9 13
From Knot Atlas
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 (KnotPlot image) |
See the full Rolfsen Knot Table. Visit 9 13's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9_13's page at Knotilus! Visit 9 13's page at the original Knot Atlas! |
[edit] Knot presentations
| Planar diagram presentation | X6271 X14,6,15,5 X16,8,17,7 X18,10,1,9 X8,18,9,17 X10,16,11,15 X2,14,3,13 X12,4,13,3 X4,12,5,11 |
| Gauss code | 1, -7, 8, -9, 2, -1, 3, -5, 4, -6, 9, -8, 7, -2, 6, -3, 5, -4 |
| Dowker-Thistlethwaite code | 6 12 14 16 18 4 2 10 8 |
| Conway Notation | [3213] |
| Minimum Braid Representative | A Morse Link Presentation | An Arc Presentation | ||||
Length is 11, width is 4, Braid index is 4 | 
|  [{3, 7}, {8, 6}, {7, 5}, {6, 4}, {5, 9}, {2, 8}, {10, 3}, {9, 11}, {1, 10}, {11, 2}, {4, 1}] |
[edit Notes on presentations of 9 13]
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[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
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In[3]:=
| K = Knot["9 13"];
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In[4]:=
| PD[K]
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
| X6271 X14,6,15,5 X16,8,17,7 X18,10,1,9 X8,18,9,17 X10,16,11,15 X2,14,3,13 X12,4,13,3 X4,12,5,11 |
In[5]:=
| GaussCode[K]
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Out[5]=
| 1, -7, 8, -9, 2, -1, 3, -5, 4, -6, 9, -8, 7, -2, 6, -3, 5, -4 |
In[6]:=
| DTCode[K]
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Out[6]=
| 6 12 14 16 18 4 2 10 8 |
(The path below may be different on your system)
In[7]:=
| AppendTo[$Path, "C:/bin/LinKnot/"];
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In[8]:=
| ConwayNotation[K]
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Out[8]=
| [3213] |
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,{1,1,1,1,2,−1,2,2,3,−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, 7}, {8, 6}, {7, 5}, {6, 4}, {5, 9}, {2, 8}, {10, 3}, {9, 11}, {1, 10}, {11, 2}, {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 | 4t2−9t + 11−9t−1 + 4t−2 |
| Conway polynomial | 4z4 + 7z2 + 1 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 37, 4 } |
| Jones polynomial | −q11 + 2q10−4q9 + 5q8−6q7 + 7q6−5q5 + 4q4−2q3 + q2 |
| HOMFLY-PT polynomial (db, data sources) | z4a−4 + 2z4a−6 + z4a−8 + 2z2a−4 + 5z2a−6 + z2a−8−z2a−10 + 3a−6−a−8−a−10 |
| Kauffman polynomial (db, data sources) | z8a−8 + z8a−10 + 2z7a−7 + 4z7a−9 + 2z7a−11 + 3z6a−6 + z6a−8 + 2z6a−12 + 2z5a−5−2z5a−7−9z5a−9−4z5a−11 + z5a−13 + z4a−4−7z4a−6−4z4a−8−z4a−10−5z4a−12−3z3a−5 + z3a−7 + 9z3a−9 + 2z3a−11−3z3a−13−2z2a−4 + 8z2a−6 + 6z2a−8−2z2a−10 + 2z2a−12 + za−7−3za−9−2za−11 + 2za−13−3a−6−a−8 + a−10 |
| The A2 invariant | q−6−q−8 + q−10 + 3q−16 + q−18 + 2q−20−q−24−2q−28−q−34 |
| The G2 invariant | q−30−q−32 + 2q−34−3q−36 + 2q−38−q−40−2q−42 + 7q−44−8q−46 + 11q−48−9q−50 + 4q−52 + 4q−54−12q−56 + 19q−58−20q−60 + 17q−62−8q−64−4q−66 + 18q−68−22q−70 + 23q−72−13q−74 + q−76 + 10q−78−15q−80 + 13q−82−q−84−8q−86 + 22q−88−18q−90 + 6q−92 + 13q−94−27q−96 + 36q−98−32q−100 + 16q−102 + 4q−104−21q−106 + 34q−108−36q−110 + 24q−112−9q−114−11q−116 + 18q−118−22q−120 + 14q−122−2q−124−10q−126 + 15q−128−14q−130 + 2q−132 + 12q−134−24q−136 + 24q−138−17q−140 + q−142 + 13q−144−23q−146 + 26q−148−20q−150 + 9q−152 + 2q−154−12q−156 + 14q−158−12q−160 + 9q−162−3q−164−q−166 + 3q−168−4q−170 + 3q−172−q−174 + q−176 |
Further Quantum Invariants
Further quantum knot invariants for 9_13.
