9 5
From Knot Atlas
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 (KnotPlot image) |
See the full Rolfsen Knot Table. Visit 9 5's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9_5's page at Knotilus! Visit 9 5's page at the original Knot Atlas! |
[edit] Knot presentations
| Planar diagram presentation | X6271 X14,6,15,5 X18,8,1,7 X16,10,17,9 X10,16,11,15 X8,18,9,17 X2,14,3,13 X12,4,13,3 X4,12,5,11 |
| Gauss code | 1, -7, 8, -9, 2, -1, 3, -6, 4, -5, 9, -8, 7, -2, 5, -4, 6, -3 |
| Dowker-Thistlethwaite code | 6 12 14 18 16 4 2 10 8 |
| Conway Notation | [513] |
| Minimum Braid Representative | A Morse Link Presentation | An Arc Presentation | |||||
Length is 12, width is 5, Braid index is 5 | 
|  [{3, 5}, {6, 4}, {5, 7}, {8, 6}, {7, 9}, {2, 8}, {10, 3}, {9, 11}, {1, 10}, {11, 2}, {4, 1}] |
[edit Notes on presentations of 9 5]
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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[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
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In[3]:=
| K = Knot["9 5"];
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In[4]:=
| PD[K]
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
| X6271 X14,6,15,5 X18,8,1,7 X16,10,17,9 X10,16,11,15 X8,18,9,17 X2,14,3,13 X12,4,13,3 X4,12,5,11 |
In[5]:=
| GaussCode[K]
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Out[5]=
| 1, -7, 8, -9, 2, -1, 3, -6, 4, -5, 9, -8, 7, -2, 5, -4, 6, -3 |
In[6]:=
| DTCode[K]
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Out[6]=
| 6 12 14 18 16 4 2 10 8 |
(The path below may be different on your system)
In[7]:=
| AppendTo[$Path, "C:/bin/LinKnot/"];
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In[8]:=
| ConwayNotation[K]
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Out[8]=
| [513] |
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[{3, 5}, {6, 4}, {5, 7}, {8, 6}, {7, 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 | 6t−11 + 6t−1 |
| Conway polynomial | 6z2 + 1 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 23, 2 } |
| Jones polynomial | −q10 + q9−2q8 + 3q7−3q6 + 4q5−3q4 + 3q3−2q2 + q |
| HOMFLY-PT polynomial (db, data sources) | z2a−2 + 2z2a−4 + 2z2a−6 + z2a−8 + a−4 + a−6−a−10 |
| Kauffman polynomial (db, data sources) | z8a−8 + z8a−10 + 2z7a−7 + 3z7a−9 + z7a−11 + 3z6a−6−2z6a−8−5z6a−10 + 3z5a−5−5z5a−7−14z5a−9−6z5a−11 + 3z4a−4−7z4a−6−3z4a−8 + 7z4a−10 + 2z3a−3−4z3a−5 + z3a−7 + 18z3a−9 + 11z3a−11 + z2a−2−3z2a−4 + 3z2a−6 + 4z2a−8−3z2a−10−6za−9−6za−11 + a−4−a−6 + a−10 |
| The A2 invariant | q−2−q−4 + q−8 + q−12 + q−14 + q−16 + q−18 + q−22−q−26−q−30−q−32 |
| The G2 invariant | q−10−q−12 + q−14−q−16−q−22 + 3q−24−2q−26 + 2q−28−q−30 + q−34−q−36 + 3q−38−2q−40 + q−42 + 2q−48−q−50 + q−52−q−54 + q−56−q−60 + q−62 + q−66 + q−72 + q−74 + 2q−78−2q−80 + 5q−82−q−84−q−86 + 5q−88−4q−90 + 6q−92−2q−94−q−96 + 3q−98−2q−100 + 4q−102−3q−104−q−110 + q−112−2q−114−q−116 + q−118−3q−120−2q−124−2q−126 + 3q−128−6q−130 + 3q−132−2q−134−2q−136 + 4q−138−5q−140 + 3q−142−q−144 + q−148−2q−150 + 2q−152 + q−156 |
Further Quantum Invariants
Further quantum knot invariants for 9_5.
A1 Invariants.
