9 44

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 44"];

In[4]:=

PD[K]

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

Out[4]=

X1425 X5,10,6,11 X3948 X9,3,10,2 X14,8,15,7 X18,15,1,16 X16,11,17,12 X12,17,13,18 X6,14,7,13

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

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

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[22,21,2-]

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,−2,3})

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Further quantum knot invariants for 9_44.

A1 Invariants.

Weight	Invariant
1	−q11 + q9 + q5 + q−1−q−3 + q−5
2	q32−q30−2q28 + 2q26 + q24−2q22 + 2q18−q16−q14 + 2q12−q8 + q6 + q4−q2−1 + 3q−2−2q−6 + 2q−8 + q−10−q−12
3	−q63 + q61 + 2q59−3q55−3q53 + 3q51 + 4q49−4q45−3q43 + 3q41 + 5q39−q37−6q35−3q33 + 6q31 + 4q29−4q27−4q25 + 4q23 + 5q21−3q19−4q17 + 2q15 + 3q13−2q11−3q9 + 3q5 + 3q3−2q−4q−1 + 2q−3 + 7q−5 + 3q−7−7q−9−4q−11 + 5q−13 + 4q−15−2q−17−5q−19 + 3q−23 + q−25−q−29
4	q104−q102−2q100 + q96 + 5q94 + q92−3q90−5q88−6q86 + 5q84 + 7q82 + 5q80−10q76−7q74−q72 + 9q70 + 14q68 + q66−11q64−16q62−5q60 + 15q58 + 17q56 + 2q54−18q52−18q50 + 5q48 + 21q46 + 14q44−9q42−20q40−5q38 + 14q36 + 13q34−4q32−15q30−5q28 + 9q26 + 8q24−2q22−9q20−3q18 + 7q16 + 6q14 + 2q12−4q10−7q8−q6 + 6q4 + 13q2 + 3−15q−2−15q−4−q−6 + 22q−8 + 20q−10−9q−12−25q−14−15q−16 + 15q−18 + 26q−20 + 5q−22−14q−24−18q−26−2q−28 + 13q−30 + 9q−32−7q−36−5q−38 + q−40 + 2q−42 + 2q−44−q−48
5	−q155 + q153 + 2q151−q147−3q145−3q143−q141 + 5q139 + 7q137 + 4q135−q133−7q131−11q129−8q127 + 5q125 + 11q123 + 13q121 + 9q119−4q117−16q115−20q113−10q111 + 6q109 + 24q107 + 29q105 + 13q103−16q101−37q99−34q97−8q95 + 32q93 + 50q91 + 32q89−13q87−54q85−54q83−12q81 + 43q79 + 67q77 + 38q75−25q73−67q71−53q69 + 3q67 + 59q65 + 62q63 + 13q61−46q59−62q57−22q55 + 32q53 + 54q51 + 25q49−23q47−45q45−23q43 + 18q41 + 34q39 + 17q37−13q35−26q33−10q31 + 12q29 + 19q27 + 8q25−9q23−16q21−10q19 + 4q17 + 17q15 + 17q13 + 6q11−14q9−29q7−21q5 + 9q3 + 41q + 40q−1 + 5q−3−44q−5−63q−7−27q−9 + 42q−11 + 81q−13 + 49q−15−23q−17−82q−19−70q−21 + q−23 + 69q−25 + 78q−27 + 19q−29−44q−31−66q−33−36q−35 + 18q−37 + 47q−39 + 36q−41 + q−43−23q−45−26q−47−11q−49 + 6q−51 + 14q−53 + 9q−55−3q−59−4q−61−2q−63 + q−65 + q−67

A2 Invariants.

Weight	Invariant
1,0	−q16 + 2q8 + q6 + q4−1−q−4 + q−6 + q−8
1,1	q44−2q42 + 4q40−8q38 + 11q36−12q34 + 12q32−12q30 + 4q28 + 2q26−8q24 + 16q22−19q20 + 22q18−20q16 + 20q14−14q12 + 10q10−2q8−4q6 + 8q4−14q2 + 14−12q−2 + 11q−4−4q−6 + 4q−8 + 2q−10−q−12−2q−16 + q−20
2,0	q42−q38−q36 + q32−q30−q28 + 2q18 + 3q16 + q14 + q12−q8−2q6−q4−2q2 + 2q−2 + 3q−4 + 2q−6 + 2q−10−2q−14−q−16 + q−20

A3 Invariants.