A1 Invariants.
| Weight | Invariant |
|---|---|
| 1 | q−3−q−5 + 2q−7−q−9 + 2q−11 + q−13−q−15 + q−17−2q−19 + q−21−q−23 |
| 2 | q−6−q−8 + 4q−12−2q−14−2q−16 + 7q−18−3q−20−5q−22 + 9q−24−q−26−6q−28 + 5q−30 + 2q−32−2q−34−3q−36 + 4q−38 + q−40−8q−42 + 3q−44 + 4q−46−8q−48 + q−50 + 6q−52−5q−54−q−56 + 4q−58−q−60−q−62 + q−64 |
| 3 | q−9−q−11 + 2q−15 + 2q−17−2q−19−2q−21 + 4q−23 + 5q−25−5q−27−6q−29 + 7q−31 + 12q−33−8q−35−17q−37 + 7q−39 + 24q−41−3q−43−27q−45−q−47 + 27q−49 + 5q−51−22q−53−11q−55 + 16q−57 + 12q−59−7q−61−12q−63−4q−65 + 14q−67 + 8q−69−14q−71−18q−73 + 13q−75 + 19q−77−10q−79−24q−81 + 7q−83 + 25q−85−q−87−24q−89−4q−91 + 21q−93 + 11q−95−15q−97−13q−99 + 10q−101 + 12q−103−3q−105−10q−107 + 6q−111 + q−113−3q−115−q−117 + q−119 + q−121−q−123 |
| 4 | q−12−q−14 + 2q−18 + 2q−22−3q−24 + 5q−28−2q−30 + 4q−32−6q−34 + 11q−38−2q−40 + 2q−42−19q−44−4q−46 + 25q−48 + 14q−50 + 9q−52−45q−54−34q−56 + 28q−58 + 50q−60 + 47q−62−57q−64−82q−66−5q−68 + 66q−70 + 97q−72−31q−74−100q−76−48q−78 + 39q−80 + 106q−82 + 12q−84−66q−86−63q−88−5q−90 + 70q−92 + 38q−94−15q−96−47q−98−33q−100 + 16q−102 + 47q−104 + 29q−106−32q−108−54q−110−22q−112 + 57q−114 + 62q−116−21q−118−69q−120−53q−122 + 60q−124 + 89q−126−71q−130−84q−132 + 38q−134 + 95q−136 + 39q−138−40q−140−100q−142−10q−144 + 62q−146 + 61q−148 + 14q−150−73q−152−43q−154 + 7q−156 + 45q−158 + 45q−160−25q−162−32q−164−21q−166 + 10q−168 + 33q−170 + 2q−172−7q−174−16q−176−5q−178 + 12q−180 + 2q−182 + 2q−184−4q−186−3q−188 + 3q−190 + q−194−q−196−q−198 + q−200 |
| 5 | q−15−q−17 + 2q−21 + q−27−q−29 + 3q−33 + q−35−3q−37 + 2q−39 + 3q−41 + 3q−43 + 2q−45−4q−47−11q−49−5q−51 + 8q−53 + 20q−55 + 19q−57−3q−59−31q−61−41q−63−17q−65 + 37q−67 + 76q−69 + 55q−71−22q−73−105q−75−120q−77−24q−79 + 126q−81 + 195q−83 + 100q−85−107q−87−264q−89−210q−91 + 54q−93 + 305q−95 + 313q−97 + 39q−99−299q−101−394q−103−148q−105 + 245q−107 + 429q−109 + 