9 47

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["9 47"];

In[4]:=

PD[K]

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

Out[4]=

X6271 X16,8,17,7 X8394 X2,15,3,16 X14,9,15,10 X10,6,11,5 X4,14,5,13 X11,1,12,18 X17,13,18,12

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

6 8 10 16 14 -18 4 2 -12

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[8*-20]

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,{−1,2,−1,2,3,2,−1,2,3})

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, 9, 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[{5, 9}, {8, 1}, {9, 3}, {2, 7}, {4, 8}, {3, 6}, {1, 5}, {7, 4}, {6, 2}]

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Further quantum knot invariants for 9_47.

A1 Invariants.

Weight	Invariant
1	−q5 + 2q3 + 2q−1−q−5−2q−9 + 2q−11
2	q16−2q14−3q12 + 5q10 + 2q8−5q6 + 3q4 + 6q2−4−q−2 + 4q−4−3q−6−4q−8 + 2q−10 + 2q−12−3q−14−q−16 + 7q−18−q−20−5q−22 + 6q−24−5q−28 + q−30 + q−32
3	−q33 + 2q31 + 3q29−2q27−8q25−5q23 + 11q21 + 11q19−5q17−17q15−2q13 + 20q11 + 15q9−14q7−20q5 + 6q3 + 24q−q−1−24q−3−7q−5 + 19q−7 + 10q−9−18q−11−8q−13 + 13q−15 + 12q−17−9q−19−10q−21 + 4q−23 + 16q−25 + 3q−27−16q−29−13q−31 + 13q−33 + 21q−35−10q−37−27q−39 + q−41 + 27q−43 + 5q−45−17q−47−12q−49 + 11q−51 + 10q−53−3q−55−4q−57−2q−59 + 2q−61
4	q56−2q54−3q52 + 2q50 + 5q48 + 11q46−2q44−16q42−18q40−10q38 + 29q36 + 33q34 + 10q32−26q30−58q28−16q26 + 35q24 + 70q22 + 43q20−51q18−83q16−45q14 + 56q12 + 109q10 + 32q8−71q6−116q4−24q2 + 98 + 97q−2−5q−4−109q−6−72q−8 + 49q−10 + 97q−12 + 32q−14−69q−16−68q−18 + 17q−20 + 70q−22 + 30q−24−42q−26−53q−28 + 54q−32 + 32q−34−20q−36−52q−38−34q−40 + 32q−42 + 60q−44 + 38q−46−47q−48−95q−50−23q−52 + 75q−54 + 116q−56 + 9q−58−120q−60−102q−62 + 18q−64 + 135q−66 + 84q−68−58q−70−108q−72−55q−74 + 60q−76 + 84q−78 + 16q−80−38q−82−49q−84−6q−86 + 22q−88 + 17q−90 + 4q−92−8q−94−5q−96 + q−98 + q−100
5	−q85 + 2q83 + 3q81−2q79−5q77−8q75−4q73 + 7q71 + 24q69 + 24q67 + 3q65−27q63−51q61−48q59−5q57 + 66q55 + 95q53 + 67q51−13q49−112q47−156q45−90q43 + 58q41 + 190q39 + 220q37 + 93q35−137q33−303q31−264q29−17q27 + 278q25 + 409q23 + 230q21−146q19−452q17−434q15−65q13 + 380q11 + 548q9 + 284q7−221q5−572q3−451q + 40q−1 + 495q−3 + 539q−5 + 130q−7−371q−9−539q−11−236q−13 + 254q−15 + 480q−17 + 282q−19−146q−21−402q−23−276q−25 + 86q−27 + 317q−29 + 229q−31−48q−33−254q−35−192q−37 + 48q−39 + 206q−41 + 151q−43−34q−45−183q−47−159q−49 + 13q−51 + 173q−53 + 191q−55 + 55q−57−150q−59−255q−61−160q−63 + 92q−65 + 319q−67 + 312q−69 + 19q−71−345q−73−469q−75−191q−77 + 296q−79 + 590q−81 + 398q−83−168q−85−624q−87−572q−89−30q−91 + 534q−93 + 663q−95 + 242q−97−364q−99−619q−101−370q−103 + 129q−105 + 472q−107 + 404q−109 + 42q−111−276q−113−319q−115−135q−117 + 99q−119 + 199q−121 + 136q−123 + q−125−89q−127−78q−129−31q−131 + 18q−133 + 35q−135 + 18q−137−3q−139−6q−141−4q−143−2q−145 + 2q−147

A2 Invariants.

