10 96
From Knot Atlas
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 (KnotPlot image) |
See the full Rolfsen Knot Table. Visit 10 96's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 10_96's page at Knotilus! Visit 10 96's page at the original Knot Atlas! |
[edit] Knot presentations
| Planar diagram presentation | X1425 X5,18,6,19 X3948 X9,3,10,2 X11,17,12,16 X7,12,8,13 X15,6,16,7 X17,11,18,10 X13,1,14,20 X19,15,20,14 |
| Gauss code | -1, 4, -3, 1, -2, 7, -6, 3, -4, 8, -5, 6, -9, 10, -7, 5, -8, 2, -10, 9 |
| Dowker-Thistlethwaite code | 4 8 18 12 2 16 20 6 10 14 |
| Conway Notation | [.2.21.2] |
| Minimum Braid Representative | A Morse Link Presentation | An Arc Presentation | |||||
Length is 12, width is 5, Braid index is 5 | 
|  [{6, 2}, {12, 7}, {8, 5}, {7, 9}, {1, 8}, {10, 6}, {9, 11}, {3, 10}, {5, 12}, {2, 4}, {11, 3}, {4, 1}] |
[edit Notes on presentations of 10 96]
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[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
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In[3]:=
| K = Knot["10 96"];
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In[4]:=
| PD[K]
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
| X1425 X5,18,6,19 X3948 X9,3,10,2 X11,17,12,16 X7,12,8,13 X15,6,16,7 X17,11,18,10 X13,1,14,20 X19,15,20,14 |
In[5]:=
| GaussCode[K]
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Out[5]=
| -1, 4, -3, 1, -2, 7, -6, 3, -4, 8, -5, 6, -9, 10, -7, 5, -8, 2, -10, 9 |
In[6]:=
| DTCode[K]
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Out[6]=
| 4 8 18 12 2 16 20 6 10 14 |
(The path below may be different on your system)
In[7]:=
| AppendTo[$Path, "C:/bin/LinKnot/"];
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In[8]:=
| ConwayNotation[K]
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Out[8]=
| [.2.21.2] |
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,−3,2,1,−3,4,−3,2,−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[{6, 2}, {12, 7}, {8, 5}, {7, 9}, {1, 8}, {10, 6}, {9, 11}, {3, 10}, {5, 12}, {2, 4}, {11, 3}, {4, 1}] |
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 + 7t2−22t + 33−22t−1 + 7t−2−t−3 |
| Conway polynomial | −z6 + z4−3z2 + 1 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 93, 0 } |
| Jones polynomial | q6−3q5 + 7q4−11q3 + 14q2−16q + 15−12q−1 + 9q−2−4q−3 + q−4 |
| HOMFLY-PT polynomial (db, data sources) | −z6 + a2z4 + 3z4a−2−3z4 + a2z2 + 5z2a−2−3z2a−4−6z2 + 2a2 + 3a−2−2a−4 + a−6−3 |
