10 31
From Knot Atlas
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 (KnotPlot image) |
See the full Rolfsen Knot Table. Visit 10 31's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 10_31's page at Knotilus! Visit 10 31's page at the original Knot Atlas! |
[edit] Knot presentations
| Planar diagram presentation | X1425 X3,12,4,13 X9,14,10,15 X13,10,14,11 X15,1,16,20 X5,17,6,16 X19,7,20,6 X7,19,8,18 X17,9,18,8 X11,2,12,3 |
| Gauss code | -1, 10, -2, 1, -6, 7, -8, 9, -3, 4, -10, 2, -4, 3, -5, 6, -9, 8, -7, 5 |
| Dowker-Thistlethwaite code | 4 12 16 18 14 2 10 20 8 6 |
| Conway Notation | [31132] |
| Minimum Braid Representative | A Morse Link Presentation | An Arc Presentation | |||||
Length is 12, width is 5, Braid index is 5 | 
|  [{12, 5}, {1, 10}, {9, 11}, {10, 12}, {11, 8}, {6, 9}, {8, 4}, {5, 2}, {3, 1}, {4, 7}, {2, 6}, {7, 3}] |
[edit Notes on presentations of 10 31]
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[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
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In[3]:=
| K = Knot["10 31"];
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In[4]:=
| PD[K]
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
| X1425 X3,12,4,13 X9,14,10,15 X13,10,14,11 X15,1,16,20 X5,17,6,16 X19,7,20,6 X7,19,8,18 X17,9,18,8 X11,2,12,3 |
In[5]:=
| GaussCode[K]
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Out[5]=
| -1, 10, -2, 1, -6, 7, -8, 9, -3, 4, -10, 2, -4, 3, -5, 6, -9, 8, -7, 5 |
In[6]:=
| DTCode[K]
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Out[6]=
| 4 12 16 18 14 2 10 20 8 6 |
(The path below may be different on your system)
In[7]:=
| AppendTo[$Path, "C:/bin/LinKnot/"];
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In[8]:=
| ConwayNotation[K]
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Out[8]=
| [31132] |
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,−1,−1,−2,1,3,−2,3,3,4,−3,4}) |
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, 5}, {1, 10}, {9, 11}, {10, 12}, {11, 8}, {6, 9}, {8, 4}, {5, 2}, {3, 1}, {4, 7}, {2, 6}, {7, 3}] |
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 | 4t2−14t + 21−14t−1 + 4t−2 |
| Conway polynomial | 4z4 + 2z2 + 1 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 57, 0 } |
| Jones polynomial | −q5 + 2q4−4q3 + 7q2−8q + 10−9q−1 + 7q−2−5q−3 + 3q−4−q−5 |
| HOMFLY-PT polynomial (db, data sources) | −z2a4 + z4a2−a2 + 2z4 + 3z2 + 2 + z4a−2 + z2a−2 + a−2−z2a−4−a−4 |
| Kauffman polynomial (db, data sources) | az9 + z9a−1 + 3a2z8 + 2z8a−2 + 5z8 + 4a3z7 + 3az7 + z7a−1 + 2z7a−3 + 3a4z6−6a2z6−3z6a−2 + 2z6a−4−14z6 + a5z5−10a3z5−12az5−4z5a−1−2z5a−3 + z5a−5−7a4z4 + 5a2z4 + 3z4a−2−5z4a−4 + 20z4−2a5z3 + 7a3z3 + 15az3 + 6z3a−1−3z3a−3−3z3a−5 + 2a4z2−3a2z2−2z2a−2 + 3z2a−4−10z2−2a3z−4az−2za−1 + 2za−3 + 2za−5 + a2−a−2−a−4 + 2 |
| The A2 invariant | −q16 + q14 + q12−2q10 + q8−q6−q4 + 2q2 + 3q−2 + q−6 + 2q−8−2q−10−q−16 |
| The G2 invariant | q80−2q78 + 4q76−7q74 + 6q72−4q70−3q68 + 14q66−21q64 + 28q62−28q60 + 14q58 + 6q56−32q54 + 53q52−58q50 + 48q48−19q46−17q44 + 51q42−68q40 + 61q38−34q36−8q34 + 37q32−49q30 + 35q28−4q26−27q24 + 49q22−47q20 + 15q18 + 25q16−67q14 + 87q12−75q10 + 39q8 + 14q6−60q4 + 93q2−92 + 66q−2−20q−4−28q−6 + 61q−8−62q−10 + 44q−12−6q−14−22q−16 + 39q−18−32q−20 + 6q−22 + 27q−24−50q−26 + 56q−28−36q−30 + q−32 + 34q−34−54q−36 + 61q−38−47q−40 + 22q−42 + 3q−44−27q−46 + 36q−48−37q−50 + 28q−52−15q−54 + 2q−56 + 7q−58−15q−60 + 15q−62−13q−64 + 8q−66−3q−68−2q−70 + 3q−72−4q−74 + 3q−76−q−78 + q−80 |
Further Quantum Invariants
Further quantum knot invariants for 10_31.
