q55−44q53−20q51 + 45q49 + 32q47−37q45−39q43 + 26q41 + 39q39−13q37−34q35−2q33 + 27q31 + 15q29−17q27−27q25 + 11q23 + 35q21−q19−43q17−11q15 + 47q13 + 17q11−43q9−29q7 + 38q5 + 36q3−23q−36q−1 + 12q−3 + 34q−5 + q−7−24q−9−8q−11 + 13q−13 + 8q−15−5q−17−8q−19 + q−21 + 2q−23 + 2q−25

A2 Invariants.

Weight	Invariant
1,0	q22−q20−q18 + 2q16 + 2q12−2q8−q6−3q4 + 2q2 + 1 + q−2 + 2q−4
1,1	q60−4q58 + 10q56−20q54 + 34q52−54q50 + 80q48−104q46 + 124q44−138q42 + 138q40−116q38 + 73q36−12q34−56q32 + 132q30−203q28 + 250q26−280q24 + 274q22−250q20 + 200q18−132q16 + 64q14 + 20q12−72q10 + 122q8−146q6 + 150q4−142q2 + 106−82q−2 + 51q−4−28q−6 + 14q−8 + 2q−12 + 2q−14
2,0	q56−q54−2q52 + q50 + 4q48 + q46−6q44−2q42 + 5q40 + q38−3q36 + 3q34 + 7q32−q30−7q28−3q26−3q24−7q22 + 4q18 + 2q16 + 7q14 + 11q12 + 4q10−5q8−q6 + 2q4−6q2−9 + 3q−4−q−6 + 2q−10 + 4q−12 + q−14

A3 Invariants.

Weight	Invariant
0,1,0	q48−2q46 + 4q42−5q40 + q38 + 7q36−9q34−q32 + 7q30−6q28 + 7q24 + q22 + 3q16−2q14−8q12 + 5q10−q8−10q6 + 6q4 + 2q2−5 + 6q−2 + 3q−4−q−6 + 3q−8
1,0,0	q29−q27−q23 + 2q21 + 3q17 + q15−2q11−3q9−2q7−3q5 + 2q3 + q + 3q−1 + q−3 + 2q−5

A4 Invariants.

Weight	Invariant
0,1,0,0	q62−q60−2q58 + 2q56 + 3q54−2q52−3q50 + 4q48 + 3q46−7q44−4q42 + 7q40 + q38−6q36 + 4q34 + 6q32−4q30−3q28 + 5q26 + q24−4q22 + 7q20 + 9q18−5q16−3q14 + 5q12−6q10−14q8−4q6 + 2q4−2q2−1 + 7q−2 + 7q−4 + 2q−6 + 2q−8 + 3q−10
1,0,0,0	q36−q34−q28 + 2q26 + 3q22 + 2q20 + q18−2q14−3q12−4q10−2q8−3q6 + 2q4 + q2 + 3 + 3q−2 + q−4 + 2q−6

B2 Invariants.

Weight	Invariant
0,1	q48−2q46 + 4q44−6q42 + 7q40−9q38 + 9q36−7q34 + 5q32−q30−2q28 + 8q26−11q24 + 15q22−16q20 + 16q18−15q16 + 10q14−8q12 + q10 + q8−6q6 + 8q4−8q2 + 9−6q−2 + 7q−4−3q−6 + 3q−8
1,0	q78−2q74−2q72 + 2q70 + 5q68−6q64−4q62 + 6q60 + 8q58−3q56−10q54−3q52 + 8q50 + 5q48−5q46−6q44 + 4q42 + 7q40−6q36 + 7q32 + 2q30−6q28−4q26 + 5q24 + 4q22−4q20−7q18 + 3q16 + 8q14−10q10−5q8 + 7q6 + 7q4−3q2−8−q−2 + 6q−4 + 5q−6−2q−8−2q−10 + q−12 + 3q−14

D4 Invariants.

Weight	Invariant
1,0,0,0	q66−2q64 + 2q62−3q60 + 5q58−6q56 + 6q54−6q52 + 8q50−7q48 + 3q46−4q44 + 2q42 + q40−5q38 + 6q36−5q34 + 14q32−9q30 + 14q28−11q26 + 14q24−10q22 + 6q20−11q18 + q16−4q14−4q12−q10−7q8 + 7q6−5q4 + 8q2−4 + 9q−2−2q−4 + 6q−6−2q−8 + 3q−10

G2 Invariants.