| Kauffman polynomial (db, data sources) | 2z9a−1 + 2z9a−3 + 11z8a−2 + 4z8a−4 + 7z8 + 11az7 + 14z7a−1 + 6z7a−3 + 3z7a−5 + 9a2z6−17z6a−2−7z6a−4 + z6a−6 + 4a3z5−15az5−34z5a−1−23z5a−3−8z5a−5 + a4z4−10a2z4−4z4a−2−z4a−4−3z4a−6−17z4−a3z3 + 7az3 + 17z3a−1 + 16z3a−3 + 7z3a−5 + 5a2z2 + 10z2a−2 + 6z2a−4 + 3z2a−6 + 12z2−az−za−1−2za−3−2za−5−2a2−3a−2−2a−4−a−6−3 |
| The A2 invariant | q12−2q10 + 3q8 + 2q6−3q4 + 3q2−3 + q−2−q−6 + 3q−8−3q−10 + q−12 + q−14−2q−16 + q−18 + q−20 |
| The G2 invariant | q66−3q64 + 6q62−10q60 + 11q58−10q56 + 4q54 + 15q52−37q50 + 63q48−80q46 + 68q44−36q42−30q40 + 119q38−191q36 + 229q34−189q32 + 78q30 + 83q28−238q26 + 334q24−324q22 + 195q20 + 4q18−197q16 + 307q14−272q12 + 120q10 + 86q8−245q6 + 269q4−161q2−63 + 294q−2−425q−4 + 394q−6−202q−8−84q−10 + 357q−12−513q−14 + 493q−16−322q−18 + 52q−20 + 219q−22−388q−24 + 414q−26−276q−28 + 57q−30 + 160q−32−281q−34 + 251q−36−98q−38−109q−40 + 278q−42−326q−44 + 228q−46−21q−48−206q−50 + 357q−52−373q−54 + 255q−56−70q−58−122q−60 + 239q−62−260q−64 + 201q−66−91q−68−11q−70 + 76q−72−97q−74 + 81q−76−47q−78 + 18q−80 + 4q−82−13q−84 + 13q−86−10q−88 + 6q−90−2q−92 + q−94 |
Further Quantum Invariants
Further quantum knot invariants for 10_96.
A1 Invariants.
| Weight | Invariant |
|---|---|
| 1 | q9−3q7 + 5q5−3q3 + 3q−q−1−2q−3 + 3q−5−4q−7 + 4q−9−2q−11 + q−13 |
| 2 | q26−3q24 + 2q22 + 9q20−18q18 + 32q14−31q12−13q10 + 49q8−22q6−29q4 + 36q2 + 3−28q−2 + 3q−4 + 26q−6−10q−8−29q−10 + 34q−12 + 13q−14−47q−16 + 21q−18 + 30q−20−38q−22 + 26q−26−14q−28−7q−30 + 9q−32−q−34−2q−36 + q−38 |
| 3 | q51−3q49 + 2q47 + 6q45−6q43−15q41 + 9q39 + 42q37−17q35−83q33 + 17q31 + 138q29 + 14q27−216q25−66q23 + 273q21 + 153q19−292q17−247q15 + 260q13 + 325q11−180q9−361q7 + 72q5 + 341q3 + 46q−288q−1−142q−3 + 203q−5 + 225q−7−116q−9−275q−11 + 22q−13 + 313q−15 + 72q−17−329q−19−160q−21 + 310q−23 + 250q−25−262q−27−321q−29 + 177q−31 + 356q−33−72q−35−343q−37−34q−39 + 288q−41 + 107q−43−195q−45−138q−47 + 102q−49 + 128q−51−35q−53−88q−55−5q−57 + 48q−59 + 15q−61−20q−63−10q−65 + 6q−67 + 4q−69−q−71−2q−73 + q−75 |
| 4 | q84−3q82 + 2q80 + 6q78−9q76−3q74−6q72 + 25q70 + 32q68−58q66−49q64−19q62 + 143q60 + 177q58−185q56−317q54−183q52 + 471q50 + 752q48−180q46−992q44−988q42 + 660q40 + 1977q38 + 686q36−1526q34−2615q32−221q30 + 2933q28 + 2513q26−723q24−3792q22−2158q20 + 2197q18 + 3763q16 + 1268q14−3046q12−3458q10 + 76q8 + 3107q6 + 2746q4−912q2−3050−1735q−2 + 1271q−4 + 2849q−6 + 1033q−8−1732q−10−2562q−12−383q−14 + 2294q−16 + 