10 75
From Knot Atlas
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 (KnotPlot image) |
See the full Rolfsen Knot Table. Visit 10 75's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 10_75's page at Knotilus! Visit 10 75's page at the original Knot Atlas! |
[edit] Knot presentations
| Planar diagram presentation | X1425 X5,12,6,13 X3,11,4,10 X11,3,12,2 X13,16,14,17 X7,15,8,14 X15,7,16,6 X17,20,18,1 X9,19,10,18 X19,9,20,8 |
| Gauss code | -1, 4, -3, 1, -2, 7, -6, 10, -9, 3, -4, 2, -5, 6, -7, 5, -8, 9, -10, 8 |
| Dowker-Thistlethwaite code | 4 10 12 14 18 2 16 6 20 8 |
| Conway Notation | [21,21,21+] |
| Minimum Braid Representative | A Morse Link Presentation | An Arc Presentation | |||||
Length is 12, width is 5, Braid index is 5 | 
|  [{12, 3}, {2, 10}, {11, 4}, {3, 6}, {10, 12}, {7, 5}, {6, 8}, {4, 7}, {5, 1}, {9, 2}, {8, 11}, {1, 9}] |
[edit Notes on presentations of 10 75]
Computer Talk
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[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
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In[3]:=
| K = Knot["10 75"];
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In[4]:=
| PD[K]
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
| X1425 X5,12,6,13 X3,11,4,10 X11,3,12,2 X13,16,14,17 X7,15,8,14 X15,7,16,6 X17,20,18,1 X9,19,10,18 X19,9,20,8 |
In[5]:=
| GaussCode[K]
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Out[5]=
| -1, 4, -3, 1, -2, 7, -6, 10, -9, 3, -4, 2, -5, 6, -7, 5, -8, 9, -10, 8 |
In[6]:=
| DTCode[K]
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Out[6]=
| 4 10 12 14 18 2 16 6 20 8 |
(The path below may be different on your system)
In[7]:=
| AppendTo[$Path, "C:/bin/LinKnot/"];
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In[8]:=
| ConwayNotation[K]
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Out[8]=
| [21,21,21+] |
In[9]:=
| br = BR[K]
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KnotTheory::credits: The minimum braids representing the knots with up to 10 crossings were provided by Thomas Gittings. See arXiv:math.GT/0401051.
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Out[9]=
| BR(5,{1,−2,1,−2,3,−2,−2,4,−3,2,4,3}) |
In[10]:=
| {First[br], Crossings[br], BraidIndex[K]}
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KnotTheory::credits: The braid index data known to KnotTheory` is taken from Charles Livingston's http://www.indiana.edu/~knotinfo/.
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KnotTheory::loading: Loading precomputed data in IndianaData`.
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Out[10]=
| { 5, 12, 5 } |
In[11]:=
| Show[BraidPlot[br]]
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Out[11]=
| -Graphics- |
In[12]:=
| Show[DrawMorseLink[K]]
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KnotTheory::credits: "MorseLink was added to KnotTheory` by Siddarth Sankaran at the University of Toronto in the summer of 2005."
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KnotTheory::credits: "DrawMorseLink was written by Siddarth Sankaran at the University of Toronto in the summer of 2005."