| Weight | Invariant |
|---|---|
| 1 | q−1−q−3 + q−5 + q−9 + q−11 + q−15−q−17−q−21 |
| 2 | q−2−q−4 + 2q−8−q−10 + q−12 + q−14−q−16 + q−18 + q−20 + 2q−26−q−30 + q−34−q−36−q−38 + q−40−q−42−2q−44 + q−46−2q−50 + q−52 + q−54−q−56 + q−60 |
| 3 | q−3−q−5 + q−9 + q−11−q−15 + q−17 + q−19 + q−21−q−25 + q−27 + 2q−29 + q−31−3q−33 + 2q−37 + 2q−39−q−43−q−45 + q−47 + 3q−49 + q−51−3q−53−2q−55 + 2q−57−q−59−3q−61−2q−63 + q−65−q−71 + q−73 + q−75−2q−79−q−81 + q−83 + 2q−85−q−87−2q−89 + 3q−93 + 2q−95−2q−97−2q−99 + q−101 + 3q−103−2q−107−q−109 + q−111 + q−113−q−117 |
| 4 | q−4−q−6 + q−10 + 2q−14−2q−16 + q−20 + q−22 + 4q−24−4q−26−2q−28 + q−30 + 4q−32 + 5q−34−5q−36−5q−38 + 7q−42 + 8q−44−4q−46−8q−48−3q−50 + 6q−52 + 10q−54−7q−58−8q−60−q−62 + 7q−64 + 4q−66 + 2q−68−3q−70−6q−72−q−74 + 4q−76 + 6q−78 + 2q−80−6q−82−6q−84−q−86 + 4q−88 + 3q−90−4q−92−6q−94 + 3q−98 + q−100−4q−102−5q−104 + 2q−106 + 5q−108 + 3q−110−3q−112−5q−114 + 2q−116 + 5q−118 + 5q−120 + q−122−5q−124−2q−126−q−128 + 3q−130 + 4q−132−2q−134−2q−136−4q−138−q−140 + 5q−142 + 3q−144 + 3q−146−3q−148−5q−150−q−152 + q−154 + 6q−156 + 2q−158−3q−160−4q−162−4q−164 + 3q−166 + 4q−168 + 2q−170−q−172−5q−174−q−176 + q−178 + 2q−180 + 2q−182−q−184−q−186−q−188 + q−192 |
| 5 | q−5−q−7 + q−11 + q−15−q−19 + 2q−23 + 2q−25−2q−29−3q−31 + q−33 + 5q−35 + 6q−37−2q−39−7q−41−5q−43 + 3q−45 + 11q−47 + 8q−49−4q−51−13q−53−7q−55 + 7q−57 + 14q−59 + 8q−61−7q−63−17q−65−11q−67 + 8q−69 + 21q−71 + 13q−73−4q−75−17q−77−18q−79−q−81 + 17q−83 + 17q−85 + 5q−87−8q−89−17q−91−12q−93−q−95 + 10q−97 + 14q−99 + 8q−101−2q−103−13q−105−15q−107−6q−109 + 8q−111 + 15q−113 + 10q−115−3q−117−13q−119−11q−121 + q−123 + 8q−125 + 7q−127−2q−129−8q−131−5q−133 + 2q−135 + 5q−137 + 4q−139−6q−141−10q−143−3q−145 + 7q−147 + 12q−149 + 8q−151−6q−153−14q−155−8q−157 + 5q−159 + 15q−161 + 14q−163 + q−165−13q−167−14q−169−3q−171 + 10q−173 + 17q−175 + 8q−177−6q−179−14q−181−12q−183 + q−185 + 10q−187 + 11q−189 + 4q−191−5q−193−11q−195−10q−197−q−199 + 6q−201 + 8q−203 + 7q−205 + q−207−6q−209−8q−211−4q−213 + 6q−217 + 8q−219 + 5q−221−2q−223−7q−225−8q−227−5q−229 + 2q−231 + 8q−233 + 8q−235 + 3q−237−4q−239−8q−241−7q−243−q−245 + 5q−247 + 8q−249 + 5q−251−q−253−5q−255−6q−257−3q−259 + 2q−261 + 5q−263 + 3q−265 + q−267−q−269−3q−271−2q−273 + q−277 + q−279 + q−281−q−285 |
A2 Invariants.
| Weight | Invariant |
|---|---|
| 1,0 | q−2−q−4 + q−8 + q−12 + q−14 + q−16 + q−18 + q−22−q−26−q−30−q−32 |
| 1,1 | q−4−2q−6 + 2q−8−2q−10 + 5q−12−4q−14 + 4q−16−2q−18 + 5q−20−4q−22 + 4q−24 + 6q−28 + 4q−32 + 2q−34 + q−36 + 2q−38−4q−40 + 2q−42−11q−44 + 10q−46−16q−48 + 12q−50−16q−52 + 14q−54−10q−56 + 8q−58−3q−60 + 2q−62 + 4q−64−8q−66 + 7q−68−12q−70 + 10q−72−10q−74 + 6q−76−4q−78 + 4q−80 + q−84 |
| 2,0 | q−4−q−6−q−8 + 2q−10 + q−12−q−14 + 2q−18 + q−20−2q−22 + 3q−26 + 2q−28 + q−30 + 2q−32 + q−34 + 2q−36 + q−38−q−42 + q−46−2q−50−q−52−q−56−3q−58−2q−60−2q−66−q−68 + q−70 + q−72−q−76 + q−78 + q−80 + q−82 |
A3 Invariants.
| Weight | Invariant |
|---|---|
| 0,1,0 | q−4−q−6−q−8 + 2q−10−q−14 + 2q−16 + q−18 + 2q−22 + 2q−24 + q−28 + q−30 + q−32 + 2q−36 + 3q−38 + q−40 + q−44−3q−46−3q−48−3q−50−4q−52−2q−54 + q−60 + 2q−62 + q−66 |
| 1,0,0 | q−3−q−5 + q−11 + q−15 + q−17 + q−19 + q−21 + q−23 + q−25 + q−29−q−35−q−39−q−41−q−43 |
B2 Invariants.
