10 153
From Knot Atlas
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 (KnotPlot image) |
See the full Rolfsen Knot Table. Visit 10 153's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 10_153's page at Knotilus! Visit 10 153's page at the original Knot Atlas! |
10_153 is not k-colourable for any k. See The Determinant and the Signature.
[edit] Knot presentations
| Planar diagram presentation | X4251 X8493 X12,6,13,5 X13,18,14,19 X9,16,10,17 X17,10,18,11 X15,20,16,1 X19,14,20,15 X6,12,7,11 X2837 |
| Gauss code | 1, -10, 2, -1, 3, -9, 10, -2, -5, 6, 9, -3, -4, 8, -7, 5, -6, 4, -8, 7 |
| Dowker-Thistlethwaite code | 4 8 12 2 -16 6 -18 -20 -10 -14 |
| Conway Notation | [(3,2)-(21,2)] |
| Minimum Braid Representative | A Morse Link Presentation | An Arc Presentation | ||||
Length is 11, width is 4, Braid index is 4 | 
|  [{3, 9}, {2, 4}, {1, 3}, {10, 5}, {9, 2}, {11, 6}, {5, 7}, {4, 10}, {6, 8}, {7, 11}, {8, 1}] |
[edit Notes on presentations of 10 153]
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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[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
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In[3]:=
| K = Knot["10 153"];
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In[4]:=
| PD[K]
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
| X4251 X8493 X12,6,13,5 X13,18,14,19 X9,16,10,17 X17,10,18,11 X15,20,16,1 X19,14,20,15 X6,12,7,11 X2837 |
In[5]:=
| GaussCode[K]
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Out[5]=
| 1, -10, 2, -1, 3, -9, 10, -2, -5, 6, 9, -3, -4, 8, -7, 5, -6, 4, -8, 7 |
In[6]:=
| DTCode[K]
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Out[6]=
| 4 8 12 2 -16 6 -18 -20 -10 -14 |
(The path below may be different on your system)
In[7]:=
| AppendTo[$Path, "C:/bin/LinKnot/"];
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In[8]:=
| ConwayNotation[K]
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Out[8]=
| [(3,2)-(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(4,{−1,−1,−1,−2,−1,−1,3,2,2,2,3}) |
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[{3, 9}, {2, 4}, {1, 3}, {10, 5}, {9, 2}, {11, 6}, {5, 7}, {4, 10}, {6, 8}, {7, 11}, {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 | t3−t2−t + 3−t−1−t−2 + t−3 |
| Conway polynomial | z6 + 5z4 + 4z2 + 1 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 1, 0 } |
| Jones polynomial | −q4 + q3−q2 + q + 1 + q−2−q−3 + q−4−q−5 |
| HOMFLY-PT polynomial (db, data sources) | z6−z4a−2 + 6z4−a4z2−a2z2−4z2a−2 + 10z2−a4−a2−3a−2 + 6 |
| Kauffman polynomial (db, data sources) | z8a−2 + z8 + az7 + 2z7a−1 + z7a−3 + a4z6−6z6a−2−7z6 + a5z5 + a3z5−7az5−13z5a−1−6z5a−3−4a4z4 + 10z4a−2 + 14z4−4a5z3−4a3z3 + 12az3 + 22z3a−1 + 10z3a−3 + 3a4z2−2a2z2−7z2a−2−12z2 + 3a5z + 2a3z−6az−10za−1−5za−3−a4 + a2 + 3a−2 + 6 |
| The A2 invariant | −q16−q12−q10 + 2q4 + 2q2 + 3 + 2q−2−q−8−q−10−q−12 |
| The G2 invariant | q80 + q76−q74 + q70−2q68 + q64−3q62 + q60−q58−4q56 + 5q54−5q52 + q50 + q48−5q46 + 5q44−3q42−2q40 + 2q38−4q36 + q34 + 3q32−5q30 + 3q28−q24 + 2q22−2q20 + 2q18 + 4q14−2q12 + 3q10 + 4q8−q6 + 5q4−q2 + 2 + 6q−2−q−4 + 2q−6 + 4q−8−2q−10 + 6q−12−q−14−3q−16 + 4q−18−4q−20 + 2q−22−3q−26 + q−28−q−30−3q−32−2q−36−q−38−3q−42−q−48−q−52 + q−56 + q−60 |
Further Quantum Invariants
Further quantum knot invariants for 10_153.