Weight	Invariant
1,0	−q6 + q4 + q2 + 2 + 2q−2−q−4 + q−6−2q−8−q−12−q−14 + q−16 + q−20
1,1	q20−4q18 + 10q16−22q14 + 32q12−46q10 + 56q8−52q6 + 48q4−22q2 + 4 + 36q−2−58q−4 + 74q−6−94q−8 + 90q−10−94q−12 + 70q−14−52q−16 + 28q−18 + 5q−20−24q−22 + 50q−24−56q−26 + 58q−28−46q−30 + 32q−32−22q−34 + 10q−36−2q−38−4q−40 + 2q−46
2,0	q18−q16−3q14−q12 + 2q10 + 2q8 + 4q4 + 6q2 + 3−2q−2 + q−4−3q−6−3q−8−2q−10−2q−12−2q−14−2q−16 + 3q−18 + 3q−24 + 5q−26−q−28−q−30 + q−32−2q−38−q−48 + q−52

A3 Invariants.

Weight	Invariant
0,1,0	q14−2q12 + q8−4q6 + 5q4 + 3q2 + 1 + 7q−2 + 3q−4−4q−6−2q−8−3q−10−5q−12−2q−14 + 5q−18 + q−20 + q−22 + 5q−24−4q−26−3q−28 + 3q−30−3q−32−q−34 + 3q−36
1,0,0	−q7 + q5 + 3q + q−1 + 2q−3−q−11−2q−15−2q−19 + q−21 + q−25 + q−27

A4 Invariants.

Weight	Invariant
0,1,0,0	q16−q14−q12−2q8 + 4q4 + 3q2 + 3 + 9q−2 + 7q−4 + q−6−2q−8 + q−10−6q−12−12q−14−5q−16−6q−20 + q−22 + 9q−24 + 2q−26 + 2q−28 + 6q−30 + 3q−32−4q−34−q−36 + 2q−38−3q−40−5q−42 + q−44 + 2q−46−2q−48 + q−50 + 2q−52
1,0,0,0	−q8 + q6 + 2q2 + 2 + q−2 + 2q−4 + q−8−q−10 + q−12−q−14−2q−18−q−20−q−22−2q−24 + q−26 + q−30 + q−32 + q−34

B2 Invariants.

Weight	Invariant
0,1	−q14 + 2q12−4q10 + 5q8−4q6 + 5q4−3q2 + 3 + q−2−q−4 + 6q−6−6q−8 + 9q−10−9q−12 + 8q−14−8q−16 + 3q−18−3q−20−q−22 + q−24−4q−26 + 5q−28−5q−30 + 5q−32−3q−34 + 3q−36
1,0	q24−2q20−2q18 + 2q16 + 3q14−3q12−4q10 + q8 + 7q6 + 3q4−3q2−1 + 6q−2 + 5q−4 + q−6−4q−8−q−10 + q−12−q−14−6q−16−4q−18 + q−20 + q−22−2q−24−4q−26 + 3q−28 + 6q−30 + 2q−32−3q−34 + q−36 + 5q−38 + 3q−40−4q−42−5q−44 + q−46 + 4q−48−q−50−4q−52−2q−54 + q−56 + 3q−58

D4 Invariants.

Weight	Invariant
1,0,0,0	q18−2q16 + 2q14−4q12 + 4q10−4q8 + 4q6−2q4 + 6q2 + 2 + 3q−2 + 4q−4 + 3q−8−6q−10 + 4q−12−9q−14 + 4q−16−9q−18 + 5q−20−5q−22 + 6q−24−2q−26 + 2q−28 + q−30 + 2q−34−4q−36 + 2q−38−5q−40 + 5q−42−3q−44 + 2q−46−2q−48 + 3q−50

G2 Invariants.

Weight

Invariant

1,0

q32−2q30 + 4q28−7q26 + 4q24−q22−8q20 + 16q18−16q16 + 16q14−4q12−11q10 + 20q8−18q6 + 14q4−2q2−13 + 21q−2−8q−4 + 13q−8−19q−10 + 18q−12−4q−14−9q−16 + 12q−18−21q−20 + 25q−22−13q−24 + 9q−28−19q−30 + 23q−32−18q−34 + 4q−36 + 3q−38−14q−40 + 19q−42−11q−44−3q−46 + 15q−48−19q−50 + 12q−52 + q−54−17q−56 + 20q−58−17q−60 + 11q−62 + 2q−64−11q−66 + 14q−68−12q−70 + 8q−72−5q−76 + 2q−78−2q−80 + 2q−82−q−84 + 2q−86 + q−88

.