A1 Invariants.
| Weight | Invariant |
|---|---|
| 1 | −q11 + 2q9−2q7 + 2q5−2q3 + q + 2q−1−q−3 + 3q−5−2q−7 + q−9−q−11 |
| 2 | q32−2q30−q28 + 6q26−4q24−6q22 + 11q20−2q18−13q16 + 12q14 + 4q12−14q10 + 8q8 + 7q6−8q4−2q2 + 6 + 4q−2−11q−4 + 3q−6 + 13q−8−12q−10−2q−12 + 14q−14−8q−16−5q−18 + 8q−20−3q−22−3q−24 + 3q−26−q−28−q−30 + q−32 |
| 3 | −q63 + 2q61 + q59−3q57−3q55 + 4q53 + 8q51−7q49−13q47 + 7q45 + 21q43−2q41−31q39−8q37 + 39q35 + 22q33−40q31−40q29 + 34q27 + 54q25−25q23−62q21 + 12q19 + 63q17 + 2q15−56q13−12q11 + 43q9 + 25q7−30q5−28q3 + 10q + 39q−1 + 7q−3−43q−5−24q−7 + 43q−9 + 40q−11−40q−13−51q−15 + 30q−17 + 59q−19−17q−21−54q−23 + q−25 + 49q−27 + 7q−29−33q−31−15q−33 + 21q−35 + 12q−37−11q−39−9q−41 + 5q−43 + 5q−45−3q−47−2q−49 + 2q−51 + q−53−2q−55 + q−59 + q−61−q−63 |
| 4 | q104−2q102−q100 + 3q98 + 3q94−7q92−3q90 + 9q88 + 9q84−20q82−15q80 + 21q78 + 15q76 + 30q74−39q72−54q70 + 4q68 + 34q66 + 102q64−7q62−104q60−91q58−16q56 + 184q54 + 121q52−58q50−199q48−175q46 + 161q44 + 257q42 + 100q40−198q38−326q36 + 22q34 + 272q32 + 244q30−85q28−343q26−108q24 + 172q22 + 267q20 + 37q18−236q16−163q14 + 46q12 + 202q10 + 108q8−95q6−170q4−62q2 + 117 + 163q−2 + 47q−4−179q−6−175q−8 + 31q−10 + 219q−12 + 194q−14−153q−16−275q−18−96q−20 + 212q−22 + 323q−24−43q−26−279q−28−225q−30 + 89q−32 + 334q−34 + 99q−36−144q−38−246q−40−68q−42 + 201q−44 + 141q−46 + 14q−48−141q−50−122q−52 + 50q−54 + 77q−56 + 71q−58−31q−60−74q−62−9q−64 + 8q−66 + 44q−68 + 9q−70−22q−72−6q−74−12q−76 + 13q−78 + 6q−80−3q−82 + 4q−84−7q−86 + q−88−q−92 + 4q−94−q−96−q−100−q−102 + q−104 |
| 5 | −q155 + 2q153 + q151−3q149 + 2q141 + 2q139−5q137−4q135 + 7q133 + 8q131 + 3q129−7q127−18q125−22q123 + 7q121 + 45q119 + 43q117 + 7q115−53q113−90q111−59q109 + 46q107 + 145q105 + 136q103 + 20q101−151q99−254q97−168q95 + 91q93 + 343q91 + 371q89 + 100q87−338q85−597q83−409q81 + 186q79 + 757q77 + 783q75 + 141q73−754q71−1150q69−598q67 + 565q65 + 1389q63 + 1088q61−199q59−1434q57−1513q55−266q53 + 1281q51 + 1770q49 + 718q47−970q45−1814q43−1072q41 + 581q39 + 1676q37 + 1267q35−211q33−1399q31−1290q29−101q27 + 1059q25 + 1210q23 + 310q21−744q19−1033q17−441q15 + 436q13 + 885q11 + 544q9−225q7−733q5−635q3−4q + 659q−1 + 780q−3 + 216q−5−604q−7−961q−9−477q−11 + 527q−13 + 1177q−15 + 799q−17−390q−19−1363q−21−1155q−23 + 145q−25 + 1439q−27 + 1513q−29 + 204q−31−1359q−33−1769q−35−621q−37 + 1090q−39 + 1859q−41 + 1011q−43−664q−45−1720q−47−1306q−49 + 178q−51 + 1403q−53 + 1375q−55 + 277q−57−927q−59−1281q−61−585q−63 + 470q−65 + 994q−67 + 703q−69−59q−71−662q−73−670q−75−179q−77 + 339q−79 + 514q−81 + 287q−83−104q−85−335q−87−278q−89−33q−91 + 181q−93 + 213q−95 + 85q−97−71q−99−138q−101−89q−103 + 13q−105 + 78q−107 + 68q−109 + 11q−111−37q−113−41q−115−20q−117 + 12q−119 + 27q−121 + 14q−123−3q−125−8q−127−10q−129−6q−131 + 5q−133 + 6q−135 + q−137 + 2q−139−q−141−4q−143−q−145 + q−147 + q−151 + q−153−q−155 |
A2 Invariants.