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Out[12]=
| -Graphics- |
In[13]:=
| ap = ArcPresentation[K]
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Out[13]=
| ArcPresentation[{12, 3}, {2, 10}, {11, 4}, {3, 6}, {10, 12}, {7, 5}, {6, 8}, {4, 7}, {5, 1}, {9, 2}, {8, 11}, {1, 9}] |
In[14]:=
| Draw[ap]
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Out[14]=
| -Graphics- |
[edit] Three dimensional invariants
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[edit] Four dimensional invariants
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[edit] Polynomial invariants
| Alexander polynomial | −t3 + 7t2−19t + 27−19t−1 + 7t−2−t−3 |
| Conway polynomial | −z6 + z4 + 1 |
| 2nd Alexander ideal (db, data sources) | {3,t + 1} |
| Determinant and Signature | { 81, 0 } |
| Jones polynomial | q6−3q5 + 6q4−10q3 + 12q2−13q + 14−10q−1 + 7q−2−4q−3 + q−4 |
| HOMFLY-PT polynomial (db, data sources) | −z6 + a2z4 + 3z4a−2−3z4 + a2z2 + 6z2a−2−3z2a−4−4z2 + 3a−2−3a−4 + a−6 |
| Kauffman polynomial (db, data sources) | z9a−1 + z9a−3 + 7z8a−2 + 3z8a−4 + 4z8 + 7az7 + 13z7a−1 + 9z7a−3 + 3z7a−5 + 7a2z6−4z6a−2−3z6a−4 + z6a−6 + 7z6 + 4a3z5−5az5−29z5a−1−29z5a−3−9z5a−5 + a4z4−8a2z4−21z4a−2−9z4a−4−3z4a−6−24z4−3a3z3−az3 + 17z3a−1 + 24z3a−3 + 9z3a−5 + 4a2z2 + 20z2a−2 + 12z2a−4 + 3z2a−6 + 15z2−az−5za−1−7za−3−3za−5−3a−2−3a−4−a−6 |
| The A2 invariant | q12−2q10 + q8−3q4 + 4q2 + 3q−2 + q−4−q−6 + 2q−8−3q−10−2q−16 + q−18 + q−20 |
| The G2 invariant | q66−3q64 + 6q62−10q60 + 9q58−6q56−2q54 + 19q52−32q50 + 50q48−55q46 + 39q44−9q42−40q40 + 87q38−126q36 + 134q34−107q32 + 37q30 + 62q28−144q26 + 193q24−183q22 + 114q20−13q18−96q16 + 154q14−150q12 + 86q10 + 31q8−114q6 + 133q4−79q2−29 + 154q−2−237q−4 + 223q−6−125q−8−26q−10 + 202q−12−306q−14 + 309q−16−212q−18 + 55q−20 + 106q−22−224q−24 + 248q−26−183q−28 + 72q−30 + 62q−32−144q−34 + 145q−36−70q−38−41q−40 + 132q−42−177q−44 + 133q−46−32q−48−92q−50 + 197q−52−228q−54 + 181q−56−76q−58−50q−60 + 136q−62−175q−64 + 153q−66−90q−68 + 16q−70 + 43q−72−71q−74 + 70q−76−46q−78 + 22q−80−11q−84 + 12q−86−10q−88 + 6q−90−2q−92 + q−94 |
Further Quantum Invariants
Further quantum knot invariants for 10_75.
A1 Invariants.
| Weight | Invariant |
|---|---|
| 1 | q9−3q7 + 3q5−3q3 + 4q + q−1−q−3 + 2q−5−4q−7 + 3q−9−2q−11 + q−13 |
| 2 | q26−3q24 + 9q20−11q18−2q16 + 22q14−22q12−10q10 + 33q8−18q6−18q4 + 29q2 + 2−16q−2 + 4q−4 + 16q−6−8q−8−21q−10 + 22q−12 + 7q−14−33q−16 + 17q−18 + 20q−20−28q−22 + 4q−24 + 19q−26−13q−28−4q−30 + 8q−32−2q−34−2q−36 + q−38 |
| 3 | q51−3q49 + 6q45 + q43−11q41−5q39 + 26q37 + 2q35−43q33−9q31 + 72q29 + 23q27−105q25−42q23 + 133q21 + 78q19−156q17−116q15 + 147q13 + 156q11−126q9−173q7 + 70q5 + 184q3−10q−157q−1−44q−3 + 120q−5 + 102q−7−79q−9−137q−11 + 22q−13 + 166q−15 + 18q−17−174q−19−74q−21 + 174q−23 + 117q−25−154q−27−154q−29 + 122q−31 + 179q−33−72q−35−183q−37 + 20q−39 + 170q−41 + 20q−43−131q−45−53q−47 + 88q−49 + 62q−51−48q−53−54q−55 + 16q−57 + 37q−59 + q−61−20q−63−4q−65 + 8q−67 + 3q−69−2q−71−2q−73 + q−75 |
