10 16
From Knot Atlas
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 (KnotPlot image) |
See the full Rolfsen Knot Table. Visit 10 16's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 10_16's page at Knotilus! Visit 10 16's page at the original Knot Atlas! |
[edit] Knot presentations
| Planar diagram presentation | X6271 X16,8,17,7 X12,5,13,6 X14,3,15,4 X4,13,5,14 X2,15,3,16 X20,12,1,11 X8,20,9,19 X18,10,19,9 X10,18,11,17 |
| Gauss code | 1, -6, 4, -5, 3, -1, 2, -8, 9, -10, 7, -3, 5, -4, 6, -2, 10, -9, 8, -7 |
| Dowker-Thistlethwaite code | 6 14 12 16 18 20 4 2 10 8 |
| Conway Notation | [4123] |
| Minimum Braid Representative | A Morse Link Presentation | An Arc Presentation | |||||
Length is 12, width is 5, Braid index is 5 | 
|  [{2, 12}, {1, 7}, {11, 4}, {12, 10}, {9, 11}, {10, 8}, {5, 3}, {4, 6}, {7, 5}, {6, 2}, {3, 9}, {8, 1}] |
[edit Notes on presentations of 10 16]
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[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
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In[3]:=
| K = Knot["10 16"];
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In[4]:=
| PD[K]
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
| X6271 X16,8,17,7 X12,5,13,6 X14,3,15,4 X4,13,5,14 X2,15,3,16 X20,12,1,11 X8,20,9,19 X18,10,19,9 X10,18,11,17 |
In[5]:=
| GaussCode[K]
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Out[5]=
| 1, -6, 4, -5, 3, -1, 2, -8, 9, -10, 7, -3, 5, -4, 6, -2, 10, -9, 8, -7 |
In[6]:=
| DTCode[K]
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Out[6]=
| 6 14 12 16 18 20 4 2 10 8 |
(The path below may be different on your system)
In[7]:=
| AppendTo[$Path, "C:/bin/LinKnot/"];
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In[8]:=
| ConwayNotation[K]
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Out[8]=
| [4123] |
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,{1,1,2,−1,2,2,−3,2,−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[{2, 12}, {1, 7}, {11, 4}, {12, 10}, {9, 11}, {10, 8}, {5, 3}, {4, 6}, {7, 5}, {6, 2}, {3, 9}, {8, 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 | −4t2 + 12t−15 + 12t−1−4t−2 |
| Conway polynomial | −4z4−4z2 + 1 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 47, 2 } |
| Jones polynomial | q7−2q6 + 4q5−6q4 + 7q3−8q2 + 7q−5 + 4q−1−2q−2 + q−3 |
| HOMFLY-PT polynomial (db, data sources) | −2z4a−2−z4a−4−z4 + a2z2−4z2a−2−z2a−4 + z2a−6−z2 + a2−2a−2 + a−6 + 1 |
| Kauffman polynomial (db, data sources) | z9a−1 + z9a−3 + 4z8a−2 + 2z8a−4 + 2z8 + 2az7−z7a−1 + 3z7a−5 + a2z6−13z6a−2−3z6a−4 + 3z6a−6−6z6−7az5−2z5a−1−4z5a−3−7z5a−5 + 2z5a−7−4a2z4 + 17z4a−2 + 2z4a−4−6z4a−6 + z4a−8 + 4z4 + 5az3 + 8z3a−3 + 10z3a−5−3z3a−7 + 4a2z2−11z2a−2 + 2z2a−4 + 5z2a−6−2z2a−8−2z2−4za−3−4za−5−a2 + 2a−2−a−6 + 1 |
| The A2 invariant | q10 + 2q4 + 1 + q−2−2q−4−2q−8−q−14 + 2q−16 + q−22 |
| The G2 invariant | q46−q44 + 3q42−4q40 + 4q38−2q36−q34 + 9q32−13q30 + 17q28−16q26 + 7q24 + 5q22−20q20 + 31q18−31q16 + 24q14−5q12−14q10 + 30q8−33q6 + 26q4−7q2−10 + 22q−2−20q−4 + 11q−6 + 8q−8−19q−10 + 23q−12−18q−14 + q−16 + 15q−18−34q−20 + 39q−22−32q−24 + 13q−26 + 8q−28−33q−30 + 42q−32−42q−34 + 25q−36−6q−38−18q−40 + 31q−42−29q−44 + 16q−46 + q−48−13q−50 + 16q−52−10q−54−3q−56 + 15q−58−18q−60 + 20q−62−10q−64−3q−66 + 16q−68−22q−70 + 25q−72−19q−74 + 11q−76−10q−80 + 17q−82−20q−84 + 18q−86−10q−88 + 2q−90 + 4q−92−9q−94 + 10q−96−8q−98 + 6q−100−2q−102−q−104 + 2q−106−3q−108 + 2q−110−q−112 + q−114 |
Further Quantum Invariants
Further quantum knot invariants for 10_16.