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["9 47"];

In[4]:=

Alexander[K][t]

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

Out[4]=

t3−4t2 + 6t−5 + 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]=

{3,t + 1}

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 27, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

2q5−4q4 + 4q3−5q2 + 5q−3 + 3q−1−q−2

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

z6a−2 + 4z4a−2−z4a−4−z4 + 4z2a−2−3z2a−4−2z2 + a−2−2a−4 + a−6 + 1

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

2z7a−1 + 2z7a−3 + 6z6a−2 + 3z6a−4 + 3z6 + az5−4z5a−1−4z5a−3 + z5a−5−16z4a−2−7z4a−4−9z4−2az3 + z3a−1 + 6z3a−3 + 3z3a−5 + 11z2a−2 + 9z2a−4 + 3z2a−6 + 5z2−2za−1−5za−3−3za−5−a−2−2a−4−a−6 + 1

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["9 47"];

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−5 + 6t−1−4t−2 + t−3, 2q5−4q4 + 4q3−5q2 + 5q−3 + 3q−1−q−2 }

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]=

{}

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	q15−6q13 + 6q12 + 6q11−17q10 + 10q9 + 14q8−25q7 + 8q6 + 19q5−25q4 + 2q3 + 20q2−18q−3 + 17q−1−8q−2−6q−3 + 9q−4−q−5−3q−6 + q−7
3	2q29−4q28−2q27 + q26 + 15q25−3q24−25q23−4q22 + 37q21 + 19q20−51q19−32q18 + 54q17 + 50q16−59q15−58q14 + 51q13 + 69q12−46q11−70q10 + 37q9 + 70q8−25q7−69q6 + 16q5 + 60q4 + 3q3−60q2−10q + 43 + 26q−1−35q−2−28q−3 + 17q−4 + 32q−5−6q−6−23q−7−5q−8 + 17q−9 + 6q−10−7q−11−5q−12 + q−13 + 3q−14−q−15
4	q48−6q46−3q45 + 12q44 + 14q43 + 5q42−34q41−46q40 + 23q39 + 68q38 + 73q37−58q36−161q35−30q34 + 118q33 + 215q32−7q31−278q30−150q29 + 100q28 + 344q27 + 100q26−319q25−248q24 + 28q23 + 392q22 + 185q21−297q20−276q19−38q18 + 374q17 + 217q16−245q15−254q14−92q13 + 321q12 + 228q11−173q10−214q9−145q8 + 236q7 + 227q6−72q5−149q4−193q3 + 115q2 + 190q + 32−47q−1−192q−2−7q−3 + 98q−4 + 77q−5 + 56q−6−115q−7−60q−8−3q−9 + 39q−10 + 88q−11−21q−12−33q−13−38q−14−12q−15 + 46q−16 + 11q−17 + 3q−18−15q−19−16q−20 + 7q−21 + 3q−22 + 5q−23−q−24−3q−25 + q−26
5	2q71−4q70−2q69−2q68 + 3q67 + 21q66 + 19q65−21q64−51q63−49q62−8q61 + 111q60 + 154q59 + 42q58−151q57−283q56−192q55 + 154q54 + 472q53 + 404q52−83q51−626q50−691q49−95q48 + 727q47 + 1010q46 + 338q45−755q44−1255q43−637q42 + 675q41 + 1466q40 + 904q39−563q38−1549q37−1124q36 + 397q35 + 1590q34 + 1268q33−270q32−1542q31−1351q30 + 145q29 + 1495q28 + 1373q27−65q26−1406q25−1369q24−15q23 + 1323q22 + 1349q21 + 84q20−1221q19−1314q18−173q17 + 1083q16 + 1287q15 + 290q14−944q13−1226q12−404q11 + 721q10 + 1161q9 + 546q8−516q7−1028q6−630q5 + 231q4 + 858q3 + 714q2−15q−619−674q−1−224q−2 + 367q−3 + 593q−4 + 336q−5−114q−6−410q−7−392q−8−78q−9 + 224q−10 + 318q−11 + 192q−12−34q−13−213q−14−209q−15−71q−16 + 71q−17 + 153q−18 + 132q−19 + 17q−20−82q−21−101q−22−61q−23 + 5q−24 + 66q−25 + 61q−26 + 17q−27−21q−28−33q−29−24q−30−5q−31 + 18q−32 + 14q−33 + 3q−34−3q−35−3q−36−5q−37 + q−38 + 3q−39−q−40