| 4 | q84−3q82 + 6q78−2q76 + q74−14q72 + 5q70 + 26q68−16q66−9q64−42q62 + 37q60 + 102q58−42q56−89q54−143q52 + 115q50 + 316q48−16q46−283q44−423q42 + 169q40 + 724q38 + 224q36−493q34−943q32−14q30 + 1135q28 + 760q26−408q24−1459q22−553q20 + 1126q18 + 1297q16 + 139q14−1448q12−1119q10 + 498q8 + 1325q6 + 808q4−776q2−1241−341q−2 + 787q−4 + 1111q−6 + 110q−8−892q−10−941q−12 + 83q−14 + 1040q−16 + 800q−18−407q−20−1235q−22−485q−24 + 811q−26 + 1258q−28 + 74q−30−1310q−32−950q−34 + 436q−36 + 1509q−38 + 612q−40−1086q−42−1303q−44−172q−46 + 1402q−48 + 1119q−50−483q−52−1295q−54−819q−56 + 816q−58 + 1244q−60 + 256q−62−785q−64−1077q−66 + 56q−68 + 830q−70 + 627q−72−99q−74−759q−76−350q−78 + 221q−80 + 464q−82 + 254q−84−257q−86−278q−88−93q−90 + 137q−92 + 197q−94−2q−96−75q−98−84q−100−7q−102 + 58q−104 + 19q−106 + q−108−20q−110−11q−112 + 8q−114 + 3q−116 + 3q−118−2q−120−2q−122 + q−124 |
| 5 | q125−3q123 + 6q119−2q117−2q115−2q113−4q111 + 5q109 + 14q107−6q105−27q103−11q101 + 30q99 + 60q97 + 21q95−65q93−149q91−80q89 + 171q87 + 321q85 + 147q83−276q81−612q79−391q77 + 452q75 + 1136q73 + 772q71−598q69−1843q67−1513q65 + 611q63 + 2804q61 + 2676q59−373q57−3862q55−4272q53−408q51 + 4782q49 + 6318q47 + 1824q45−5292q43−8403q41−3932q39 + 4942q37 + 10239q35 + 6485q33−3625q31−11185q29−9089q27 + 1333q25 + 10980q23 + 11083q21 + 1516q19−9338q17−12117q15−4443q13 + 6725q11 + 11765q9 + 6815q7−3402q5−10244q3−8354q + 140q−1 + 7907q−3 + 8861q−5 + 2721q−7−5196q−9−8601q−11−4950q−13 + 2662q−15 + 7894q−17 + 6446q−19−426q−21−7047q−23−7609q−25−1330q−27 + 6335q−29 + 8415q−31 + 2851q−33−5657q−35−9279q−37−4256q−39 + 5029q−41 + 9984q−43 + 5791q−45−4091q−47−10585q−49−7436q−51 + 2722q−53 + 10720q−55 + 9145q−57−773q−59−10198q−61−10594q−63−1627q−65 + 8759q−67 + 11476q−69 + 4246q−71−6478q−73−11431q−75−6592q−77 + 3532q−79 + 10242q−81 + 8235q−83−365q−85−8093q−87−8765q−89−2409q−91 + 5236q−93 + 8065q−95 + 4393q−97−2297q−99−6419q−101−5148q−103−187q−105 + 4199q−107 + 4831q−109 + 1816q−111−2045q−113−3733q−115−2428q−117 + 365q−119 + 2334q−121 + 2244q−123 + 615q−125−1094q−127−1633q−129−913q−131 + 249q−133 + 926q−135 + 794q−137 + 169q−139−397q−141−517q−143−250q−145 + 96q−147 + 252q−149 + 192q−151 + 29q−153−100q−155−108q−157−37q−159 + 26q−161 + 40q−163 + 28q−165 + q−167−20q−169−11q−171 + q−173 + 3q−175 + 3q−177 + 3q−179−2q−181−2q−183 + q−185 |
A2 Invariants.
| Weight | Invariant |
|---|---|
| 1,0 | q12−2q10 + q8−3q4 + 4q2 + 3q−2 + q−4−q−6 + 2q−8−3q−10−2q−16 + q−18 + q−20 |
| 2,0 | q32−2q30−q28 + 4q26−q24−5q22 + 3q20 + 11q18−6q16−15q14 + 8q12 + 16q10−15q8−17q6 + 15q4 + 11q2−6−3q−2 + 14q−4 + q−6−3q−8 + 8q−10−q−12−12q−14 + 3q−16 + 8q−18−16q−20−11q−22 + 11q−24 + 11q−26−11q−28−7q−30 + 14q−32 + 9q−34−8q−36−7q−38 + 5q−40 + 6q−42−2q−44−5q−46−q−48 + q−50 + q−52 |
A3 Invariants.