2268q−18−527q−20−2906q−22−1613q−24 + 1659q−26 + 3193q−28 + 657q−30−2957q−32−2812q−34 + 590q−36 + 3703q−38 + 2169q−40−2095q−42−3634q−44−1211q−46 + 2985q−48 + 3378q−50−124q−52−3072q−54−2775q−56 + 941q−58 + 3049q−60 + 1677q−62−1121q−64−2690q−66−911q−68 + 1297q−70 + 1830q−72 + 567q−74−1223q−76−1182q−78−161q−80 + 780q−82 + 805q−84−70q−86−470q−88−395q−90 + 30q−92 + 311q−94 + 138q−96−22q−98−133q−100−69q−102 + 41q−104 + 34q−106 + 23q−108−14q−110−16q−112 + 3q−114 + q−116 + 4q−118−q−120−2q−122 + q−124 |
| 5 | q125−3q123 + 2q121 + 6q119−9q117−6q115 + 6q113 + 10q111 + 15q109−3q107−48q105−56q103 + 42q101 + 145q99 + 119q97−95q95−346q93−336q91 + 136q89 + 823q87 + 867q85−140q83−1585q81−1997q79−294q77 + 2663q75 + 4140q73 + 1652q71−3769q69−7394q67−4631q65 + 3967q63 + 11549q61 + 9927q59−2279q57−15649q55−17196q53−2539q51 + 17911q49 + 25619q47 + 10696q45−16770q43−32921q41−21228q39 + 11069q37 + 36814q35 + 32043q33−1366q31−35574q29−40327q27−10445q25 + 28877q23 + 43868q21 + 21724q19−18214q17−41801q15−29894q13 + 5958q11 + 34897q9 + 33650q7 + 5317q5−25118q3−33074q−13948q−1 + 14857q−3 + 29481q−5 + 19294q−7−5762q−9−24644q−11−22168q−13−1138q−15 + 20180q−17 + 23554q−19 + 6143q−21−16835q−23−24883q−25−10068q−27 + 14747q−29 + 26768q−31 + 13965q−33−13016q−35−29348q−37−18765q−39 + 10575q−41 + 32004q−43 + 24653q−45−6324q−47−33394q−49−31233q−51−408q−53 + 32267q−55 + 37240q−57 + 9286q−59−27460q−61−40930q−63−19220q−65 + 18847q−67 + 40696q−69 + 28154q−71−7396q−73−35691q−75−33819q−77−4838q−79 + 26289q−81 + 34643q−83 + 15255q−85−14398q−87−30268q−89−21479q−91 + 2578q−93 + 21920q−95 + 22620q−97 + 6502q−99−12116q−101−19138q−103−11260q−105 + 3337q−107 + 13010q−109 + 11772q−111 + 2499q−113−6637q−115−9240q−117−4935q−119 + 1714q−121 + 5626q−123 + 4792q−125 + 934q−127−2486q−129−3294q−131−1704q−133 + 514q−135 + 1707q−137 + 1410q−139 + 287q−141−623q−143−800q−145−393q−147 + 96q−149 + 334q−151 + 265q−153 + 43q−155−105q−157−110q−159−43q−161 + 11q−163 + 40q−165 + 26q−167−6q−169−10q−171−3q−173−2q−175 + q−177 + 4q−179−q−181−2q−183 + q−185 |
A2 Invariants.
| Weight | Invariant |
|---|---|
| 1,0 | q12−2q10 + 3q8 + 2q6−3q4 + 3q2−3 + q−2−q−6 + 3q−8−3q−10 + q−12 + q−14−2q−16 + q−18 + q−20 |
| 2,0 | q32−2q30 + q28 + 6q26−4q24−9q22 + 6q20 + 17q18−9q16−25q14 + 11q12 + 26q10−12q8−20q6 + 18q4 + 14q2−11−10q−2 + 7q−4−3q−6−7q−8 + 11q−10−3q−12−12q−14 + 12q−16 + 19q−18−15q−20−15q−22 + 16q−24 + 15q−26−16q−28−17q−30 + 14q−32 + 13q−34−8q−36−11q−38 + 3q−40 + 8q−42−4q−46−q−48 + q−50 + q−52 |
A3 Invariants.