237q−111−158q−113−408q−115−294q−117 + 54q−119 + 337q−121 + 315q−123 + 28q−125−245q−127−282q−129−94q−131 + 139q−133 + 235q−135 + 129q−137−59q−139−174q−141−144q−143−9q−145 + 115q−147 + 159q−149 + 67q−151−91q−153−167q−155−104q−157 + 60q−159 + 199q−161 + 152q−163−59q−165−232q−167−192q−169 + 37q−171 + 268q−173 + 258q−175−16q−177−292q−179−309q−181−38q−183 + 295q−185 + 368q−187 + 106q−189−254q−191−395q−193−189q−195 + 181q−197 + 388q−199 + 258q−201−75q−203−334q−205−308q−207−38q−209 + 241q−211 + 304q−213 + 128q−215−122q−217−258q−219−188q−221 + 15q−223 + 178q−225 + 193q−227 + 66q−229−84q−231−159q−233−110q−235 + 14q−237 + 102q−239 + 103q−241 + 35q−243−44q−245−79q−247−49q−249 + 8q−251 + 44q−253 + 42q−255 + 10q−257−17q−259−26q−261−15q−263 + 5q−265 + 13q−267 + 8q−269−2q−273−5q−275−2q−277 + 3q−279 + q−281−q−283−q−289 + q−291 + q−293−q−295 |
A2 Invariants.
| Weight | Invariant |
|---|---|
| 1,0 | q−6−q−8 + q−10 + 3q−16 + q−18 + 2q−20−q−24−2q−28−q−34 |
| 1,1 | q−12−2q−14 + 4q−16−8q−18 + 17q−20−20q−22 + 32q−24−42q−26 + 56q−28−58q−30 + 70q−32−70q−34 + 69q−36−46q−38 + 30q−40−40q−44 + 74q−46−114q−48 + 130q−50−153q−52 + 148q−54−140q−56 + 120q−58−87q−60 + 52q−62−12q−64−18q−66 + 47q−68−72q−70 + 80q−72−78q−74 + 70q−76−62q−78 + 46q−80−30q−82 + 21q−84−12q−86 + 6q−88−2q−90 + q−92 |
| 2,0 | q−12−q−14−q−16 + 3q−18 + 2q−20−3q−22 + 5q−26 + 2q−28−3q−30 + q−32 + 5q−34−q−36−2q−38 + 5q−40 + 3q−42 + q−44 + 2q−46 + 2q−48−3q−50−3q−52−3q−56−7q−58−2q−60 + 2q−62−2q−64−3q−66 + q−68 + 3q−70−2q−74 + q−76 + 2q−78 + q−86 |
A3 Invariants.
| Weight | Invariant |
|---|---|
| 0,1,0 | q−12−q−14 + 2q−18−3q−20 + q−22 + 7q−24−3q−26 + q−28 + 9q−30−2q−32−q−34 + 7q−36−q−38−q−40 + q−44−3q−46−6q−48 + 2q−50−q−52−8q−54 + 3q−56 + 3q−58−6q−60 + 3q−62 + 2q−64−3q−66 + 2q−68 + q−70−q−72 + q−74 |
| 1,0,0 | q−9−q−11 + q−13−q−15 + q−17 + 3q−21 + 2q−23 + 2q−25 + 2q−27−2q−33−2q−37−q−41−q−45 |
A4 Invariants.