Weight	Invariant
0,1,0	q34−q32−2q26 + q24−q22−q20 + q18 + q16 + 2q12 + 2q10 + q8 + q6 + q2−1−q−2 + q−4−2q−6 + 2q−10 + q−16
1,0,0	−q21−q17 + 2q11 + 2q9 + 2q7 + q5−q−2q−1−q−5 + q−7 + q−9 + q−11

A4 Invariants.

Weight	Invariant
0,1,0,0	q44 + q38−q36−3q34−2q32−q30−3q28−q26 + 3q24 + 5q22 + 2q20 + 3q18 + 4q16−q14−2q12−q8−q6 + 2q4 + 3q2 + 1 + q−4−3q−8−q−10 + q−12 + q−18 + q−20 + q−22
1,0,0,0	−q26−q22−q20 + 2q14 + 2q12 + 3q10 + 2q8 + q6−q2−2−2q−2−q−6 + q−8 + q−10 + q−12 + q−14

B2 Invariants.

Weight	Invariant
0,1	−q34 + q32−2q30 + 2q28−2q26 + q24−q22 + q20 + q18−q16 + 4q14−2q12 + 4q10−3q8 + 3q6−2q4 + q2−1−q−2 + q−4−2q−6 + 2q−8−2q−10 + 2q−12 + q−16
1,0	q56−q52−q50 + q48 + q46−2q44−2q42 + q40 + 2q38−2q34−q32 + 2q30 + q28−q24 + q22 + q20 + q18−q16 + 2q12 + q10−q8−q6 + q4 + 2q2−2q−2 + 2q−6−2q−10−q−12 + q−14 + 2q−16−q−20 + q−26

D4 Invariants.

Weight	Invariant
1,0,0,0	q46−q44 + q42−2q40 + 2q38−2q36−2q32−q28−q26 + q24−q22 + 3q20 + 5q16 + 5q12−q10 + 3q8−q6 + q4−2q2−1−q−2−2q−4 + q−6−2q−8 + q−10−q−12 + 3q−14 + q−18 + q−22

G2 Invariants.

Weight	Invariant
1,0	q80−q78 + 2q76−3q74 + q72−4q68 + 6q66−5q64 + 3q62−q60−4q58 + 5q56−4q54 + 2q50−4q48 + 4q46−4q42 + 7q40−5q38 + 3q36−2q32 + 6q30−4q28 + 6q26−2q24 + 2q22 + 4q20−4q18 + 4q16−2q14 + q12 + 3q10−4q8 + 2q6−4q2 + 5−6q−2 + 2q−6−6q−8 + 5q−10−3q−12 + q−14 + q−16−3q−18 + 2q−20 + q−24 + q−26 + q−32 + q−38

.

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 44"];

In[4]:=

Alexander[K][t]

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

Out[4]=

t2−4t + 7−4t−1 + t−2

In[5]:=

Conway[K][z]

Out[5]=

z4 + 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]=

{ 17, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

q2−2q + 3−3q−1 + 3q−2−2q−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 + 3z2a2 + 3a2−2z2−2 + a−2

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

a3z7 + az7 + 2a4z6 + 3a2z6 + z6 + a5z5−2a3z5−3az5−7a4z4−10a2z4−3z4−3a5z3−a3z3 + 4az3 + 2z3a−1 + 5a4z2 + 10a2z2 + z2a−2 + 6z2 + a5z + a3z−az−za−1−a4−3a2−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["9 44"];

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]=

{ t2−4t + 7−4t−1 + t−2, q2−2q + 3−3q−1 + 3q−2−2q−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]=