A1 Invariants.
| Weight | Invariant |
|---|---|
| 1 | q7−q5 + 2q3−q + 2q−1−q−3−q−5 + q−7−2q−9 + 2q−11−q−13 + q−15 |
| 2 | q22−q20−q18 + 4q16−q14−5q12 + 6q10 + 2q8−8q6 + 4q4 + 5q2−8 + q−2 + 6q−4−5q−6−2q−8 + 5q−10 + 2q−12−4q−14−q−16 + 8q−18−4q−20−5q−22 + 9q−24−3q−26−5q−28 + 5q−30−2q−32−3q−34 + 3q−36−q−40 + q−42 |
| 3 | q45−q43−q41 + q39 + 3q37−q35−6q33 + 10q29 + 3q27−11q25−10q23 + 12q21 + 16q19−8q17−21q15 + 22q11 + 8q9−21q7−15q5 + 16q3 + 21q−8q−1−22q−3 + 5q−5 + 23q−7 + q−9−23q−11−2q−13 + 19q−15 + 8q−17−16q−19−11q−21 + 9q−23 + 15q−25−q−27−19q−29−10q−31 + 19q−33 + 16q−35−16q−37−20q−39 + 10q−41 + 21q−43−7q−45−14q−47 + 2q−49 + 11q−51−4q−55 + q−57 + 2q−59−2q−61−2q−63 + 2q−65 + q−67−2q−69−2q−71 + q−73 + q−75−q−79 + q−81 |
| 4 | q76−q74−q72 + q70 + 3q66−3q64−4q62 + 2q60 + 2q58 + 12q56−5q54−16q52−6q50 + 2q48 + 32q46 + 10q44−20q42−29q40−26q38 + 40q36 + 43q34 + 10q32−32q30−69q28 + q26 + 43q24 + 59q22 + 20q20−71q18−53q16−12q14 + 64q12 + 83q10−14q8−65q6−81q4 + 16q2 + 103 + 49q−2−32q−4−106q−6−34q−8 + 80q−10 + 73q−12−q−14−89q−16−47q−18 + 51q−20 + 66q−22 + 9q−24−64q−26−46q−28 + 24q−30 + 61q−32 + 21q−34−35q−36−52q−38−21q−40 + 48q−42 + 55q−44 + 23q−46−55q−48−86q−50 + 5q−52 + 71q−54 + 91q−56−14q−58−114q−60−55q−62 + 30q−64 + 114q−66 + 44q−68−75q−70−69q−72−30q−74 + 68q−76 + 60q−78−14q−80−33q−82−49q−84 + 13q−86 + 34q−88 + 13q−90 + 6q−92−30q−94−7q−96 + 7q−98 + 7q−100 + 15q−102−10q−104−3q−106−3q−108−2q−110 + 8q−112−3q−114 + q−116−q−118−3q−120 + 2q−122−q−124 + q−126−q−130 + q−132 |
| 5 | q115−q113−q111 + q109 + q103−q101−3q99 + 2q97 + 5q95 + 2q93−8q89−13q87−5q85 + 16q83 + 26q81 + 16q79−10q77−42q75−43q73−7q71 + 48q69 + 76q67 + 42q65−31q63−96q61−96q59−18q57 + 94q55 + 141q53 + 81q51−39q49−149q47−157q45−46q43 + 108q41 + 190q39 + 144q37−4q35−167q33−222q31−122q29 + 78q27 + 241q25 + 244q23 + 68q21−189q19−327q17−223q15 + 76q13 + 344q11 + 352q9 + 71q7−296q5−439q3−215q + 205q−1 + 461q−3 + 324q−5−91q−7−428q−9−391q−11−9q−13 + 369q−15 + 398q−17 + 79q−19−282q−21−372q−23−117q−25 + 221q−27 + 318q−29 + 115q−31−161q−33−263q−35−109q−37 + 136q−39 + 224q−41 + 91q−43−107q−45−199q−47−110q−49 + 76q−51 + 190q−53 + 155q−55−13q−57−184q−59−221q−61−90q−63 + 143q−65 + 300q−67 + 226q−69−66q−71−348q−73−365q−75−68q−77 + 336q−79 + 489q−81 + 224q−83−265q−85−550q−87−365q−89 + 134q−91 + 526q−93 + 471q−95 + 15q−97−441q−99−493q−101−134q−103 + 290q−105 + 453q−107 + 223q−109−157q−111−353q−113−242q−115 + 38q−117 + 241q−119 + 223q−121 + 37q−123−140q−125−176q−127−73q−129 + 63q−131 + 121q−133 + 76q−135−14q−137−77q−139−67q−141−10q−143 + 41q−145 + 48q−147 + 20q−149−17q−151−30q−153−20q−155 + 5q−157 + 19q−159 + 13q−161 + 2q−163−6q−165−10q−167−3q−169 + 5q−171 + 2q−173 + q−175 + q−177−2q−179−2q−181 + q−183−q−187 + q−189−q−193 + q−195 |
A2 Invariants.
| Weight | Invariant |
|---|---|
| 1,0 | q10 + 2q4 + 1 + q−2−2q−4−2q−8−q−14 + 2q−16 + q−22 |
| 1,1 | q28−2q26 + 6q24−12q22 + 23q20−34q18 + 54q16−72q14 + 90q12−102q10 + 110q8−104q6 + 81q4−52q2 + 12 + 36q−2−84q−4 + 122q−6−156q−8 + 176q−10−189q−12 + 178q−14−158q−16 + 136q−18−91q−20 + 62q−22−16q−24−10q−26 + 35q−28−56q−30 + 58q−32−70q−34 + 70q−36−68q−38 + 60q−40−56q−42 + 50q−44−38q−46 + 28q−48−20q−50 + 15q−52−8q−54 + 4q−56−2q−58 + q−60 |
| 2,0 | q28−q24 + 3q20 + 3q18−2q16−2q14 + 3q12 + 3q10−2q8−5q6 + 2q4 + 3q2−4−5q−2 + 3q−4 + q−6−3q−8 + 2q−12 + q−14 + 5q−18 + q−20−2q−22 + 5q−24 + 5q−26−5q−28−4q−30 + 4q−32 + 2q−34−5q−36−4q−38 + 2q−40−3q−44 + 2q−48 + q−50 + q−56 |
A3 Invariants.
| Weight | Invariant |
|---|---|
| 0,1,0 | q20−q18 + q16 + 2q14−2q12 + 3q10 + 3q8−4q6 + 5q4 + 2q2−6 + 4q−2 + 3q−4−8q−6 + 2q−10−5q−12−2q−14 + q−16 + 4q−18−2q−20 + q−22 + 8q−24−4q−26−4q−28 + 7q−30−3q−32−5q−34 + 6q−36−3q−40 + 2q−42−q−46 + q−48 |
| 1,0,0 | q13 + q9 + 2q5 + 2q + q−3−2q−5−q−7−q−9−2q−11−q−15 + q−17−q−19 + 2q−21 + q−25 + q−29 |
B2 Invariants.
| Weight | Invariant |
|---|---|
| 0,1 | q20−q18 + 3q16−4q14 + 6q12−7q10 + 9q8−8q6 + 9q4−6q2 + 4−3q−4 + 8q−6−12q−8 + 14q−10−17q−12 + 16q−14−17q−16 + 12q−18−10q−20 + 5q−22−2q−24−2q−26 + 6q−28−7q−30 + 9q−32−7q−34 + 8q−36−6q−38 + 5q−40−4q−42 + 2q−44−q−46 + q−48 |
| 1,0 | q34−q30−q28 + 2q26 + 3q24−q22−4q20 + 6q16 + 5q14−5q12−7q10 + 2q8 + 10q6 + 3q4−8q2−7 + 4q−2 + 8q−4 + q−6−8q−8−4q−10 + 4q−12 + 3q−14−4q−16−5q−18 + 3q−20 + 4q−22−2q−24−6q−26 + 2q−28 + 7q−30 + 2q−32−6q−34−2q−36 + 7q−38 + 6q−40−4q−42−8q−44 + 8q−48 + 4q−50−6q−52−7q−54 + 7q−58 + 3q−60−3q−62−4q−64 + 3q−68 + q−70−q−72−q−74 + q−78 |
G2 Invariants.
