9 37

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 37"];

In[4]:=

PD[K]

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

Out[4]=

X1425 X7,12,8,13 X3,11,4,10 X11,3,12,2 X5,14,6,15 X13,6,14,7 X15,18,16,1 X9,17,10,16 X17,9,18,8

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

4 10 14 12 16 2 6 18 8

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[3,21,21]

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(5,{−1,−1,2,−1,−3,2,1,4,−3,2,−3,4})

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]=

{ 5, 12, 5 }

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Further quantum knot invariants for 9_37.

A1 Invariants.

Weight	Invariant
1	−q11 + 2q9−q7 + 3q5−q3−q−2q−3 + 3q−5−q−7 + q−9
2	q32−2q30−2q28 + 6q26−2q24−7q22 + 10q20 + 3q18−12q16 + 8q14 + 6q12−13q10 + 6q6−3q4−4q2 + 5 + 9q−2−8q−4−2q−6 + 13q−8−9q−10−7q−12 + 11q−14−3q−16−5q−18 + 4q−20−q−24 + q−26
3	−q63 + 2q61 + 2q59−3q57−6q55 + 2q53 + 13q51−q49−19q47−7q45 + 26q43 + 19q41−25q39−34q37 + 22q35 + 46q33−8q31−56q29−5q27 + 54q25 + 15q23−51q21−30q19 + 45q17 + 32q15−26q13−33q11 + 18q9 + 34q7 + 3q5−32q3−18q + 26q−1 + 32q−3−20q−5−51q−7 + 10q−9 + 56q−11 + 4q−13−58q−15−11q−17 + 49q−19 + 24q−21−38q−23−25q−25 + 25q−27 + 21q−29−10q−31−16q−33 + 4q−35 + 8q−37−2q−39−4q−41 + q−43 + q−45−q−49 + q−51
4	q104−2q102−2q100 + 3q98 + 3q96 + 6q94−9q92−13q90 + 2q88 + 10q86 + 31q84−8q82−41q80−26q78 + 5q76 + 82q74 + 38q72−48q70−91q68−69q66 + 104q64 + 137q62 + 39q60−122q58−208q56 + 11q54 + 186q52 + 197q50−30q48−287q46−149q44 + 110q42 + 288q40 + 120q38−235q36−241q34−11q32 + 262q30 + 200q28−122q26−225q24−87q22 + 164q20 + 187q18−17q16−167q14−127q12 + 61q10 + 153q8 + 85q6−91q4−160q2−60 + 115q−2 + 209q−4 + 16q−6−183q−8−204q−10 + 30q−12 + 292q−14 + 149q−16−128q−18−296q−20−105q−22 + 253q−24 + 229q−26 + 7q−28−247q−30−196q−32 + 108q−34 + 185q−36 + 105q−38−106q−40−159q−42−5q−44 + 68q−46 + 92q−48−3q−50−63q−52−22q−54 + 34q−58 + 9q−60−13q−62−2q−64−6q−66 + 6q−68 + q−70−3q−72 + 2q−74−2q−76 + q−78−q−82 + q−84
5	−q155 + 2q153 + 2q151−3q149−3q147−3q145 + q143 + 9q141 + 13q139−2q137−20q135−22q133−7q131 + 24q129 + 49q127 + 36q125−30q123−87q121−77q119 + q117 + 113q115 + 163q113 + 70q111−122q109−251q107−195q105 + 42q103 + 322q101 + 387q99 + 126q97−304q95−569q93−409q91 + 149q89 + 686q87 + 733q85 + 168q83−652q81−1042q79−583q77 + 432q75 + 1201q73 + 1037q71−51q69−1202q67−1392q65−396q63 + 997q61 + 1595q59 + 828q57−676q55−1618q53−1126q51 + 330q49 + 1462q47 + 1286q45 + 2q43−1231q41−1298q39−210q37 + 938q35 + 1177q33 + 372q31−702q29−1041q27−424q25 + 476q23 + 875q21 + 485q19−293q17−756q15−533q13 + 112q11 + 662q9 + 652q7 + 95q5−573q3−803q−357q−1 + 462q−3 + 977q−5 + 682q−7−284q−9−1125q−11−1033q−13 + 10q−15 + 1166q−17 + 1377q−19 + 345q−21−1089q−23−1605q−25−733q−27 + 812q−29 + 1696q−31 + 1087q−33−457q−35−1549q−37−1305q−39 + 15q−41 + 1246q−43 + 1358q−45 + 342q−47−825q−49−1191q−51−576q−53 + 399q−55 + 907q−57 + 640q−59−75q−61−579q−63−543q−65−127q−67 + 285q−69 + 390q−71 + 185q−73−99q−75−223q−77−155q−79−2q−81 + 105q−83 + 96q−85 + 29q−87−36q−89−49q−91−20q−93 + 9q−95 + 16q−97 + 11q−99 + 4q−101−8q−103−5q−105 + 2q−107−q−109 + 3q−113−q−115−2q−117 + q−119−q−123 + q−125

A2 Invariants.