6	q99−6q97−3q96 + 6q95 + 11q94 + 14q93 + 12q92−7q91−64q90−83q89−18q88 + 78q87 + 162q86 + 204q85 + 92q84−220q83−477q82−434q81−65q80 + 445q79 + 935q78 + 897q77 + 32q76−1075q75−1656q74−1225q73 + 77q72 + 1873q71 + 2748q70 + 1631q69−829q68−3059q67−3524q66−1819q65 + 1792q64 + 4634q63 + 4325q62 + 970q61−3303q60−5637q59−4594q58 + 244q57 + 5213q56 + 6574q55 + 3386q54−2186q53−6367q52−6680q51−1701q50 + 4511q49 + 7418q48 + 5032q47−767q46−5949q45−7440q44−2940q43 + 3523q42 + 7244q41 + 5580q40 + 139q39−5253q38−7358q37−3392q36 + 2818q35 + 6777q34 + 5549q33 + 612q32−4638q31−7027q30−3574q29 + 2217q28 + 6240q27 + 5439q26 + 1127q25−3892q24−6593q23−3894q22 + 1292q21 + 5430q20 + 5347q19 + 1991q18−2664q17−5850q16−4346q15−152q14 + 4030q13 + 4979q12 + 3060q11−844q10−4434q9−4462q8−1809q7 + 1944q6 + 3828q5 + 3677q4 + 1147q3−2236q2−3594q−2831−290q−1 + 1775q−2 + 3078q−3 + 2319q−4 + 68q−5−1695q−6−2423q−7−1547q−8−320q−9 + 1362q−10 + 1925q−11 + 1247q−12 + 161q−13−917q−14−1199q−15−1184q−16−195q−17 + 578q−18 + 881q−19 + 759q−20 + 271q−21−126q−22−684q−23−536q−24−284q−25 + 48q−26 + 284q−27 + 368q−28 + 360q−29−15q−30−124q−31−236q−32−196q−33−118q−34 + 28q−35 + 186q−36 + 103q−37 + 89q−38−2q−39−48q−40−94q−41−62q−42 + 12q−43 + 10q−44 + 39q−45 + 25q−46 + 17q−47−16q−48−17q−49−q−50−7q−51 + 3q−52 + 3q−53 + 5q−54−q−55−3q−56 + q−57
7	2q131−4q130−2q129−2q128 + 15q126 + 19q125 + 15q124−13q123−47q122−70q121−68q120−28q119 + 109q118 + 246q117 + 278q116 + 144q115−160q114−489q113−725q112−649q111−75q110 + 846q109 + 1567q108 + 1653q107 + 791q106−812q105−2516q104−3461q103−2611q102 + 49q101 + 3396q100 + 5762q99 + 5505q98 + 2098q97−3244q96−8154q95−9574q94−5881q93 + 1670q92 + 9855q91 + 13991q90 + 11087q89 + 1803q88−9990q87−17999q86−17161q85−6983q84 + 8363q83 + 20776q82 + 22927q81 + 13091q80−4888q79−21648q78−27727q77−19347q76 + 351q75 + 20885q74 + 30785q73 + 24610q72 + 4578q71−18666q70−32143q69−28605q68−8988q67 + 15921q66 + 32023q65 + 30941q64 + 12461q63−13064q62−31045q61−32019q60−14775q59 + 10730q58 + 29642q57 + 32091q56 + 16131q55−8934q54−28287q53−31718q52−16723q51 + 7756q50 + 27068q49 + 31078q48 + 16987q47−6870q46−26075q45−30510q44−17103q43 + 6142q42 + 25133q41 + 29948q40 + 17342q39−5240q38−24091q37−29454q36−17828q35 + 4011q34 + 22811q33 + 28917q32 + 18523q31−2320q30−20988q29−28182q28−19533q27 + 19q26 + 18678q25 + 27138q24 + 20527q23 + 2746q22−15509q21−25433q20−21539q19−6000q18 + 11708q17 + 23066q16 + 21943q15 + 9245q14−7100q13−19589q12−21648q11−12361q10 + 2229q9 + 15299q8 + 20080q7 + 14458q6 + 2675q5−10035q4−17269q3−15407q2−6824q + 4661 + 13133q−1 + 14552q−2 + 9688q−3 + 496q−4−8266q−5−12190q−6−10696q−7−4404q−8 + 3309q−9 + 8477q−10 + 9890q−11 + 6668q−12 + 794q−13−4385q−14−7487q−15−6944q−16−3494q−17 + 670q−18 + 4387q−19 + 5691q−20 + 4371q−21 + 1803q−22−1381q−23−3472q−24−3781q−25−2888q−26−721q−27 + 1299q−28 + 2355q−29 + 2579q−30 + 1629q−31 + 318q−32−772q−33−1648q−34−1589q−35−970q−36−256q−37 + 579q−38 + 907q−39 + 926q−40 + 734q−41 + 138q−42−303q−43−529q−44−605q−45−368q−46−146q−47 + 94q−48 + 360q−49 + 332q−50 + 236q−51 + 88q−52−96q−53−140q−54−175q−55−152q−56−30q−57 + 42q−58 + 84q−59 + 90q−60 + 32q−61 + 21q−62−12q−63−42q−64−30q−65−18q−66 + 4q−67 + 15q−68 + 4q−69 + 5q−70 + 7q−71−3q−72−3q−73−5q−74 + q−75 + 3q−76−q−77