| Weight | Invariant |
|---|---|
| 1,0 | −q16 + q14 + q12−2q10 + q8−q6−q4 + 2q2 + 3q−2 + q−6 + 2q−8−2q−10−q−16 |
| 1,1 | q44−4q42 + 10q40−22q38 + 42q36−66q34 + 98q32−142q30 + 182q28−214q26 + 240q24−248q22 + 228q20−176q18 + 94q16 + 2q14−118q12 + 238q10−342q8 + 424q6−469q4 + 480q2−448 + 388q−2−294q−4 + 186q−6−70q−8−26q−10 + 111q−12−174q−14 + 210q−16−214q−18 + 203q−20−184q−22 + 154q−24−124q−26 + 94q−28−72q−30 + 50q−32−34q−34 + 21q−36−12q−38 + 6q−40−2q−42 + q−44 |
| 2,0 | q42−q40−2q38 + q36 + 4q34 + q32−7q30−2q28 + 7q26 + 2q24−8q22−3q20 + 9q18 + 3q16−9q14−q12 + 8q10−2q8−4q6 + 2q4 + q2−1 + 3q−2 + 4q−4−5q−6 + 11q−10 + 3q−12−10q−14 + 2q−16 + 9q−18−q−20−9q−22−2q−24 + 5q−26−q−28−4q−30 + 2q−34−q−38 + q−42 |
A3 Invariants.
| Weight | Invariant |
|---|---|
| 0,1,0 | q34−2q32 + 3q28−5q26 + 3q24 + 6q22−10q20 + 3q18 + 8q16−13q14 + 9q10−8q8−2q6 + 6q4 + q2−2 + 10q−4−q−6−5q−8 + 14q−10 + q−12−11q−14 + 9q−16−q−18−10q−20 + 3q−22−4q−26 + 2q−28 + q−30−q−32 + q−34 |
| 1,0,0 | −q21 + q19 + q15−2q13 + q11−2q9−q5 + 2q3 + q + q−1 + 3q−3 + 2q−7 + 2q−11−2q−13−q−17−q−21 |
B2 Invariants.
| Weight | Invariant |
|---|---|
| 0,1 | −q34 + 2q32−4q30 + 7q28−9q26 + 11q24−14q22 + 14q20−13q18 + 10q16−5q14 + 7q10−14q8 + 20q6−24q4 + 27q2−26 + 24q−2−18q−4 + 13q−6−5q−8 + 7q−12−9q−14 + 13q−16−13q−18 + 12q−20−11q−22 + 8q−24−6q−26 + 4q−28−3q−30 + q−32−q−34 |
| 1,0 | q56−2q52−2q50 + 2q48 + 5q46−q44−7q42−3q40 + 8q38 + 10q36−5q34−14q32−3q30 + 13q28 + 10q26−9q24−14q22 + q20 + 13q18 + 4q16−10q14−6q12 + 7q10 + 7q8−5q6−8q4 + 4q2 + 9−q−2−9q−4 + q−6 + 11q−8 + 5q−10−9q−12−6q−14 + 10q−16 + 14q−18−3q−20−14q−22−5q−24 + 12q−26 + 9q−28−6q−30−12q−32−3q−34 + 7q−36 + 4q−38−3q−40−5q−42−q−44 + 3q−46 + 2q−48−q−50−q−52 + q−56 |
G2 Invariants.
| Weight | Invariant |
|---|---|
| 1,0 | q80−2q78 + 4q76−7q74 + 6q72−4q70−3q68 + 14q66−21q64 + 28q62−28q60 + 14q58 + 6q56−32q54 + 53q52−58q50 + 48q48−19q46−17q44 + 51q42−68q40 + 61q38−34q36−8q34 + 37q32−49q30 + 35q28−4q26−27q24 + 49q22−47q20 + 15q18 + 25q16−67q14 + 87q12−75q10 + 39q8 + 14q6−60q4 + 93q2−92 + 66q−2−20q−4−28q−6 + 61q−8−62q−10 + 44q−12−6q−14−22q−16 + 39q−18−32q−20 + 6q−22 + 27q−24−50q−26 + 56q−28−36q−30 + q−32 + 34q−34−54q−36 + 61q−38−47q−40 + 22q−42 + 3q−44−27q−46 + 36q−48−37q−50 + 28q−52−15q−54 + 2q−56 + 7q−58−15q−60 + 15q−62−13q−64 + 8q−66−3q−68−2q−70 + 3q−72−4q−74 + 3q−76−q−78 + q−80 |
.