| Weight | Invariant |
|---|---|
| 0,1,0 | q28−3q26 + 10q22−11q20−6q18 + 26q16−18q14−12q12 + 38q10−18q8−18q6 + 31q4−12q2−17 + 10q−2 + 5q−4−4q−6−12q−8 + 18q−10 + 12q−12−30q−14 + 15q−16 + 20q−18−35q−20 + 12q−22 + 17q−24−25q−26 + 9q−28 + 8q−30−10q−32 + 4q−34 + 2q−36−2q−38 + q−40 |
| 1,0,0 | q15−2q13 + 4q11 + 3q7−3q5 + 2q3−3q−q−1 + 2q−7−q−9 + 4q−11−3q−13 + q−15−2q−17 + q−19−2q−21 + q−23 + q−25 + q−27 |
B2 Invariants.
| Weight | Invariant |
|---|---|
| 0,1 | q28−3q26 + 6q24−12q22 + 21q20−28q18 + 36q16−40q14 + 42q12−36q10 + 26q8−10q6−9q4 + 30q2−51 + 66q−2−77q−4 + 80q−6−74q−8 + 62q−10−44q−12 + 24q−14−3q−16−16q−18 + 29q−20−38q−22 + 41q−24−39q−26 + 33q−28−26q−30 + 18q−32−10q−34 + 6q−36−2q−38 + q−40 |
| 1,0 | q46−3q42−3q40 + 3q38 + 11q36 + 4q34−16q32−17q30 + 9q28 + 31q26 + 8q24−33q22−27q20 + 23q18 + 42q16−q14−42q12−16q10 + 33q8 + 27q6−23q4−32q2 + 10 + 30q−2−2q−4−30q−6−5q−8 + 27q−10 + 11q−12−24q−14−15q−16 + 25q−18 + 26q−20−19q−22−37q−24 + 7q−26 + 44q−28 + 12q−30−39q−32−32q−34 + 23q−36 + 40q−38−3q−40−34q−42−14q−44 + 20q−46 + 19q−48−6q−50−14q−52−2q−54 + 7q−56 + 4q−58−2q−60−2q−62 + q−66 |
G2 Invariants.
| Weight | Invariant |
|---|---|
| 1,0 | q66−3q64 + 6q62−10q60 + 11q58−10q56 + 4q54 + 15q52−37q50 + 63q48−80q46 + 68q44−36q42−30q40 + 119q38−191q36 + 229q34−189q32 + 78q30 + 83q28−238q26 + 334q24−324q22 + 195q20 + 4q18−197q16 + 307q14−272q12 + 120q10 + 86q8−245q6 + 269q4−161q2−63 + 294q−2−425q−4 + 394q−6−202q−8−84q−10 + 357q−12−513q−14 + 493q−16−322q−18 + 52q−20 + 219q−22−388q−24 + 414q−26−276q−28 + 57q−30 + 160q−32−281q−34 + 251q−36−98q−38−109q−40 + 278q−42−326q−44 + 228q−46−21q−48−206q−50 + 357q−52−373q−54 + 255q−56−70q−58−122q−60 + 239q−62−260q−64 + 201q−66−91q−68−11q−70 + 76q−72−97q−74 + 81q−76−47q−78 + 18q−80 + 4q−82−13q−84 + 13q−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 96"];
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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 + 7t2−22t + 33−22t−1 + 7t−2−t−3 |
In[5]:=
| Conway[K][z]
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Out[5]=
| −z6 + z4−3z2 + 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.