| Weight | Invariant |
|---|---|
| 0,1 | q−4−q−6 + q−8−2q−10 + 2q−12−q−14 + 2q−16−q−18 + 2q−20 + 2q−26−q−28 + 3q−30−3q−32 + 4q−34−4q−36 + 3q−38−3q−40 + 2q−42−q−44 + q−46 + q−48−q−50 + 2q−52−2q−54 + 2q−56−2q−58 + q−60−2q−62−q−66 |
| 1,0 | q−6−q−10−q−12 + 2q−16 + q−18−q−20−q−22 + 2q−26 + q−28−q−32 + q−34 + 2q−36 + q−38−q−40 + q−44 + 2q−46−q−50 + 2q−54 + q−56 + q−60 + 2q−62 + q−64−q−66−q−68 + q−70 + q−72−q−74−4q−76−2q−78−2q−84−3q−86−q−88 + q−90 + q−92−q−94−q−96 + q−98 + 2q−100 + q−108 |
G2 Invariants.
| Weight | Invariant |
|---|---|
| 1,0 | q−10−q−12 + q−14−q−16−q−22 + 3q−24−2q−26 + 2q−28−q−30 + q−34−q−36 + 3q−38−2q−40 + q−42 + 2q−48−q−50 + q−52−q−54 + q−56−q−60 + q−62 + q−66 + q−72 + q−74 + 2q−78−2q−80 + 5q−82−q−84−q−86 + 5q−88−4q−90 + 6q−92−2q−94−q−96 + 3q−98−2q−100 + 4q−102−3q−104−q−110 + q−112−2q−114−q−116 + q−118−3q−120−2q−124−2q−126 + 3q−128−6q−130 + 3q−132−2q−134−2q−136 + 4q−138−5q−140 + 3q−142−q−144 + q−148−2q−150 + 2q−152 + q−156 |
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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["9 5"];
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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]=
| 6t−11 + 6t−1 |
In[5]:=
| Conway[K][z]
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Out[5]=
| 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, 2 } |
In[8]:=
| Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
| −q10 + q9−2q8 + 3q7−3q6 + 4q5−3q4 + 3q3−2q2 + q |
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]=
| z2a−2 + 2z2a−4 + 2z2a−6 + z2a−8 + a−4 + a−6−a−10 |
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 + 2z7a−7 + 3z7a−9 + z7a−11 + 3z6a−6−2z6a−8−5z6a−10 + 3z5a−5−5z5a−7−14z5a−9−6z5a−11 + 3z4a−4−7z4a−6−3z4a−8 + 7z4a−10 + 2z3a−3−4z3a−5 + z3a−7 + 18z3a−9 + 11z3a−11 + z2a−2−3z2a−4 + 3z2a−6 + 4z2a−8−3z2a−10−6za−9−6za−11 + a−4−a−6 + 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`
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Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
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In[3]:=
| K = Knot["9 5"];
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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]=
| { 6t−11 + 6t−1, −q10 + q9−2q8 + 3q7−3q6 + 4q5−3q4 + 3q3−2q2 + q } |
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 = 2 is the signature of 9 5. 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−q28−q27 + 3q26−q25−4q24 + 5q23−7q21 + 6q20 + 2q19−9q18 + 6q17 + 4q16−10q15 + 5q14 + 5q13−8q12 + 3q11 + 5q10−7q9 + 3q8 + 3q7−5q6 + 3q5 + q4−2q3 + q2 |
| 3 | −q57 + q56 + q55−3q53 + 3q51 + 3q50−5q49−3q48 + 3q47 + 7q46−4q45−6q44 + q43 + 8q42−q41−7q40−q39 + 7q38 + q37−6q36−q35 + 5q34 + 2q33−6q32 + 2q30 + q29−4q28 + 3q27−2q26 + 5q23−4q22−2q21 + 6q19−2q18−2q17−2q16 + 3q15 + 2q14−q13−3q12 + q11 + 3q10−3q8 + q7 + q6 + q5−2q4 + q3 |
| 4 | q94−q93−q92 + 4q89−q88−2q87−2q86−4q85 + 8q84 + 2q83−3q81−11q80 + 8q79 + 3q78 + 5q77 + q76−16q75 + 6q74−q73 + 7q72 + 7q71−16q70 + 8q69−7q68 + 4q67 + 9q66−16q65 + 14q64−8q63 + 8q61−19q60 + 20q59−4q58 + 5q56−26q55 + 22q54 + 2q53 + 2q52 + 2q51−33q50 + 23q49 + 7q48 + 4q47−q46−39q45 + 25q44 + 14q43 + 5q42−6q41−44q40 + 25q39 + 22q38 + 9q37−8q36−49q35 + 20q34 + 25q33 + 14q32−6q31−46q30 + 12q29 + 18q28 + 15q27 + q26−36q25 + 8q24 + 9q23 + 10q22 + 5q21−24q20 + 7q19 + 2q18 + 5q17 + 5q16−14q15 + 6q14−q13 + 2q12 + 3q11−6q10 + 3q9−q8 + q7 + q6−2q5 + q4 |
[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. |
|