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 31"];
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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]=
| 4t2−14t + 21−14t−1 + 4t−2 |
In[5]:=
| Conway[K][z]
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Out[5]=
| 4z4 + 2z2 + 1 |
In[6]:=
| Alexander[K, 2][t]
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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]=
| {1} |
In[7]:=
| {KnotDet[K], KnotSignature[K]}
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Out[7]=
| { 57, 0 } |
In[8]:=
| Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
| −q5 + 2q4−4q3 + 7q2−8q + 10−9q−1 + 7q−2−5q−3 + 3q−4−q−5 |
In[9]:=
| HOMFLYPT[K][a, z]
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KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
|
Out[9]=
| −z2a4 + z4a2−a2 + 2z4 + 3z2 + 2 + z4a−2 + z2a−2 + a−2−z2a−4−a−4 |
In[10]:=
| Kauffman[K][a, z]
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KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
|
Out[10]=
| az9 + z9a−1 + 3a2z8 + 2z8a−2 + 5z8 + 4a3z7 + 3az7 + z7a−1 + 2z7a−3 + 3a4z6−6a2z6−3z6a−2 + 2z6a−4−14z6 + a5z5−10a3z5−12az5−4z5a−1−2z5a−3 + z5a−5−7a4z4 + 5a2z4 + 3z4a−2−5z4a−4 + 20z4−2a5z3 + 7a3z3 + 15az3 + 6z3a−1−3z3a−3−3z3a−5 + 2a4z2−3a2z2−2z2a−2 + 3z2a−4−10z2−2a3z−4az−2za−1 + 2za−3 + 2za−5 + a2−a−2−a−4 + 2 |
[edit] "Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {10_68,}
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 31"];
|
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]=
| { 4t2−14t + 21−14t−1 + 4t−2, −q5 + 2q4−4q3 + 7q2−8q + 10−9q−1 + 7q−2−5q−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]=
| {10_68,} |
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 31. 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 | q15−2q14 + 5q12−8q11 + 16q9−21q8−3q7 + 38q6−37q5−13q4 + 63q3−47q2−27q + 78−45q−1−35q−2 + 72q−3−30q−4−34q−5 + 50q−6−12q−7−26q−8 + 25q−9−q−10−13q−11 + 8q−12 + q−13−3q−14 + q−15 |
| 3 | −q30 + 2q29−q27−3q26 + 5q25 + q24−5q23−4q22 + 13q21 + q20−19q19−6q18 + 36q17 + 10q16−55q15−24q14 + 76q13 + 52q12−103q11−79q10 + 113q9 + 128q8−132q7−160q6 + 124q5 + 208q4−129q3−227q2 + 105q + 258−97q−1−256q−2 + 67q−3 + 256q−4−42q−5−238q−6 + 12q−7 + 212q−8 + 16q−9−177q−10−39q−11 + 138q−12 + 53q−13−98q−14−59q−15 + 64q−16 + 53q−17−36q−18−42q−19 + 17q−20 + 30q−21−7q−22−19q−23 + 3q−24 + 10q−25−q−26−4q−27−q−28 + 3q−29−q−30 |
| 4 | q50−2q49 + q47−q46 + 6q45−7q44 + q43 + 2q42−9q41 + 17q40−14q39 + 10q38 + 9q37−34q36 + 23q35−30q34 + 41q33 + 44q32−70q31 + 6q30−95q29 + 84q28 + 146q27−64q26−21q25−267q24 + 65q23 + 301q22 + 63q21 + 39q20−536q19−113q18 + 403q17 + 306q16 + 274q15−781q14−427q13 + 349q12 + 542q11 + 640q10−892q9−735q8 + 170q7 + 664q6 + 987q5−867q4−923q3−36q2 + 660q + 1213−751q−1−969q−2−215q−3 + 552q−4 + 1288q−5−548q−6−875q−7−371q−8 + 343q−9 + 1215q−10−275q−11−645q−12−466q−13 + 63q−14 + 980q−15−17q−16−316q−17−438q−18−187q−19 + 632q−20 + 111q−21−18q−22−281q−23−283q−24 + 296q−25 + 87q−26 + 123q−27−102q−28−220q−29 + 96q−30 + 12q−31 + 110q−32−5q−33−111q−34 + 28q−35−18q−36 + 52q−37 + 10q−38−42q−39 + 13q−40−12q−41 + 16q−42 + 5q−43−13q−44 + 4q−45−3q−46 + 4q−47 + q−48−3q−49 + q−50 |
[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. |
|