| Weight | Invariant |
|---|---|
| 1,0 | q46−q44 + 3q42−4q40 + 4q38−2q36−q34 + 9q32−13q30 + 17q28−16q26 + 7q24 + 5q22−20q20 + 31q18−31q16 + 24q14−5q12−14q10 + 30q8−33q6 + 26q4−7q2−10 + 22q−2−20q−4 + 11q−6 + 8q−8−19q−10 + 23q−12−18q−14 + q−16 + 15q−18−34q−20 + 39q−22−32q−24 + 13q−26 + 8q−28−33q−30 + 42q−32−42q−34 + 25q−36−6q−38−18q−40 + 31q−42−29q−44 + 16q−46 + q−48−13q−50 + 16q−52−10q−54−3q−56 + 15q−58−18q−60 + 20q−62−10q−64−3q−66 + 16q−68−22q−70 + 25q−72−19q−74 + 11q−76−10q−80 + 17q−82−20q−84 + 18q−86−10q−88 + 2q−90 + 4q−92−9q−94 + 10q−96−8q−98 + 6q−100−2q−102−q−104 + 2q−106−3q−108 + 2q−110−q−112 + q−114 |
.
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 16"];
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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 + 12t−15 + 12t−1−4t−2 |
In[5]:=
| Conway[K][z]
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Out[5]=
| −4z4−4z2 + 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]=
| { 47, 2 } |
In[8]:=
| Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
| q7−2q6 + 4q5−6q4 + 7q3−8q2 + 7q−5 + 4q−1−2q−2 + q−3 |
In[9]:=
| HOMFLYPT[K][a, z]
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KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
|
Out[9]=
| −2z4a−2−z4a−4−z4 + a2z2−4z2a−2−z2a−4 + z2a−6−z2 + a2−2a−2 + a−6 + 1 |
In[10]:=
| Kauffman[K][a, z]
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KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
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Out[10]=
| z9a−1 + z9a−3 + 4z8a−2 + 2z8a−4 + 2z8 + 2az7−z7a−1 + 3z7a−5 + a2z6−13z6a−2−3z6a−4 + 3z6a−6−6z6−7az5−2z5a−1−4z5a−3−7z5a−5 + 2z5a−7−4a2z4 + 17z4a−2 + 2z4a−4−6z4a−6 + z4a−8 + 4z4 + 5az3 + 8z3a−3 + 10z3a−5−3z3a−7 + 4a2z2−11z2a−2 + 2z2a−4 + 5z2a−6−2z2a−8−2z2−4za−3−4za−5−a2 + 2a−2−a−6 + 1 |
[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 16"];
|
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 + 12t−15 + 12t−1−4t−2, q7−2q6 + 4q5−6q4 + 7q3−8q2 + 7q−5 + 4q−1−2q−2 + q−3 } |
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 = 2 is the signature of 10 16. 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 | q20−2q19 + q18 + 4q17−8q16 + 2q15 + 11q14−18q13 + 4q12 + 23q11−32q10 + 5q9 + 35q8−41q7 + 2q6 + 41q5−38q4−5q3 + 38q2−27q−10 + 29q−1−14q−2−11q−3 + 17q−4−4q−5−7q−6 + 6q−7−2q−9 + q−10 |
| 3 | q39−2q38 + q37 + q36 + q35−5q34 + q33 + 4q32 + 2q31−9q30 + q29 + 8q28 + q27−14q26 + 5q25 + 19q24−8q23−30q22 + 12q21 + 47q20−19q19−60q18 + 16q17 + 79q16−16q15−89q14 + 7q13 + 97q12−95q10−13q9 + 92q8 + 24q7−84q6−34q5 + 71q4 + 48q3−62q2−52q + 44 + 62q−1−33q−2−57q−3 + 13q−4 + 56q−5−4q−6−43q−7−9q−8 + 35q−9 + 9q−10−19q−11−13q−12 + 13q−13 + 8q−14−5q−15−6q−16 + 3q−17 + 2q−18−2q−20 + q−21 |
[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. |
|