| Weight | Invariant |
|---|---|
| 0,1,0 | q28−3q26 + 8q22−9q20−2q18 + 19q16−15q14−12q12 + 24q10−16q8−16q6 + 27q4−2q2−6 + 14q−2 + 8q−4−6q−6−13q−8 + 9q−10 + 4q−12−24q−14 + 12q−16 + 17q−18−24q−20 + 9q−22 + 14q−24−19q−26 + 5q−28 + 7q−30−9q−32 + 3q−34 + 2q−36−2q−38 + q−40 |
| 1,0,0 | q15−2q13 + 2q11−2q9 + q7−3q5 + 3q3 + 2q−1 + 2q−3 + q−5 + 2q−7−q−9 + 3q−11−3q−13−3q−17−2q−21 + q−23 + q−25 + q−27 |
B2 Invariants.
| Weight | Invariant |
|---|---|
| 0,1 | q28−3q26 + 6q24−10q22 + 15q20−20q18 + 25q16−29q14 + 28q12−26q10 + 18q8−8q6−5q4 + 22q2−34 + 48q−2−52q−4 + 58q−6−53q−8 + 47q−10−34q−12 + 20q−14−6q−16−9q−18 + 18q−20−27q−22 + 28q−24−29q−26 + 25q−28−21q−30 + 15q−32−9q−34 + 6q−36−2q−38 + q−40 |
| 1,0 | q46−3q42−3q40 + 3q38 + 9q36 + 2q34−12q32−11q30 + 9q28 + 22q26 + 4q24−25q22−20q20 + 13q18 + 28q16−2q14−31q12−13q10 + 23q8 + 23q6−12q4−19q2 + 9 + 26q−2 + 2q−4−18q−6−5q−8 + 17q−10 + 6q−12−18q−14−14q−16 + 14q−18 + 16q−20−14q−22−26q−24 + 4q−26 + 32q−28 + 11q−30−26q−32−24q−34 + 16q−36 + 30q−38 + q−40−25q−42−13q−44 + 14q−46 + 16q−48−4q−50−12q−52−3q−54 + 6q−56 + 4q−58−2q−60−2q−62 + q−66 |
G2 Invariants.
| Weight | Invariant |
|---|---|
| 1,0 | q66−3q64 + 6q62−10q60 + 9q58−6q56−2q54 + 19q52−32q50 + 50q48−55q46 + 39q44−9q42−40q40 + 87q38−126q36 + 134q34−107q32 + 37q30 + 62q28−144q26 + 193q24−183q22 + 114q20−13q18−96q16 + 154q14−150q12 + 86q10 + 31q8−114q6 + 133q4−79q2−29 + 154q−2−237q−4 + 223q−6−125q−8−26q−10 + 202q−12−306q−14 + 309q−16−212q−18 + 55q−20 + 106q−22−224q−24 + 248q−26−183q−28 + 72q−30 + 62q−32−144q−34 + 145q−36−70q−38−41q−40 + 132q−42−177q−44 + 133q−46−32q−48−92q−50 + 197q−52−228q−54 + 181q−56−76q−58−50q−60 + 136q−62−175q−64 + 153q−66−90q−68 + 16q−70 + 43q−72−71q−74 + 70q−76−46q−78 + 22q−80−11q−84 + 12q−86−10q−88 + 6q−90−2q−92 + q−94 |
.
Computer Talk
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.
|
In[3]:=
| K = Knot["10 75"];
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In[4]:=
| Alexander[K][t]
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
| −t3 + 7t2−19t + 27−19t−1 + 7t−2−t−3 |
In[5]:=
| Conway[K][z]
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Out[5]=
| −z6 + 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.
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Out[6]=
| {3,t + 1} |
In[7]:=
| {KnotDet[K], KnotSignature[K]}
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Out[7]=
| { 81, 0 } |
In[8]:=
| Jones[K][q]
|
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
|
Out[8]=
| q6−3q5 + 6q4−10q3 + 12q2−13q + 14−10q−1 + 7q−2−4q−3 + q−4 |
In[9]:=
| HOMFLYPT[K][a, z]
|
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
|
Out[9]=
| −z6 + a2z4 + 3z4a−2−3z4 + a2z2 + 6z2a−2−3z2a−4−4z2 + 3a−2−3a−4 + a−6 |
In[10]:=
| Kauffman[K][a, z]
|
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
|
Out[10]=
| z9a−1 + z9a−3 + 7z8a−2 + 3z8a−4 + 4z8 + 7az7 + 13z7a−1 + 9z7a−3 + 3z7a−5 + 7a2z6−4z6a−2−3z6a−4 + z6a−6 + 7z6 + 4a3z5−5az5−29z5a−1−29z5a−3−9z5a−5 + a4z4−8a2z4−21z4a−2−9z4a−4−3z4a−6−24z4−3a3z3−az3 + 17z3a−1 + 24z3a−3 + 9z3a−5 + 4a2z2 + 20z2a−2 + 12z2a−4 + 3z2a−6 + 15z2−az−5za−1−7za−3−3za−5−3a−2−3a−4−a−6 |
[edit] "Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {10_42,}
Same Jones Polynomial (up to mirroring, 
):
{}
Computer Talk
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.