A1 Invariants.
| Weight | Invariant |
|---|---|
| 1 | −q11 + q3 + q + 2q−1−q−9 |
| 2 | q32−q28 + q24−3q20−q18 + 2q16−2q14−q12 + 2q10 + q6 + q4 + 2q2 + 1 + q−2 + 2q−4−2q−8 + q−10 + q−12−2q−14−q−16−q−24 + q−28 |
| 3 | −q63 + q59 + q57−2q53−q51 + q49 + 4q47 + 3q45−2q43−5q41−2q39 + 5q37 + 4q35−4q33−7q31−q29 + 5q27 + q25−4q23−2q21 + q19 + q17 + q15 + q9 + q7−q5 + q3 + 5q + 4q−1−2q−3−3q−5 + 6q−7 + 5q−9−3q−11−7q−13−q−15 + 5q−17 + 2q−19−3q−21−5q−23−2q−25 + 3q−27 + 3q−29−3q−33−2q−35 + q−37 + q−39 + q−41−q−47 + q−51 + q−53−q−57 |
A2 Invariants.
| Weight | Invariant |
|---|---|
| 1,0 | −q16−q12−q10 + 2q4 + 2q2 + 3 + 2q−2−q−8−q−10−q−12 |
| 1,1 | q44 + 2q40−2q38 + 2q34−4q32 + 6q30−8q28 + 4q26−2q24−4q22 + 2q20−12q18 + 4q16−6q14 + 3q12 + 6q8 + 4q6 + 5q4 + 8q2 + 10q−2−2q−4 + 4q−6−4q−10 + 4q−12−6q−14−2q−16−2q−18−4q−20 + 2q−22−4q−24 + 2q−32 + q−36 |
| 2,0 | q42 + q36 + q34 + q32−q28−2q26−5q24−2q22−3q20−3q18−q16 + q12 + 3q8 + 3q6 + 6q4 + 5q2 + 8 + 4q−2 + 2q−4 + q−6−2q−8−2q−10−2q−12−2q−14−2q−16−3q−18−2q−20−q−22−q−24 + q−30 + q−32 + q−34 |
A3 Invariants.
| Weight | Invariant |
|---|---|
| 0,1,0 | q34 + q30−q24−3q22−q20−3q18−3q16−q14−2q12−2q10 + q8 + 3q6 + 5q4 + 8q2 + 10 + 7q−2 + 4q−4−q−6−2q−8−7q−10−5q−12−3q−14−3q−16 + q−20 + q−22 + q−26 |
| 1,0,0 | −q21−q17−q15−q13−q11 + 2q5 + 3q3 + 4q + 3q−1 + 3q−3−q−7−q−9−2q−11−q−13−q−15 |
A4 Invariants.
| Weight | Invariant |
|---|---|
| 0,1,0,0 | q44 + q40 + 2q38 + q36 + q34 + q32−q30−3q28−3q26−5q24−6q22−7q20−6q18−7q16−8q14−2q12 + q10 + 4q8 + 13q6 + 19q4 + 20q2 + 20 + 17q−2 + 8q−4−8q−8−11q−10−15q−12−13q−14−8q−16−5q−18−2q−20 + 2q−22 + 3q−24 + 2q−26 + 2q−28 + q−30 + q−32 |
| 1,0,0,0 | −q26−q22−q20−q18−q16−q14−q12 + 2q6 + 3q4 + 5q2 + 4 + 4q−2 + 3q−4−q−8−2q−10−2q−12−2q−14−q−16−q−18 |
B2 Invariants.
| Weight | Invariant |
|---|---|
| 0,1 | −q34−q30−q24 + q22−q20 + q18−q16 + q14 + q8−q6 + 3q4 + 2 + q−2 + 2q−4 + q−6 + q−10−q−12 + q−14−q−16−q−20−q−22−q−26 |
| 1,0 | q56 + q48−q44 + q40−q38−2q36−q34−q30−3q28−2q26−2q20−q18 + q14 + q12 + q10 + 2q8 + 4q6 + 3q4 + 4q2 + 3 + 4q−2 + 3q−4 + 3q−6−q−8−q−14−3q−16−3q−18−q−20−q−22−3q−24−2q−26 + q−36 + q−44 |
D4 Invariants.
| Weight | Invariant |
|---|---|
| 1,0,0,0 | q46 + q42 + q38−2q34−q32−3q30−q28−3q26−q24−3q22−q20−q18−2q16−2q14−2q12 + q10 + q8 + 6q6 + 6q4 + 10q2 + 10 + 10q−2 + 6q−4 + 4q−6−4q−10−5q−12−7q−14−5q−16−6q−18−2q−20−2q−22 + q−26 + q−28 + q−30 + q−34 |
G2 Invariants.
