10 134
From Knot Atlas
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 (KnotPlot image) |
See the full Rolfsen Knot Table. Visit 10 134's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 10_134's page at Knotilus! Visit 10 134's page at the original Knot Atlas! |
[edit] Knot presentations
| Planar diagram presentation | X4251 X8493 X9,15,10,14 X5,13,6,12 X13,7,14,6 X11,19,12,18 X15,1,16,20 X19,17,20,16 X17,11,18,10 X2837 |
| Gauss code | 1, -10, 2, -1, -4, 5, 10, -2, -3, 9, -6, 4, -5, 3, -7, 8, -9, 6, -8, 7 |
| Dowker-Thistlethwaite code | 4 8 -12 2 -14 -18 -6 -20 -10 -16 |
| Conway Notation | [221,3,2-] |
| Minimum Braid Representative | A Morse Link Presentation | An Arc Presentation | ||||
Length is 11, width is 4, Braid index is 4 | 
|  [{4, 12}, {3, 5}, {1, 4}, {6, 10}, {5, 8}, {2, 6}, {12, 3}, {11, 9}, {10, 7}, {8, 2}, {7, 11}, {9, 1}] |
[edit Notes on presentations of 10 134]
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[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
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In[3]:=
| K = Knot["10 134"];
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In[4]:=
| PD[K]
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
| X4251 X8493 X9,15,10,14 X5,13,6,12 X13,7,14,6 X11,19,12,18 X15,1,16,20 X19,17,20,16 X17,11,18,10 X2837 |
In[5]:=
| GaussCode[K]
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Out[5]=
| 1, -10, 2, -1, -4, 5, 10, -2, -3, 9, -6, 4, -5, 3, -7, 8, -9, 6, -8, 7 |
In[6]:=
| DTCode[K]
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Out[6]=
| 4 8 -12 2 -14 -18 -6 -20 -10 -16 |
(The path below may be different on your system)
In[7]:=
| AppendTo[$Path, "C:/bin/LinKnot/"];
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In[8]:=
| ConwayNotation[K]
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Out[8]=
| [221,3,2-] |
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(4,{1,1,1,2,1,1,2,3,−2,3,3}) |
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]=
| { 4, 11, 4 } |
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[{4, 12}, {3, 5}, {1, 4}, {6, 10}, {5, 8}, {2, 6}, {12, 3}, {11, 9}, {10, 7}, {8, 2}, {7, 11}, {9, 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 | 2t3−4t2 + 4t−3 + 4t−1−4t−2 + 2t−3 |
| Conway polynomial | 2z6 + 8z4 + 6z2 + 1 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 23, 6 } |
| Jones polynomial | q11−3q10 + 3q9−4q8 + 4q7−3q6 + 3q5−q4 + q3 |
| HOMFLY-PT polynomial (db, data sources) | z6a−6 + z6a−8 + 5z4a−6 + 4z4a−8−z4a−10 + 7z2a−6 + 3z2a−8−4z2a−10 + 3a−6−3a−10 + a−12 |
| Kauffman polynomial (db, data sources) | z8a−8 + z8a−10 + z7a−7 + 3z7a−9 + 2z7a−11 + z6a−6−3z6a−8−3z6a−10 + z6a−12−3z5a−7−11z5a−9−8z5a−11−5z4a−6 + z4a−8 + 5z4a−10−z4a−12 + 11z3a−9 + 14z3a−11 + 3z3a−13 + 7z2a−6−7z2a−10 + z2a−12 + z2a−14 + 2za−7−4za−9−8za−11−2za−13−3a−6 + 3a−10 + a−12 |
| The A2 invariant | q−10 + 2q−14 + q−16 + 2q−18 + q−20 + q−24−2q−26−q−28−2q−30−q−32 + q−38 |
| The G2 invariant | q−50 + 2q−54−q−56 + 2q−58 + 5q−64−5q−66 + 8q−68−4q−70 + 6q−74−7q−76 + 10q−78−5q−80 + 2q−82 + 5q−84−6q−86 + 6q−88−5q−92 + 9q−94−6q−96 + q−98 + 5q−100−9q−102 + 12q−104−8q−106 + 3q−108 + q−110−7q−112 + 8q−114−10q−116 + 5q−118−4q−120−3q−122 + 4q−124−8q−126 + 2q−128−q−130−6q−132 + 5q−134−6q−136−2q−138 + 6q−140−9q−142 + 9q−144−4q−146−2q−148 + 7q−150−7q−152 + 6q−154−q−156 + 3q−160−q−162 + q−164 + 2q−168−2q−174−q−180 + q−182 |
Further Quantum Invariants
Further quantum knot invariants for 10_134.