|
Out[6]=
| {1} |
In[7]:=
| {KnotDet[K], KnotSignature[K]}
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Out[7]=
| { 93, 0 } |
In[8]:=
| Jones[K][q]
|
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
|
Out[8]=
| q6−3q5 + 7q4−11q3 + 14q2−16q + 15−12q−1 + 9q−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 + 5z2a−2−3z2a−4−6z2 + 2a2 + 3a−2−2a−4 + a−6−3 |
In[10]:=
| Kauffman[K][a, z]
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KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
|
Out[10]=
| 2z9a−1 + 2z9a−3 + 11z8a−2 + 4z8a−4 + 7z8 + 11az7 + 14z7a−1 + 6z7a−3 + 3z7a−5 + 9a2z6−17z6a−2−7z6a−4 + z6a−6 + 4a3z5−15az5−34z5a−1−23z5a−3−8z5a−5 + a4z4−10a2z4−4z4a−2−z4a−4−3z4a−6−17z4−a3z3 + 7az3 + 17z3a−1 + 16z3a−3 + 7z3a−5 + 5a2z2 + 10z2a−2 + 6z2a−4 + 3z2a−6 + 12z2−az−za−1−2za−3−2za−5−2a2−3a−2−2a−4−a−6−3 |
[edit] "Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {}
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 96"];
|
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 + 7t2−22t + 33−22t−1 + 7t−2−t−3, q6−3q5 + 7q4−11q3 + 14q2−16q + 15−12q−1 + 9q−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]=
| {} |
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 96. 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 + q16 + 11q15−19q14−6q13 + 51q12−45q11−44q10 + 119q9−54q8−112q7 + 179q6−33q5−175q4 + 198q3 + 3q2−198q + 167 + 34q−1−165q−2 + 102q−3 + 41q−4−94q−5 + 40q−6 + 23q−7−31q−8 + 8q−9 + 5q−10−4q−11 + q−12 |
| 3 | q36−3q35 + q34 + 5q33 + 3q32−19q31−9q30 + 40q29 + 36q28−72q27−92q26 + 93q25 + 199q24−98q23−332q22 + 36q21 + 501q20 + 83q19−654q18−273q17 + 772q16 + 511q15−833q14−771q13 + 831q12 + 1023q11−773q10−1241q9 + 662q8 + 1424q7−532q6−1532q5 + 365q4 + 1583q3−191q2−1554q + 20 + 1437q−1 + 143q−2−1259q−3−249q−4 + 1004q−5 + 324q−6−754q−7−314q−8 + 497q−9 + 279q−10−309q−11−194q−12 + 158q−13 + 129q−14−79q−15−70q−16 + 37q−17 + 29q−18−13q−19−11q−20 + 4q−21 + 5q−22−4q−23 + q−24 |
| 4 | q60−3q59 + q58 + 5q57−3q56 + 3q55−22q54 + 3q53 + 42q52 + 8q51 + 10q50−132q49−61q48 + 153q47 + 168q46 + 183q45−413q44−486q43 + 78q42 + 568q41 + 1058q40−438q39−1427q38−943q37 + 527q36 + 2848q35 + 825q34−1960q33−3151q32−1252q31 + 4417q30 + 3623q29−588q28−5259q27−4968q26 + 4120q25 + 6571q24 + 2914q23−5652q22−9164q21 + 1697q20 + 8110q19 + 7178q18−4118q17−12277q16−1705q15 + 7965q14 + 10792q13−1582q12−13811q11−4977q10 + 6672q9 + 13171q8 + 1213q7−13785q6−7654q5 + 4493q4 + 14001q3 + 3978q2−11969q−9232 + 1487q−1 + 12686q−2 + 6116q−3−8311q−4−8871q−5−1544q−6 + 9152q−7 + 6528q−8−3997q−9−6376q−10−3110q−11 + 4797q−12 + 4894q−13−928q−14−3140q−15−2690q−16 + 1643q−17 + 2500q−18 + 161q−19−928q−20−1399q−21 + 326q−22 + 852q−23 + 157q−24−116q−25−467q−26 + 45q−27 + 198q−28 + 23q−29 + 16q−30−105q−31 + 11q−32 + 36q−33−7q−34 + 7q−35−15q−36 + 4q−37 + 5q−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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