| Weight | Invariant |
|---|---|
| 0,1,0,0 | q−18−q−20 + q−24−q−26−q−28 + 3q−30 + 3q−32−q−34 + q−36 + 6q−38 + 4q−40−2q−42 + 3q−44 + 10q−46 + 2q−48 + q−50 + 6q−52 + 5q−54−3q−56−2q−58−6q−62−9q−64−3q−66−3q−68−10q−70−3q−72 + 3q−74−2q−76−4q−78 + 3q−80 + 5q−82−q−86 + 3q−88 + 2q−90−q−92 + q−96 |
| 1,0,0,0 | q−12−q−14 + q−16−q−18 + q−22 + 3q−26 + 2q−28 + 3q−30 + 2q−32 + 2q−34−q−40−2q−42−2q−46−q−50−q−52−q−56 |
B2 Invariants.
| Weight | Invariant |
|---|---|
| 0,1 | q−12−q−14 + 2q−16−4q−18 + 5q−20−5q−22 + 7q−24−5q−26 + 7q−28−3q−30 + 2q−32 + 3q−34−5q−36 + 9q−38−11q−40 + 12q−42−13q−44 + 11q−46−10q−48 + 6q−50−3q−52 + 3q−56−5q−58 + 6q−60−7q−62 + 6q−64−5q−66 + 4q−68−3q−70 + q−72−q−74 |
| 1,0 | q−18−q−22−q−24 + q−26 + 3q−28−4q−32−2q−34 + 4q−36 + 7q−38−5q−42−3q−44 + 6q−46 + 7q−48−q−50−6q−52 + q−54 + 6q−56 + 3q−58−4q−60−2q−62 + 4q−64 + 4q−66−3q−68−5q−70 + q−72 + 3q−74−2q−76−6q−78−q−80 + 4q−82 + q−84−6q−86−6q−88 + 3q−90 + 7q−92−7q−96−4q−98 + 5q−100 + 5q−102−q−104−4q−106−q−108 + 3q−110 + 2q−112−q−114−q−116 + q−120 |
D4 Invariants.
| Weight | Invariant |
|---|---|
| 1,0,0,0 | q−18−q−20 + q−22−2q−24 + 3q−26−4q−28 + 4q−30−3q−32 + 7q−34−3q−36 + 6q−38−q−40 + 8q−42 + q−44 + q−46 + 3q−48−2q−50 + 7q−52−7q−54 + 7q−56−10q−58 + 10q−60−10q−62 + 7q−64−11q−66 + 4q−68−6q−70 + q−72−4q−74−2q−76 + 2q−78−3q−80 + 4q−82−5q−84 + 6q−86−5q−88 + 4q−90−4q−92 + 4q−94−2q−96 + 2q−98−q−100 + q−102 |
G2 Invariants.
| Weight | Invariant |
|---|---|
| 1,0 | q−30−q−32 + 2q−34−3q−36 + 2q−38−q−40−2q−42 + 7q−44−8q−46 + 11q−48−9q−50 + 4q−52 + 4q−54−12q−56 + 19q−58−20q−60 + 17q−62−8q−64−4q−66 + 18q−68−22q−70 + 23q−72−13q−74 + q−76 + 10q−78−15q−80 + 13q−82−q−84−8q−86 + 22q−88−18q−90 + 6q−92 + 13q−94−27q−96 + 36q−98−32q−100 + 16q−102 + 4q−104−21q−106 + 34q−108−36q−110 + 24q−112−9q−114−11q−116 + 18q−118−22q−120 + 14q−122−2q−124−10q−126 + 15q−128−14q−130 + 2q−132 + 12q−134−24q−136 + 24q−138−17q−140 + q−142 + 13q−144−23q−146 + 26q−148−20q−150 + 9q−152 + 2q−154−12q−156 + 14q−158−12q−160 + 9q−162−3q−164−q−166 + 3q−168−4q−170 + 3q−172−q−174 + q−176 |
.