Weight

Invariant

1,0

q114−2q112 + 4q110−6q108 + 4q106−q104−4q102 + 13q100−18q98 + 22q96−19q94 + 3q92 + 12q90−29q88 + 39q86−36q84 + 24q82−24q78 + 38q76−35q74 + 17q72 + 4q70−24q68 + 27q66−13q64−5q62 + 34q60−45q58 + 42q56−16q54−18q52 + 48q50−63q48 + 57q46−32q44 + 5q42 + 27q40−49q38 + 47q36−34q34 + 7q32 + 13q30−34q28 + 23q26−4q24−12q22 + 28q20−38q18 + 22q16 + 3q14−29q12 + 44q10−46q8 + 30q6−17q2 + 29−29q−2 + 25q−4−7q−6−4q−8 + 9q−10−10q−12 + 8q−14−q−16 + q−18 + q−20

.

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["10 144"];

In[4]:=

Alexander[K][t]

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

Out[4]=

−3t2 + 10t−13 + 10t−1−3t−2

In[5]:=

Conway[K][z]

Out[5]=

−3z4−2z2 + 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]=

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 39, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

2q−3 + 5q−1−7q−2 + 7q−3−6q−4 + 5q−5−3q−6 + q−7

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

z2a6−z4a4 + 2a4−2z4a2−5z2a2−4a2 + 2z2 + 3

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

z4a8−z2a8 + 3z5a7−4z3a7 + 4z6a6−6z4a6 + 2z2a6 + 3z7a5−4z5a5 + 4z3a5−2za5 + z8a4 + 2z6a4−2z4a4−2z2a4 + 2a4 + 4z7a3−8z5a3 + 8z3a3−2za3 + z8a2−2z6a2 + 8z4a2−12z2a2 + 4a2 + z7a−z5a + 3z4−7z2 + 3

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["10 144"];