| Weight | Invariant |
|---|---|
| 1,0 | q80 + q76−q74 + q70−2q68 + q64−3q62 + q60−q58−4q56 + 5q54−5q52 + q50 + q48−5q46 + 5q44−3q42−2q40 + 2q38−4q36 + q34 + 3q32−5q30 + 3q28−q24 + 2q22−2q20 + 2q18 + 4q14−2q12 + 3q10 + 4q8−q6 + 5q4−q2 + 2 + 6q−2−q−4 + 2q−6 + 4q−8−2q−10 + 6q−12−q−14−3q−16 + 4q−18−4q−20 + 2q−22−3q−26 + q−28−q−30−3q−32−2q−36−q−38−3q−42−q−48−q−52 + q−56 + q−60 |
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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["10 153"];
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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−t2−t + 3−t−1−t−2 + t−3 |
In[5]:=
| Conway[K][z]
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Out[5]=
| z6 + 5z4 + 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]=
| { 1, 0 } |
In[8]:=
| Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
| −q4 + q3−q2 + q + 1 + q−2−q−3 + q−4−q−5 |
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]=
| z6−z4a−2 + 6z4−a4z2−a2z2−4z2a−2 + 10z2−a4−a2−3a−2 + 6 |
In[10]:=
| Kauffman[K][a, z]
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KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
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Out[10]=
| z8a−2 + z8 + az7 + 2z7a−1 + z7a−3 + a4z6−6z6a−2−7z6 + a5z5 + a3z5−7az5−13z5a−1−6z5a−3−4a4z4 + 10z4a−2 + 14z4−4a5z3−4a3z3 + 12az3 + 22z3a−1 + 10z3a−3 + 3a4z2−2a2z2−7z2a−2−12z2 + 3a5z + 2a3z−6az−10za−1−5za−3−a4 + a2 + 3a−2 + 6 |
[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.
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In[3]:=
| K = Knot["10 153"];
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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]=
| { t3−t2−t + 3−t−1−t−2 + t−3, −q4 + q3−q2 + q + 1 + q−2−q−3 + q−4−q−5 } |
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] Vassiliev invariants
| V2 and V3: | (4, -1) |
| V2,1 through V6,9: |
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V2,1 through V6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V2 and V3.
[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 10 153. 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 | q13−q12−q11 + 2q10−q9−q8 + q7−2q6 + 2q5 + q4−5q3 + 4q2 + 3q−6 + 4q−1 + 4q−2−7q−3 + 4q−4 + 3q−5−5q−6 + q−7 + 2q−8−q−9−2q−10 + 2q−12−q−13−q−14 + q−15 |
| 3 | −q27 + q26 + q25−2q23 + 2q21−q19 + 2q17−3q16−2q15 + 3q14 + 5q13−3q12−7q11 + 7q9 + 2q8−4q7−6q6 + q5 + 6q4 + 4q3−5q2−8q + 7 + 10q−1−4q−2−12q−3 + 5q−4 + 12q−5−4q−6−13q−7 + 5q−8 + 13q−9−4q−10−13q−11 + 2q−12 + 11q−13 + q−14−9q−15−4q−16 + 5q−17 + 4q−18−q−19−3q−20−2q−21 + q−22 + 2q−23 + 2q−24−q−25−2q−26 + q−28 + q−29−q−30 |