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[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
|
Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.
|
In[3]:=
| K = Knot["9 13"];
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In[4]:=
| Alexander[K][t]
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
|
Out[4]=
| 4t2−9t + 11−9t−1 + 4t−2 |
In[5]:=
| Conway[K][z]
|
Out[5]=
| 4z4 + 7z2 + 1 |
In[6]:=
| Alexander[K, 2][t]
|
KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.
|
Out[6]=
| {1} |
In[7]:=
| {KnotDet[K], KnotSignature[K]}
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Out[7]=
| { 37, 4 } |
In[8]:=
| Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
|
Out[8]=
| −q11 + 2q10−4q9 + 5q8−6q7 + 7q6−5q5 + 4q4−2q3 + q2 |
In[9]:=
| HOMFLYPT[K][a, z]
|
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
|
Out[9]=
| z4a−4 + 2z4a−6 + z4a−8 + 2z2a−4 + 5z2a−6 + z2a−8−z2a−10 + 3a−6−a−8−a−10 |
In[10]:=
| Kauffman[K][a, z]
|
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
|
Out[10]=
| z8a−8 + z8a−10 + 2z7a−7 + 4z7a−9 + 2z7a−11 + 3z6a−6 + z6a−8 + 2z6a−12 + 2z5a−5−2z5a−7−9z5a−9−4z5a−11 + z5a−13 + z4a−4−7z4a−6−4z4a−8−z4a−10−5z4a−12−3z3a−5 + z3a−7 + 9z3a−9 + 2z3a−11−3z3a−13−2z2a−4 + 8z2a−6 + 6z2a−8−2z2a−10 + 2z2a−12 + za−7−3za−9−2za−11 + 2za−13−3a−6−a−8 + a−10 |
[edit] "Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {}
Same Jones Polynomial (up to mirroring, 
):
{}
Computer Talk
The above data is available with the Mathematica package
KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.
(The path below may be different on your system, and possibly also the KnotTheory` date)
In[1]:=
| AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
|
Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
|
In[3]:=
| K = Knot["9 13"];
|
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]=
| { 4t2−9t + 11−9t−1 + 4t−2, −q11 + 2q10−4q9 + 5q8−6q7 + 7q6−5q5 + 4q4−2q3 + q2 } |
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 = 4 is the signature of 9 13. 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 | q31−2q30 + 6q28−7q27−4q26 + 17q25−12q24−13q23 + 29q22−13q21−24q20 + 38q19−10q18−31q17 + 39q16−6q15−28q14 + 28q13−q12−18q11 + 14q10 + q9−8q8 + 5q7 + q6−2q5 + q4 |
| 3 | −q60 + 2q59−2q57−3q56 + 6q55 + 5q54−8q53−13q52 + 13q51 + 20q50−10q49−36q48 + 11q47 + 46q46−61q44−9q43 + 69q42 + 26q41−79q40−40q39 + 83q38 + 55q37−85q36−71q35 + 87q34 + 77q33−79q32−89q31 + 79q30 + 82q29−60q28−85q27 + 52q26 + 71q25−33q24−63q23 + 24q22 + 45q21−9q20−36q19 + 7q18 + 21q17−16q15 + 2q14 + 8q13 + q12−6q11 + q10 + 2q9 + q8−2q7 + q6 |
| 4 | q98−2q97 + 2q95−q94 + 4q93−8q92−q91 + 8q90−q89 + 14q88−25q87−12q86 + 17q85 + 8q84 + 45q83−48q82−43q81 + 6q80 + 15q79 + 115q78−48q77−81q76−44q75−15q74 + 202q73−q72−80q71−116q70−105q69 + 262q68 + 78q67−24q66−173q65−227q64 + 275q63 + 149q62 + 65q61−202q60−340q59 + 259q58 + 197q57 + 148q56−207q55−419q54 + 227q53 + 219q52 + 209q51−189q50−450q49 + 178q48 + 205q47 + 241q46−136q45−418q44 + 103q43 + 147q42 + 238q41−58q40−324q39 + 36q38 + 60q37 + 186q36 + 11q35−196q34 + 5q33−11q32 + 109q31 + 36q30−92q29 + 8q28−33q27 + 47q26 + 25q25−38q24 + 13q23−22q22 + 18q21 + 10q20−17q19 + 9q18−9q17 + 7q16 + 4q15−7q14 + 3q13−2q12 + 2q11 + q10−2q9 + q8 |
[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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