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]=

{ −3t2 + 10t−13 + 10t−1−3t−2, 2q−3 + 5q−1−7q−2 + 7q−3−6q−4 + 5q−5−3q−6 + q−7 }

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]=

{K11n99,}

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	q5 + 2q4−6q3 + q2 + 12q−16−5q−1 + 31q−2−24q−3−17q−4 + 49q−5−26q−6−29q−7 + 54q−8−21q−9−32q−10 + 44q−11−10q−12−25q−13 + 25q−14−q−15−13q−16 + 8q−17 + q−18−3q−19 + q−20
3	2q11−q9−9q8 + 5q7 + 13q6 + 4q5−30q4−11q3 + 38q2 + 37q−52−59q−1 + 51q−2 + 96q−3−50q−4−126q−5 + 37q−6 + 156q−7−20q−8−184q−9 + 5q−10 + 198q−11 + 16q−12−208q−13−33q−14 + 208q−15 + 48q−16−196q−17−62q−18 + 176q−19 + 69q−20−144q−21−75q−22 + 111q−23 + 71q−24−75q−25−62q−26 + 46q−27 + 47q−28−23q−29−33q−30 + 9q−31 + 21q−32−4q−33−10q−34 + q−35 + 4q−36 + q−37−3q−38 + q−39
4	q20 + 2q19−6q17−4q16−5q15 + 14q14 + 23q13−7q12−19q11−54q10 + 9q9 + 81q8 + 44q7 + 6q6−159q5−86q4 + 108q3 + 158q2 + 156q−241−273q−1 + 8q−2 + 245q−3 + 432q−4−202q−5−454q−6−218q−7 + 217q−8 + 732q−9−47q−10−547q−11−475q−12 + 91q−13 + 962q−14 + 137q−15−552q−16−676q−17−61q−18 + 1081q−19 + 294q−20−493q−21−796q−22−201q−23 + 1085q−24 + 409q−25−375q−26−814q−27−326q−28 + 943q−29 + 463q−30−183q−31−700q−32−419q−33 + 664q−34 + 415q−35 + 22q−36−462q−37−412q−38 + 345q−39 + 262q−40 + 134q−41−205q−42−289q−43 + 123q−44 + 95q−45 + 120q−46−46q−47−141q−48 + 32q−49 + 9q−50 + 59q−51 + q−52−49q−53 + 12q−54−8q−55 + 17q−56 + 4q−57−13q−58 + 4q−59−3q−60 + 4q−61 + q−62−3q−63 + q−64
5	2q31 + 2q29−3q28−9q27−9q26 + 7q25 + 9q24 + 27q23 + 24q22−24q21−58q20−48q19−19q18 + 73q17 + 152q16 + 79q15−77q14−200q13−236q12−37q11 + 293q10 + 410q9 + 208q8−221q7−615q6−537q5 + 88q4 + 733q3 + 885q2 + 279q−753−1283q−1−712q−2 + 571q−3 + 1576q−4 + 1311q−5−248q−6−1784q−7−1865q−8−246q−9 + 1810q−10 + 2432q−11 + 817q−12−1731q−13−2882q−14−1407q−15 + 1520q−16 + 3237q−17 + 1980q−18−1255q−19−3508q−20−2471q−21 + 984q−22 + 3649q−23 + 2904q−24−692q−25−3764q−26−3248q−27 + 447q−28 + 3785q−29 + 3520q−30−176q−31−3767q−32−3736q−33−86q−34 + 3667q−35 + 3881q−36 + 370q−37−3451q−38−3956q−39−695q−40 + 3138q−41 + 3912q−42 + 1020q−43−2672q−44−3740q−45−1333q−46 + 2119q−47 + 3403q−48 + 1554q−49−1481q−50−2927q−51−1665q−52 + 883q−53 + 2328q−54 + 1611q−55−347q−56−1709q−57−1424q−58−19q−59 + 1122q−60 + 1127q−61 + 239q−62−643q−63−815q−64−302q−65 + 312q−66 + 518q−67 + 267q−68−101q−69−294q−70−206q−71 + 15q−72 + 148q−73 + 123q−74 + 15q−75−56q−76−67q−77−25q−78 + 26q−79 + 33q−80 + 8q−81−9q−82−5q−83−10q−84−q−85 + 12q−86−5q−88 + q−89−3q−91 + 4q−92 + q−93−3q−94 + q−95
6	q45 + 2q44−4q41−6q40−12q39−5q38 + 14q37 + 29q36 + 29q35 + 16q34−2q33−76q32−97q31−68q30 + 38q29 + 122q28 + 187q27 + 221q26−22q25−256q24−436q23−313q22−85q21 + 349q20 + 847q19 + 669q18 + 153q17−699q16−1105q15−1254q14−453q13 + 1113q12 + 1902q11 + 1869q10 + 422q9−1150q8−2990q7−2924q6−559q5 + 2059q4 + 4106q3 + 3520q2 + 1301q−3296−5871q−1−4542q−2−647q−3 + 4580q−4 + 7057q−5 + 6315q−6−481q−7−6883q−8−9007q−9−5958q−10 + 1812q−11 + 8666q−12 + 11846q−13 + 4886q−14−4787q−15−11704q−16−11724q−17−3341q−18 + 7487q−19 + 15783q−20 + 10647q−21−634q−22−12000q−23−16032q−24−8770q−25 + 4622q−26 + 17597q−27 + 15100q−28 + 3666q−29−10874q−30−18460q−31−13001q−32 + 1665q−33 + 18028q−34 + 17905q−35 + 7003q−36−9489q−37−19604q−38−15841q−39−674q−40 + 17839q−41 + 19571q−42 + 9454q−43−8121q−44−19992q−45−17824q−46−2763q−47 + 16974q−48 + 20475q−49 + 11658q−50−6137q−51−19357q−52−19234q−53−5335q−54 + 14623q−55 + 20160q−56 + 13775q−57−2785q−58−16737q−59−19384q−60−8355q−61 + 10120q−62 + 17516q−63 + 14793q−64 + 1572q−65−11590q−66−16994q−67−10422q−68 + 4288q−69 + 12175q−70 + 13224q−71 + 5031q−72−5202q−73−11867q−74−9841q−75−486q−76 + 5819q−77 + 9032q−78 + 5731q−79−251q−80−5970q−81−6725q−82−2329q−83 + 1156q−84 + 4300q−85 + 3957q−86 + 1671q−87−1827q−88−3172q−89−1764q−90−658q−91 + 1204q−92 + 1722q−93 + 1396q−94−154q−95−960q−96−659q−97−656q−98 + 65q−99 + 423q−100 + 636q−101 + 96q−102−167q−103−71q−104−272q−105−84q−106 + 20q−107 + 199q−108 + 23q−109−26q−110 + 51q−111−64q−112−32q−113−23q−114 + 55q−115−10q−116−14q−117 + 28q−118−11q−119−4q−120−11q−121 + 19q−122−5q−123−9q−124 + 9q−125−3q−126−3q−128 + 4q−129 + q−130−3q−131 + q−132
7	2q61 + 2q59−3q57−9q56−7q55−9q54−q53 + 9q52 + 29q51 + 49q50 + 38q49−6q48−44q47−90q46−128q45−107q44−24q43 + 162q42 + 298q41 + 289q40 + 217q39−17q38−369q37−656q36−752q35−371q34 + 292q33 + 861q32 + 1352q31 + 1304q30 + 530q29−689q28−2088q27−2574q26−1898q25−420q24 + 1901q23 + 3829q22 + 4256q21 + 2823q20−697q19−4322q18−6453q17−6358q16−2686q15 + 2946q14 + 8114q13 + 10597q12 + 7692q11 + 823q10−7412q9−14103q8−14260q7−7497q6 + 3858q5 + 15672q4 + 20608q3 + 16308q2 + 3546q−13814−25625q−1−26196q−2−13969q−3 + 7879q−4 + 27326q−5 + 35465q−6 + 26853q−7 + 2077q−8−25236q−9−42533q−10−40118q−11−15237q−12 + 18660q−13 + 46212q−14 + 52700q−15 + 30243q−16−8641q−17−46145q−18−62822q−19−45426q−20−4128q−21 + 42400q−22 + 70147q−23 + 59682q−24 + 17927q−25−36031q−26−74330q−27−71801q−28−31605q−29 + 27847q−30 + 75845q−31 + 81593q−32 + 44203q−33−19186q−34−75517q−35−88915q−36−54896q−37 + 10861q−38 + 73802q−39 + 94171q−40 + 63809q−41−3410q−42−71812q−43−97919q−44−70663q−45−2766q−46 + 69606q−47 + 100458q−48 + 76233q−49 + 7962q−50−67759q−51−102489q−52−80631q−53−12238q−54 + 65923q−55 + 104008q−56 + 84545q−57 + 16348q−58−63973q−59−105234q−60−88259q−61−20680q−62 + 61376q−63 + 105793q−64 + 91890q−65 + 25794q−66−57352q−67−105234q−68−95431q−69−31979q−70 + 51480q−71 + 102845q−72 + 98134q−73 + 39138q−74−43106q−75−97776q−76−99400q−77−46793q−78 + 32413q−79 + 89477q−80 + 98054q−81 + 53890q−82−19784q−83−77718q−84−93364q−85−59228q−86 + 6475q−87 + 63046q−88 + 84781q−89 + 61556q−90 + 6102q−91−46641q−92−72739q−93−60074q−94−16168q−95 + 30171q−96 + 58100q−97 + 54737q−98 + 22765q−99−15486q−100−42763q−101−46248q−102−25158q−103 + 4022q−104 + 28217q−105 + 35978q−106 + 24029q−107 + 3551q−108−16206q−109−25632q−110−20196q−111−7265q−112 + 7378q−113 + 16421q−114 + 15166q−115 + 8105q−116−1826q−117−9380q−118−10235q−119−6983q−120−916q−121 + 4553q−122 + 6031q−123 + 5155q−124 + 1931q−125−1759q−126−3167q−127−3325q−128−1765q−129 + 407q−130 + 1320q−131 + 1850q−132 + 1307q−133 + 148q−134−401q−135−946q−136−816q−137−175q−138 + 30q−139 + 363q−140 + 400q−141 + 151q−142 + 141q−143−147q−144−227q−145−53q−146−71q−147 + 33q−148 + 48q−149 + 3q−150 + 96q−151 + 8q−152−56q−153 + 4q−154−19q−155 + 10q−156−3q−157−29q−158 + 31q−159 + 10q−160−12q−161 + 2q−162−5q−163 + 9q−164 + 2q−165−14q−166 + 5q−167 + 5q−168−3q−169−3q−171 + 4q−172 + q−173−3q−174 + q−175