| 4 | q46−q45−q44 + 3q41−q40−q39−q38−q37 + 3q36−q35−q33 + 2q32 + 3q31−3q30−3q29−5q28 + 5q27 + 6q26 + 3q25−q24−10q23−q22 + 4q20 + 6q19 + q18 + 2q17−8q16−10q15−3q14 + 8q13 + 17q12 + 7q11−17q10−22q9−5q8 + 20q7 + 32q6−5q5−29q4−24q3 + 7q2 + 48q + 10−28q−1−34q−2−4q−3 + 54q−4 + 14q−5−26q−6−36q−7−7q−8 + 55q−9 + 14q−10−26q−11−35q−12−8q−13 + 53q−14 + 16q−15−21q−16−36q−17−15q−18 + 44q−19 + 21q−20−5q−21−29q−22−26q−23 + 19q−24 + 19q−25 + 16q−26−9q−27−23q−28−4q−29 + 2q−30 + 15q−31 + 7q−32−4q−33−5q−34−6q−35 + 3q−37 + 3q−38 + 2q−39 + q−40−3q−41−2q−42−q−43 + 3q−45−q−48−q−49 + q−50 |
| 5 | −q70 + q69 + q68−q65−2q64 + 2q62 + q61 + q60−q59−q58−q57 + q55 + q54−3q53−q52 + 2q51 + 2q50 + 4q49 + 2q48−5q47−7q46−3q45 + 5q43 + 8q42 + 4q41−q40−4q39−5q38−6q37−3q36 + 4q34 + 11q33 + 11q32 + 6q31−8q30−19q29−20q28−6q27 + 13q26 + 31q25 + 26q24 + 2q23−23q22−41q21−29q20 + 5q19 + 37q18 + 49q17 + 25q16−20q15−55q14−53q13−14q12 + 49q11 + 74q10 + 40q9−23q8−80q7−75q6 + 4q5 + 83q4 + 87q3 + 22q2−72q−108−33q−1 + 73q−2 + 106q−3 + 47q−4−66q−5−117q−6−46q−7 + 68q−8 + 111q−9 + 51q−10−65q−11−118q−12−47q−13 + 68q−14 + 112q−15 + 50q−16−65q−17−118q−18−48q−19 + 67q−20 + 113q−21 + 52q−22−59q−23−114q−24−60q−25 + 48q−26 + 106q−27 + 70q−28−25q−29−92q−30−79q−31−5q−32 + 67q−33 + 78q−34 + 33q−35−32q−36−65q−37−49q−38−4q−39 + 39q−40 + 48q−41 + 29q−42−8q−43−33q−44−32q−45−16q−46 + 10q−47 + 25q−48 + 21q−49 + 6q−50−5q−51−17q−52−13q−53−q−54 + 2q−55 + 7q−56 + 8q−57 + 2q−58−2q−59−q−60−4q−61−4q−62 + q−64 + 2q−65 + 2q−66 + 2q−67−q−68−2q−69−q−70 + q−73 + q−74−q−75 |
| 6 | q99−q98−q97 + q94 + 3q92−q91−2q90−q89−q88 + 4q85−q84−2q80 + q79 + 4q78−4q77−2q76−2q75−3q73 + 6q72 + 9q71 + q70−q69−2q68−5q67−13q66 + 3q64 + 3q63 + 4q62 + 10q61 + 8q60−5q59−3q57−11q56−16q55−6q54 + 3q53 + 5q52 + 17q51 + 28q50 + 16q49−5q48−20q47−25q46−38q45−24q44 + 16q43 + 33q42 + 46q41 + 36q40 + 30q39−28q38−59q37−54q36−50q35−8q34 + 32q33 + 99q32 + 69q31 + 44q30−8q29−84q28−117q27−106q26 + 8q25 + 56q24 + 146q23 + 147q22 + 61q21−73q20−184q19−159q18−119q17 + 72q16 + 200q15 + 236q14 + 110q13−85q12−214q11−287q10−100q9 + 110q8 + 293q7 + 262q6 + 75q5−165q4−352q3−225q2−q + 276 + 325q−1 + 175q−2−116q−3−361q−4−273q−5−54q−6 + 258q−7 + 341q−8 + 208q−9−102q−10−362q−11−281q−12−65q−13 + 255q−14 + 343q−15 + 213q−16−101q−17−363q−18−282q−19−65q−20 + 255q−21 + 343q−22 + 213q−23−102q−24−359q−25−285q−26−70q−27 + 250q−28 + 346q−29 + 226q−30−93q−31−343q−32−298q−33−106q−34 + 208q−35 + 337q−36 + 270q−37−19q−38−269q−39−305q−40−194q−41 + 69q−42 + 253q−43 + 300q−44 + 121q−45−82q−46−215q−47−241q−48−116q−49 + 55q−50 + 195q−51 + 173q−52 + 109q−53−15q−54−123q−55−155q−56−97q−57−q−58 + 51q−59 + 109q−60 + 92q−61 + 39q−62−32q−63−54q−64−64q−65−58q−66−4q−67 + 28q−68 + 47q−69 + 33q−70 + 30q−71−q−72−27q−73−28q−74−22q−75−8q−76 + 18q−78 + 16q−79 + 10q−80 + 3q−81−2q−82−7q−83−8q−84−5q−85−2q−86 + q−87 + 2q−88 + 4q−89 + 3q−90 + 3q−91−q−92−q−93−2q−94−2q−95−2q−96 + 3q−98 + q−100−q−103−q−104 + q−105 |
[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. |
|