A1 Invariants.
| Weight | Invariant |
|---|---|
| 1 | q−5 + 2q−9 + q−13−q−17−2q−21 + q−23 |
| 2 | q−10 + 3q−16 + q−18−2q−20 + 3q−22 + 3q−24−2q−26−q−28 + 2q−30−2q−32−3q−34 + q−36−3q−40 + 2q−44−q−46−2q−48 + 3q−50 + q−52−3q−54 + 2q−56 + q−58−q−60 |
| 3 | q−15 + q−21 + 3q−23 + q−25−2q−27−q−29 + 5q−31 + 5q−33−q−35−6q−37 + 6q−41 + 5q−43−4q−45−9q−47−2q−49 + 7q−51 + 4q−53−8q−55−9q−57 + 3q−59 + 8q−61−3q−63−9q−65 + q−67 + 10q−69−q−71−6q−73 + 7q−77 + q−79−4q−81−4q−83 + 3q−85 + 6q−87 + 2q−89−8q−91−6q−93 + 8q−95 + 9q−97−5q−99−10q−101 + 2q−103 + 7q−105 + 3q−107−6q−109−3q−111 + q−113 + 2q−115 + q−117−q−119 |
A2 Invariants.
| Weight | Invariant |
|---|---|
| 1,0 | q−10 + 2q−14 + q−16 + 2q−18 + q−20 + q−24−2q−26−q−28−2q−30−q−32 + q−38 |
| 1,1 | q−20 + 4q−24−2q−26 + 12q−28−8q−30 + 24q−32−16q−34 + 26q−36−18q−38 + 14q−40−6q−42−16q−44 + 14q−46−38q−48 + 32q−50−45q−52 + 36q−54−32q−56 + 28q−58−12q−60 + 6q−62 + 10q−64−16q−66 + 23q−68−24q−70 + 20q−72−16q−74 + 10q−76−4q−78 + 2q−84−2q−90 + q−92 |
| 2,0 | q−20 + 2q−26 + 3q−28 + q−30 + 2q−32 + 4q−34 + 5q−36 + 2q−38 + 2q−40 + 2q−42−3q−46−2q−48−4q−50−5q−52−4q−54−5q−56−4q−58−2q−60 + q−62 + 2q−64 + q−66 + 2q−68 + 3q−70 + q−72 + q−78 + 2q−80 + q−82−q−84−q−88−q−90−q−92 + q−96 |
A3 Invariants.
| Weight | Invariant |
|---|---|
| 0,1,0 | q−20 + 2q−24 + 3q−26 + 2q−28 + 4q−30 + 4q−32 + q−34 + 4q−36−q−40−2q−44−5q−46−4q−48−5q−50−5q−52−3q−54−q−56 + 4q−58 + 2q−60 + 4q−62 + 4q−64−q−66−q−68 + q−70−2q−72−q−74 + q−76 |
| 1,0,0 | q−15 + 2q−19 + q−21 + 3q−23 + q−25 + 2q−27 + q−29−2q−35−q−37−3q−39−q−41−2q−43 + q−49 + q−51 |
A4 Invariants.
| Weight | Invariant |
|---|---|
| 0,1,0,0 | q−30 + 2q−34 + 3q−36 + 4q−38 + 4q−40 + 7q−42 + 5q−44 + 6q−46 + 5q−48 + 2q−50 + 2q−52 + 3q−54−3q−58−2q−60−4q−62−10q−64−14q−66−10q−68−11q−70−11q−72−2q−74 + 4q−76 + 4q−78 + 10q−80 + 12q−82 + 7q−84 + 3q−86 + 3q−88−5q−92−4q−94−q−98−2q−100 + q−102 + q−104 |
| 1,0,0,0 | q−20 + 2q−24 + q−26 + 3q−28 + 2q−30 + 2q−32 + 2q−34 + q−36 + q−38−q−40−2q−44−q−46−3q−48−2q−50−2q−52−2q−54 + q−60 + q−62 + q−64 |
B2 Invariants.
| Weight | Invariant |
|---|---|
| 0,1 | q−20 + 2q−24−q−26 + 4q−28−2q−30 + 4q−32−q−34 + 2q−36−q−40 + 2q−42−4q−44 + 5q−46−6q−48 + 5q−50−5q−52 + 3q−54−3q−56−2q−62 + 2q−64−3q−66 + 3q−68−3q−70 + 2q−72−q−74 + q−76 |
| 1,0 | q−30 + 2q−38 + 2q−40 + q−42−q−44 + 2q−46 + 4q−48 + 3q−50−2q−52 + 3q−56 + 4q−58−3q−62−q−64 + 2q−66−3q−70−3q−72−q−74−q−76−3q−78−4q−80−2q−82 + q−84−q−86−3q−88−2q−90 + 3q−92 + 3q−94−q−98 + 3q−100 + 4q−102 + q−104−2q−106−q−108 + q−110 + 2q−112−q−114−2q−116−q−118 + q−122 |
D4 Invariants.