{}

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 + q3−5q2 + 4q + 4−9q−1 + 4q−2 + 6q−3−9q−4 + 2q−5 + 7q−6−7q−7−q−8 + 7q−9−4q−10−3q−11 + 5q−12−q−13−2q−14 + q−15
3	−q13 + q12 + q11 + 2q10−4q9−4q8 + 4q7 + 8q6−3q5−13q4 + q3 + 18q2 + q−18−5q−1 + 20q−2 + 6q−3−18q−4−8q−5 + 17q−6 + 7q−7−13q−8−9q−9 + 11q−10 + 8q−11−5q−12−10q−13 + 3q−14 + 8q−15 + 3q−16−8q−17−6q−18 + 5q−19 + 8q−20−2q−21−8q−22−q−23 + 7q−24 + 2q−25−4q−26−2q−27 + q−28 + 2q−29−q−30
4	−q22 + q21 + 2q20−q18−7q17−q16 + 9q15 + 9q14 + 3q13−22q12−17q11 + 13q10 + 28q9 + 24q8−33q7−47q6 + 3q5 + 44q4 + 53q3−31q2−70q−11 + 44q−1 + 71q−2−21q−3−77q−4−18q−5 + 38q−6 + 74q−7−15q−8−73q−9−17q−10 + 28q−11 + 68q−12−8q−13−63q−14−16q−15 + 14q−16 + 58q−17 + 3q−18−46q−19−15q−20−5q−21 + 43q−22 + 14q−23−23q−24−8q−25−21q−26 + 20q−27 + 14q−28−3q−29 + 7q−30−23q−31 + 3q−33 + 2q−34 + 19q−35−10q−36−5q−37−7q−38−4q−39 + 16q−40−5q−43−6q−44 + 5q−45 + q−46 + 2q−47−q−48−2q−49 + q−50
5	q31−3q29−2q28 + q27 + 3q26 + 10q25 + 5q24−11q23−19q22−14q21 + 6q20 + 34q19 + 40q18−48q16−68q15−24q14 + 56q13 + 103q12 + 59q11−57q10−136q9−95q8 + 44q7 + 162q6 + 131q5−25q4−175q3−164q2 + 8q + 181 + 180q−1 + 10q−2−174q−3−196q−4−22q−5 + 173q−6 + 195q−7 + 30q−8−163q−9−196q−10−35q−11 + 159q−12 + 189q−13 + 37q−14−146q−15−185q−16−42q−17 + 137q−18 + 173q−19 + 50q−20−116q−21−168q−22−58q−23 + 96q−24 + 151q−25 + 72q−26−68q−27−139q−28−80q−29 + 42q−30 + 111q−31 + 88q−32−9q−33−90q−34−83q−35−14q−36 + 55q−37 + 74q−38 + 33q−39−27q−40−54q−41−38q−42 + 32q−44 + 33q−45 + 14q−46−9q−47−20q−48−18q−49−8q−50 + 7q−51 + 11q−52 + 12q−53 + 9q−54−2q−55−13q−56−11q−57−5q−58 + 2q−59 + 13q−60 + 10q−61−7q−63−7q−64−4q−65 + 7q−67 + 4q−68−q−69−2q−70−q−71−2q−72 + q−73 + 2q−74−q−75
6	q47−q46−q45−q42−q41 + 8q40 + 3q39−2q37−8q36−17q35−16q34 + 17q33 + 26q32 + 31q31 + 25q30−6q29−65q28−88q27−25q26 + 31q25 + 100q24 + 134q23 + 82q22−83q21−213q20−173q19−66q18 + 128q17 + 297q16 + 285q15 + 9q14−288q13−365q12−266q11 + 47q10 + 399q9 + 506q8 + 182q7−259q6−478q5−452q4−93q3 + 397q2 + 625q + 328−179q−1−490q−2−539q−3−199q−4 + 349q−5 + 645q−6 + 390q−7−120q−8−460q−9−549q−10−240q−11 + 310q−12 + 624q−13 + 399q−14−96q−15−428q−16−530q−17−248q−18 + 279q−19 + 587q−20 + 395q−21−71q−22−384q−23−500q−24−261q−25 + 221q−26 + 524q−27 + 395q−28−12q−29−301q−30−454q−31−296q−32 + 115q−33 + 420q−34 + 391q−35 + 84q−36−168q−37−372q−38−330q−39−28q−40 + 264q−41 + 347q−42 + 175q−43−q−44−234q−45−311q−46−151q−47 + 75q−48 + 231q−49 + 193q−50 + 135q−51−59q−52−203q−53−182q−54−74q−55 + 73q−56 + 109q−57 + 163q−58 + 67q−59−51q−60−105q−61−105q−62−29q−63−11q−64 + 85q−65 + 76q−66 + 36q−67−7q−68−43q−69−22q−70−58q−71 + 2q−72 + 16q−73 + 25q−74 + 18q−75 + 8q−76 + 25q−77−28q−78−10q−79−16q−80−6q−81−7q−82 + 2q−83 + 33q−84 + 7q−86−5q−87−6q−88−15q−89−10q−90 + 12q−91 + q−92 + 8q−93 + 3q−94 + 3q−95−7q−96−6q−97 + 3q−98−2q−99 + 2q−100 + q−101 + 2q−102−q−103−2q−104 + q−105
7	q63−q62−q61−q60 + 2q58 + q57 + 2q56 + 5q55 + q54−5q53−11q52−14q51−3q50 + q49 + 14q48 + 34q47 + 36q46 + 16q45−23q44−66q43−74q42−62q41−14q40 + 83q39 + 152q38 + 166q37 + 84q36−76q35−217q34−294q33−243q32−15q31 + 257q30 + 461q29 + 466q28 + 182q27−225q26−602q25−725q24−446q23 + 93q22 + 683q21 + 1000q20 + 770q19 + 109q18−684q17−1216q16−1089q15−386q14 + 595q13 + 1362q12 + 1380q11 + 672q10−460q9−1421q8−1593q7−911q6 + 284q5 + 1403q4 + 1723q3 + 1113q2−125q−1358−1784q−1−1228q−2 + 3q−3 + 1281q−4 + 1788q−5 + 1300q−6 + 89q−7−1221q−8−1778q−9−1318q−10−135q−11 + 1171q−12 + 1745q−13 + 1319q−14 + 163q−15−1133q−16−1720q−17−1309q−18−173q−19 + 1103q−20 + 1689q−21 + 1291q−22 + 187q−23−1064q−24−1655q−25−1278q−26−208q−27 + 1010q−28 + 1608q−29 + 1266q−30 + 250q−31−934q−32−1544q−33−1252q−34−313q−35 + 818q−36 + 1456q−37 + 1250q−38 + 398q−39−679q−40−1339q−41−1222q−42−503q−43 + 484q−44 + 1189q−45 + 1197q−46 + 617q−47−285q−48−1000q−49−1117q−50−717q−51 + 45q−52 + 762q−53 + 1020q−54 + 791q−55 + 171q−56−512q−57−842q−58−798q−59−377q−60 + 236q−61 + 635q−62 + 751q−63 + 500q−64 + 10q−65−380q−66−617q−67−557q−68−209q−69 + 139q−70 + 428q−71 + 514q−72 + 321q−73 + 77q−74−220q−75−402q−76−337q−77−211q−78 + 28q−79 + 238q−80 + 277q−81 + 260q−82 + 108q−83−88q−84−164q−85−227q−86−163q−87−23q−88 + 42q−89 + 146q−90 + 154q−91 + 79q−92 + 34q−93−63q−94−94q−95−63q−96−76q−97−12q−98 + 41q−99 + 37q−100 + 64q−101 + 22q−102 + 4q−103 + 15q−104−33q−105−32q−106−18q−107−23q−108 + 8q−109 + 2q−110 + 4q−111 + 37q−112 + 12q−113 + 6q−114−20q−116−7q−117−13q−118−15q−119 + 7q−120 + 9q−121 + 12q−122 + 12q−123−5q−124 + q−125−3q−126−10q−127−3q−128−2q−129 + 4q−130 + 6q−131−q−132 + 2q−134−2q−135−q−136−2q−137 + q−138 + 2q−139−q−140