Weight	Invariant
1,0	−q16 + q14 + q12−q10 + 3q8 + q6−3−2q−4 + q−6 + 2q−8−q−10 + q−12 + q−14
1,1	q44−4q42 + 10q40−22q38 + 42q36−70q34 + 100q32−136q30 + 169q28−186q26 + 186q24−156q22 + 117q20−40q18−44q16 + 138q14−229q12 + 284q10−348q8 + 352q6−343q4 + 300q2−214 + 146q−2−42q−4−30q−6 + 104q−8−154q−10 + 166q−12−170q−14 + 152q−16−132q−18 + 99q−20−72q−22 + 50q−24−32q−26 + 21q−28−10q−30 + 6q−32−2q−34 + q−36
2,0	q42−q40−2q38 + 3q34 + q32−6q30 + 8q26 + 5q24−6q22−2q20 + 8q18 + 4q16−9q14−5q12 + 2q10−5q8−4q6 + 2q2 + 2 + 8q−2 + 6q−4−3q−6−q−8 + 8q−10−11q−14 + 7q−18−8q−22−3q−24 + 4q−26 + 2q−28−q−30 + q−34 + q−36

A3 Invariants.

Weight	Invariant
0,1,0	q34−2q32 + 2q28−6q26 + 4q24 + 6q22−6q20 + 7q18 + 8q16−11q14 + 2q10−9q8−2q6 + 4q4 + 6q2 + 1 + q−2 + 8q−4−5q−6−10q−8 + 8q−10−4q−12−8q−14 + 9q−16−3q−20 + 4q−22 + q−24−q−26 + q−28
1,0,0	−q21 + q19 + q15−q13 + 3q11 + q9 + 2q7−2q−3q−1−2q−5 + q−7 + 2q−11−q−13 + q−15 + q−17 + q−19

B2 Invariants.

Weight	Invariant
0,1	−q34 + 2q32−4q30 + 6q28−8q26 + 10q24−10q22 + 10q20−7q18 + 6q16 + q14−4q12 + 12q10−15q8 + 18q6−20q4 + 18q2−19 + 11q−2−8q−4 + q−6 + 2q−8−6q−10 + 10q−12−10q−14 + 11q−16−8q−18 + 7q−20−4q−22 + 3q−24−q−26 + q−28
1,0	q56−2q52−2q50 + 2q48 + 4q46−2q44−7q42−2q40 + 9q38 + 8q36−4q34−9q32 + 2q30 + 12q28 + 7q26−9q24−9q22 + 2q20 + 7q18−4q16−10q14−2q12 + 6q10 + 2q8−6q6−q4 + 8q2 + 9−3q−2−5q−4 + 4q−6 + 9q−8−2q−10−11q−12−5q−14 + 8q−16 + 7q−18−7q−20−11q−22 + 10q−26 + 4q−28−4q−30−5q−32 + q−34 + 4q−36 + 2q−38−q−40−q−42 + q−46

G2 Invariants.

Weight

Invariant

1,0

q80−2q78 + 4q76−7q74 + 5q72−4q70−4q68 + 16q66−23q64 + 29q62−22q60 + 6q58 + 16q56−38q54 + 52q52−49q50 + 24q48 + 8q46−33q44 + 50q42−44q40 + 24q38 + 7q36−27q34 + 33q32−28q30−6q28 + 40q26−46q24 + 41q22−18q20−17q18 + 57q16−71q14 + 65q12−48q10 + 4q8 + 43q6−65q4 + 65q2−47 + 16q−2 + 18q−4−39q−6 + 32q−8−22q−10−7q−12 + 33q−14−38q−16 + 22q−18 + 6q−20−33q−22 + 50q−24−48q−26 + 29q−28−8q−30−20q−32 + 38q−34−40q−36 + 34q−38−14q−40 + 2q−42 + 9q−44−15q−46 + 15q−48−12q−50 + 8q−52−2q−54−q−56 + 3q−58−3q−60 + 3q−62−q−64 + q−66

.