|
In[3]:=
| K = Knot["10 75"];
|
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 + 7t2−19t + 27−19t−1 + 7t−2−t−3, q6−3q5 + 6q4−10q3 + 12q2−13q + 14−10q−1 + 7q−2−4q−3 + q−4 } |
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_42,} |
In[6]:=
| DeleteCases[
Select[
AllKnots[],
(J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) &
],
K
]
|
KnotTheory::loading: Loading precomputed data in Jones4Knots11`.
|
Out[6]=
| {} |
[edit] Khovanov Homology
| The coefficients of the monomials trqj are shown, along with their alternating sums χ (fixed j, alternation over r). The squares with yellow highlighting are those on the "critical diagonals", where j−2r = s + 1 or j−2r = s−1, where s = 0 is the signature of 10 75. Nonzero entries off the critical diagonals (if any exist) are highlighted in red. |
|
| Integral Khovanov Homology
(db, data source) |
|
[edit] The Coloured Jones Polynomials
| n | Jn |
| 2 | q18−3q17 + 11q15−15q14−9q13 + 43q12−30q11−41q10 + 91q9−33q8−91q7 + 131q6−18q5−134q4 + 144q3 + 6q2−146q + 124 + 24q−1−119q−2 + 77q−3 + 24q−4−68q−5 + 34q−6 + 12q−7−24q−8 + 10q−9 + 3q−10−4q−11 + q−12 |
| 3 | q36−3q35 + 5q33 + 6q32−15q31−16q30 + 26q29 + 42q28−36q27−86q26 + 32q25 + 152q24−10q23−227q22−46q21 + 303q20 + 140q19−377q18−249q17 + 414q16 + 391q15−434q14−525q13 + 414q12 + 662q11−377q10−773q9 + 314q8 + 854q7−229q6−917q5 + 155q4 + 912q3−48q2−899q−9 + 799q−1 + 99q−2−705q−3−123q−4 + 556q−5 + 146q−6−423q−7−132q−8 + 293q−9 + 106q−10−189q−11−77q−12 + 118q−13 + 43q−14−61q−15−28q−16 + 37q−17 + 9q−18−16q−19−4q−20 + 6q−21 + 3q−22−4q−23 + q−24 |
| 4 | q60−3q59 + 5q57 + 6q55−22q54−9q53 + 26q52 + 18q51 + 45q50−87q49−86q48 + 35q47 + 91q46 + 244q45−147q44−316q43−150q42 + 112q41 + 755q40 + 63q39−559q38−721q37−297q36 + 1415q35 + 789q34−356q33−1495q32−1430q31 + 1707q30 + 1830q29 + 632q28−1923q27−3065q26 + 1231q25 + 2642q24 + 2234q23−1640q22−4639q21 + 100q20 + 2859q19 + 3932q18−743q17−5712q16−1286q15 + 2499q14 + 5316q13 + 441q12−6159q11−2582q10 + 1749q9 + 6144q8 + 1648q7−5919q6−3539q5 + 725q4 + 6193q3 + 2650q2−4918q−3863−403q−1 + 5293q−2 + 3115q−3−3334q−4−3346q−5−1230q−6 + 3676q−7 + 2786q−8−1747q−9−2188q−10−1401q−11 + 1997q−12 + 1880q−13−696q−14−1020q−15−1026q−16 + 848q−17 + 951q−18−246q−19−303q−20−526q−21 + 293q−22 + 359q−23−106q−24−36q−25−194q−26 + 92q−27 + 101q−28−52q−29 + 11q−30−50q−31 + 27q−32 + 22q−33−19q−34 + 4q−35−8q−36 + 6q−37 + 3q−38−4q−39 + q−40 |
[edit] Computer Talk
Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session, or any of the Computer Talk sections above.
[edit] Modifying This Page
| Read me first: Modifying Knot Pages
See/edit the Rolfsen Knot Page master template (intermediate). See/edit the Rolfsen_Splice_Base (expert). Back to the top. |
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