| Weight | Invariant |
|---|---|
| 1,0,0,0 | q−30 + 2q−34 + q−36 + 5q−38 + q−40 + 6q−42 + q−44 + 6q−46 + q−48 + 2q−50 + q−54 + q−56−2q−58 + 2q−60−5q−62 + q−64−8q−66−q−68−10q−70−2q−72−8q−74−2q−78 + 3q−80 + 3q−82 + 3q−84 + 5q−86 + q−88 + 4q−90−2q−92 + q−94−3q−96 + 2q−98−2q−100−q−104 + q−106 |
G2 Invariants.
| Weight | Invariant |
|---|---|
| 1,0 | q−50 + 2q−54−q−56 + 2q−58 + 5q−64−5q−66 + 8q−68−4q−70 + 6q−74−7q−76 + 10q−78−5q−80 + 2q−82 + 5q−84−6q−86 + 6q−88−5q−92 + 9q−94−6q−96 + q−98 + 5q−100−9q−102 + 12q−104−8q−106 + 3q−108 + q−110−7q−112 + 8q−114−10q−116 + 5q−118−4q−120−3q−122 + 4q−124−8q−126 + 2q−128−q−130−6q−132 + 5q−134−6q−136−2q−138 + 6q−140−9q−142 + 9q−144−4q−146−2q−148 + 7q−150−7q−152 + 6q−154−q−156 + 3q−160−q−162 + q−164 + 2q−168−2q−174−q−180 + q−182 |
.
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`
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Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.
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In[3]:=
| K = Knot["10 134"];
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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]=
| 2t3−4t2 + 4t−3 + 4t−1−4t−2 + 2t−3 |
In[5]:=
| Conway[K][z]
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Out[5]=
| 2z6 + 8z4 + 6z2 + 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]=
| { 23, 6 } |
In[8]:=
| Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
| q11−3q10 + 3q9−4q8 + 4q7−3q6 + 3q5−q4 + q3 |
In[9]:=
| HOMFLYPT[K][a, z]
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KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
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Out[9]=
| z6a−6 + z6a−8 + 5z4a−6 + 4z4a−8−z4a−10 + 7z2a−6 + 3z2a−8−4z2a−10 + 3a−6−3a−10 + a−12 |
In[10]:=
| Kauffman[K][a, z]
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KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
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Out[10]=
| z8a−8 + z8a−10 + z7a−7 + 3z7a−9 + 2z7a−11 + z6a−6−3z6a−8−3z6a−10 + z6a−12−3z5a−7−11z5a−9−8z5a−11−5z4a−6 + z4a−8 + 5z4a−10−z4a−12 + 11z3a−9 + 14z3a−11 + 3z3a−13 + 7z2a−6−7z2a−10 + z2a−12 + z2a−14 + 2za−7−4za−9−8za−11−2za−13−3a−6 + 3a−10 + a−12 |
[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["10 134"];
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In[4]:=
| {A = Alexander[K][t], J = Jones[K][q]}
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[4]=
| { 2t3−4t2 + 4t−3 + 4t−1−4t−2 + 2t−3, q11−3q10 + 3q9−4q8 + 4q7−3q6 + 3q5−q4 + q3 } |
In[5]:=
| DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]
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KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.
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KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.
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Out[5]=
| {} |
In[6]:=
| DeleteCases[
Select[
AllKnots[],
(J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) &
],
K
]
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KnotTheory::loading: Loading precomputed data in Jones4Knots11`.