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 37"];

In[4]:=

Alexander[K][t]

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

Out[4]=

2t2−11t + 19−11t−1 + 2t−2

In[5]:=

Conway[K][z]

Out[5]=

2z4−3z2 + 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]=

{ 45, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

q4−2q3 + 5q2−7q + 7−8q−1 + 7q−2−4q−3 + 3q−4−q−5

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

−z2a4 + z4a2 + z2a2 + 2a2 + z4−z2−2−2z2a−2 + a−4

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

a2z8 + z8 + 3a3z7 + 6az7 + 3z7a−1 + 3a4z6 + 5a2z6 + 3z6a−2 + 5z6 + a5z5−6a3z5−13az5−4z5a−1 + 2z5a−3−8a4z4−17a2z4−3z4a−2 + z4a−4−13z4−2a5z3 + 3a3z3 + 13az3 + 6z3a−1−2z3a−3 + 5a4z2 + 14a2z2 + z2a−2−2z2a−4 + 12z2−2a3z−7az−5za−1−2a2 + a−4−2

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["9 37"];

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]=

{ 2t2−11t + 19−11t−1 + 2t−2, q4−2q3 + 5q2−7q + 7−8q−1 + 7q−2−4q−3 + 3q−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]=

{K11n100,}

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	q12−2q11 + q10 + 5q9−11q8 + 3q7 + 19q6−29q5 + q4 + 41q3−44q2−5q + 58−48q−1−14q−2 + 59q−3−39q−4−20q−5 + 46q−6−20q−7−18q−8 + 26q−9−5q−10−11q−11 + 9q−12−3q−14 + q−15
3	q24−2q23 + q22 + q21 + q20−7q19 + 3q18 + 11q17−3q16−27q15 + 9q14 + 42q13 + q12−77q11−4q10 + 104q9 + 26q8−137q7−51q6 + 166q5 + 78q4−183q3−112q2 + 197q + 130−189q−1−156q−2 + 183q−3 + 165q−4−158q−5−172q−6 + 132q−7 + 172q−8−100q−9−159q−10 + 57q−11 + 151q−12−34q−13−120q−14−2q−15 + 100q−16 + 14q−17−66q−18−26q−19 + 44q−20 + 23q−21−22q−22−19q−23 + 11q−24 + 11q−25−4q−26−5q−27 + 3q−29−q−30
4	q40−2q39 + q38 + q37−3q36 + 5q35−7q34 + 5q33 + 6q32−15q31 + 9q30−18q29 + 27q28 + 31q27−49q26−13q25−59q24 + 87q23 + 126q22−73q21−86q20−213q19 + 140q18 + 337q17 + 7q16−163q15−517q14 + 89q13 + 591q12 + 229q11−139q10−875q9−102q8 + 759q7 + 506q6 + 4q5−1137q4−336q3 + 780q2 + 705q + 197−1231q−1−511q−2 + 680q−3 + 774q−4 + 373q−5−1163q−6−603q−7 + 492q−8 + 734q−9 + 523q−10−959q−11−626q−12 + 241q−13 + 596q−14 + 626q−15−637q−16−564q−17−32q−18 + 366q−19 + 632q−20−282q−21−396q−22−210q−23 + 107q−24 + 494q−25−25q−26−169q−27−221q−28−68q−29 + 275q−30 + 61q−31−8q−32−123q−33−101q−34 + 102q−35 + 39q−36 + 35q−37−37q−38−57q−39 + 25q−40 + 8q−41 + 20q−42−4q−43−18q−44 + 4q−45 + 5q−47−3q−49 + q−50
5	q60−2q59 + q58 + q57−3q56 + q55 + 5q54−5q53 + 4q51−10q50−2q49 + 17q48 + 2q47 + 5q46−3q45−39q44−31q43 + 30q42 + 67q41 + 72q40 + 6q39−146q38−184q37−38q36 + 191q35 + 356q34 + 211q33−251q32−596q31−454q30 + 155q29 + 860q28 + 926q27 + 16q26−1104q25−1429q24−460q23 + 1226q22 + 2093q21 + 1032q20−1216q19−2660q18−1780q17 + 982q16 + 3185q15 + 2576q14−607q13−3544q12−3325q11 + 110q10 + 3701q9 + 4010q8 + 425q7−3755q6−4481q5−933q4 + 3609q3 + 4851q2 + 1391q−3460−4996q−1−1752q−2 + 3163q−3 + 5081q−4 + 2059q−5−2903q−6−4986q−7−2302q−8 + 2518q−9 + 4858q−10 + 2522q−11−2125q−12−4596q−13−2701q−14 + 1618q−15 + 4241q−16 + 2861q−17−1051q−18−3791q−19−2940q−20 + 470q−21 + 3153q−22 + 2928q−23 + 182q−24−2507q−25−2764q−26−662q−27 + 1697q−28 + 2436q−29 + 1124q−30−1003q−31−1997q−32−1260q−33 + 304q−34 + 1440q−35 + 1314q−36 + 148q−37−909q−38−1096q−39−465q−40 + 425q−41 + 855q−42 + 538q−43−89q−44−531q−45−512q−46−112q−47 + 297q−48 + 378q−49 + 176q−50−101q−51−251q−52−177q−53 + 17q−54 + 141q−55 + 120q−56 + 28q−57−59q−58−84q−59−33q−60 + 29q−61 + 42q−62 + 18q−63−2q−64−18q−65−20q−66 + 4q−67 + 11q−68 + 3q−69−5q−72 + 3q−74−q−75