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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 = 6 is the signature of 10 134. Nonzero entries off the critical diagonals (if any exist) are highlighted in red. |
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| Integral Khovanov Homology
(db, data source) |
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[edit] The Coloured Jones Polynomials
| n | Jn |
| 2 | −q29 + 2q28 + q27−6q26 + 6q25 + 3q24−11q23 + 7q22 + 6q21−13q20 + 4q19 + 9q18−12q17 + 10q15−8q14−3q13 + 9q12−3q11−3q10 + 4q9−q7 + q6 |
| 3 | −q58 + 2q57 + q56−q55−5q54−q53 + 10q52 + 3q51−10q50−13q49 + 15q48 + 17q47−11q46−27q45 + 13q44 + 27q43−7q42−30q41 + 6q40 + 27q39−2q38−24q37−q36 + 21q35 + 3q34−13q33−10q32 + 11q31 + 9q30−2q29−15q28−q27 + 10q26 + 10q25−12q24−10q23 + 3q22 + 15q21−3q20−9q19−3q18 + 9q17 + 2q16−3q15−3q14 + 3q13 + q12−q10 + q9 |
| 4 | −q94 + 2q93 + 2q92−3q91−3q90−6q89 + 8q88 + 15q87−q86−12q85−32q84 + 6q83 + 47q82 + 23q81−15q80−78q79−24q78 + 74q77 + 70q76 + 6q75−117q74−71q73 + 77q72 + 105q71 + 39q70−126q69−104q68 + 65q67 + 111q66 + 60q65−116q64−110q63 + 54q62 + 95q61 + 67q60−95q59−105q58 + 42q57 + 72q56 + 70q55−66q54−93q53 + 22q52 + 41q51 + 71q50−27q49−71q48 + q47 + 2q46 + 58q45 + 8q44−36q43−2q42−31q41 + 26q40 + 18q39−4q38 + 16q37−35q36−4q35 + 2q34 + 5q33 + 31q32−16q31−9q30−12q29−4q28 + 24q27−9q24−8q23 + 10q22 + q21 + 3q20−2q19−4q18 + 3q17 + q15−q13 + q12 |
| 5 | q136−q135−3q134 + 4q132 + 5q131 + 6q130−7q129−22q128−13q127 + 12q126 + 40q125 + 39q124−7q123−70q122−86q121−12q120 + 105q119 + 144q118 + 53q117−117q116−225q115−123q114 + 133q113 + 288q112 + 196q111−99q110−349q109−284q108 + 74q107 + 378q106 + 342q105−17q104−387q103−397q102−17q101 + 375q100 + 414q99 + 60q98−360q97−424q96−76q95 + 335q94 + 414q93 + 94q92−314q91−404q90−100q89 + 291q88 + 385q87 + 112q86−261q85−368q84−130q83 + 228q82 + 350q81 + 145q80−179q79−322q78−176q77 + 130q76 + 293q75 + 186q74−66q73−240q72−210q71 + 11q70 + 191q69 + 192q68 + 49q67−120q66−180q65−81q64 + 61q63 + 128q62 + 102q61 + q60−88q59−87q58−31q57 + 24q56 + 67q55 + 49q54 + q53−25q52−31q51−35q50−2q49 + 18q48 + 21q47 + 21q46 + 15q45−19q44−24q43−19q42−6q41 + 14q40 + 28q39 + 10q38−3q37−16q36−18q35−5q34 + 13q33 + 9q32 + 8q31−q30−9q29−7q28 + 4q27 + q26 + 3q25 + 3q24−2q23−3q22 + 2q21 + q18−q16 + q15 |
| 6 | q191−2q190−q189 + 2q188 + q187 + q186−2q185 + 7q184−5q183−9q182−q181−4q180 + q179 + 5q178 + 41q177 + 13q176−15q175−32q174−61q173−58q172−3q171 + 144q170 + 138q169 + 75q168−45q167−211q166−296q165−171q164 + 224q163 + 406q162 + 406q161 + 146q160−324q159−706q158−618q157 + 67q156 + 622q155 + 896q154 + 618q153−183q152−1024q151−1173q150−351q149 + 570q148 + 1238q147 + 1131q146 + 177q145−1046q144−1516q143−763q142 + 320q141 + 1279q140 + 1408q139 + 497q138−881q137−1568q136−955q135 + 105q134 + 1157q133 + 1437q132 + 630q131−729q130−1483q129−962q128 + 11q127 + 1032q126 + 1365q125 + 647q124−627q123−1379q122−923q121−49q120 + 912q119 + 1280q118 + 670q117−485q116−1247q115−909q114−176q113 + 720q112 + 1173q111 + 749q110−234q109−1029q108−898q107−386q106 + 411q105 + 985q104 + 832q103 + 108q102−681q101−802q100−595q99 + 13q98 + 656q97 + 798q96 + 427q95−237q94−533q93−646q92−339q91 + 213q90 + 547q89 + 545q88 + 147q87−132q86−439q85−452q84−157q83 + 153q82 + 377q81 + 266q80 + 181q79−94q78−272q77−247q76−128q75 + 89q74 + 111q73 + 214q72 + 110q71−18q70−96q69−134q68−46q67−70q66 + 62q65 + 73q64 + 65q63 + 33q62−13q61 + 8q60−84q59−27q58−23q57 + 6q56 + 19q55 + 28q54 + 62q53−15q52−2q51−30q50−26q49−27q48−3q47 + 42q46 + 7q45 + 22q44−7q42−25q41−16q40 + 13q39−2q38 + 12q37 + 7q36 + 5q35−9q34−8q33 + 5q32−4q31 + 2q30 + 2q29 + 4q28−2q27−3q26 + 3q25−q24 + q21−q19 + q18 |
[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. |
|
