9 10
From Knot Atlas
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 (KnotPlot image) |
See the full Rolfsen Knot Table. Visit 9 10's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9_10's page at Knotilus! Visit 9 10's page at the original Knot Atlas! |
[edit] Knot presentations
| Planar diagram presentation | X8291 X12,4,13,3 X18,10,1,9 X10,18,11,17 X16,8,17,7 X2,12,3,11 X4,16,5,15 X14,6,15,5 X6,14,7,13 |
| Gauss code | 1, -6, 2, -7, 8, -9, 5, -1, 3, -4, 6, -2, 9, -8, 7, -5, 4, -3 |
| Dowker-Thistlethwaite code | 8 12 14 16 18 2 6 4 10 |
| Conway Notation | [333] |
| Minimum Braid Representative | A Morse Link Presentation | An Arc Presentation | ||||
Length is 11, width is 4, Braid index is 4 | 
|  [{3, 11}, {2, 4}, {5, 3}, {4, 10}, {1, 5}, {11, 9}, {10, 6}, {7, 2}, {6, 8}, {9, 7}, {8, 1}] |
[edit Notes on presentations of 9 10]
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[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
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In[3]:=
| K = Knot["9 10"];
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In[4]:=
| PD[K]
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
| X8291 X12,4,13,3 X18,10,1,9 X10,18,11,17 X16,8,17,7 X2,12,3,11 X4,16,5,15 X14,6,15,5 X6,14,7,13 |
In[5]:=
| GaussCode[K]
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Out[5]=
| 1, -6, 2, -7, 8, -9, 5, -1, 3, -4, 6, -2, 9, -8, 7, -5, 4, -3 |
In[6]:=
| DTCode[K]
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Out[6]=
| 8 12 14 16 18 2 6 4 10 |
(The path below may be different on your system)
In[7]:=
| AppendTo[$Path, "C:/bin/LinKnot/"];
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In[8]:=
| ConwayNotation[K]
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Out[8]=
| [333] |
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,{1,1,2,−1,2,2,2,2,3,−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, 11}, {2, 4}, {5, 3}, {4, 10}, {1, 5}, {11, 9}, {10, 6}, {7, 2}, {6, 8}, {9, 7}, {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 | 4t2−8t + 9−8t−1 + 4t−2 |
| Conway polynomial | 4z4 + 8z2 + 1 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 33, 4 } |
| Jones polynomial | −q11 + q10−3q9 + 5q8−5q7 + 6q6−5q5 + 4q4−2q3 + q2 |
| HOMFLY-PT polynomial (db, data sources) | z4a−4 + 2z4a−6 + z4a−8 + 2z2a−4 + 5z2a−6 + 2z2a−8−z2a−10 + 2a−6 + a−8−2a−10 |
| Kauffman polynomial (db, data sources) | z8a−8 + z8a−10 + 2z7a−7 + 3z7a−9 + z7a−11 + 3z6a−6−z6a−8−3z6a−10 + z6a−12 + 2z5a−5−3z5a−7−7z5a−9−z5a−11 + z5a−13 + z4a−4−7z4a−6 + 3z4a−8 + 9z4a−10−2z4a−12−3z3a−5 + 3z3a−7 + 9z3a−9−z3a−11−4z3a−13−2z2a−4 + 7z2a−6−2z2a−8−11z2a−10−4za−9 + 4za−13−2a−6 + a−8 + 2a−10 |
| The A2 invariant | q−6−q−8 + q−10 + 2q−16 + 2q−20 + q−22 + q−24 + q−26−2q−28−q−30−q−32−q−34 |
| The G2 invariant | q−30−q−32 + 2q−34−3q−36 + 2q−38−q−40−2q−42 + 7q−44−9q−46 + 11q−48−8q−50 + 3q−52 + 5q−54−13q−56 + 21q−58−19q−60 + 12q−62−2q−64−10q−66 + 18q−68−17q−70 + 14q−72−2q−74−7q−76 + 13q−78−9q−80−q−82 + 12q−84−19q−86 + 18q−88−9q−90−4q−92 + 21q−94−28q−96 + 31q−98−20q−100 + 6q−102 + 11q−104−21q−106 + 26q−108−18q−110 + 12q−112 + 2q−114−10q−116 + 15q−118−10q−120−2q−122 + 9q−124−16q−126 + 10q−128−3q−130−12q−132 + 18q−134−20q−136 + 15q−138−10q−140−9q−142 + 13q−144−16q−146 + 13q−148−9q−150 + 2q−152 + 3q−154−4q−156 + 6q−158−6q−160 + 4q−162−q−164 + q−168−2q−170 + 2q−172 + q−176 |
Further Quantum Invariants
Further quantum knot invariants for 9_10.
A1 Invariants.
| Weight | Invariant |
|---|---|
| 1 | q−3−q−5 + 2q−7−q−9 + q−11 + q−13 + 2q−17−2q−19−q−23 |
| 2 | q−6−q−8 + 4q−12−2q−14−3q−16 + 7q−18−q−20−6q−22 + 7q−24 + q−26−5q−28 + 2q−30 + 2q−32−3q−36 + 4q−38 + 3q−40−7q−42 + q−44 + 5q−46−7q−48−2q−50 + 4q−52−3q−54−q−56 + 2q−58 + q−64 |
| 3 | q−9−q−11 + 2q−15 + 2q−17−2q−19−3q−21 + 4q−23 + 7q−25−4q−27−10q−29 + 2q−31 + 17q−33 + 2q−35−20q−37−6q−39 + 21q−41 + 12q−43−20q−45−14q−47 + 17q−49 + 15q−51−9q−53−14q−55 + 3q−57 + 9q−59 + 3q−61−9q−63−9q−65 + 6q−67 + 15q−69−2q−71−20q−73 + q−75 + 20q−77 + 3q−79−23q−81−9q−83 + 16q−85 + 13q−87−16q−89−13q−91 + 8q−93 + 14q−95−2q−97−10q−99 + q−101 + 7q−103 + 2q−105−3q−107 + q−111 + q−113−q−115−q−123 |
| 4 | q−12−q−14 + 2q−18 + 2q−22−3q−24−q−26 + 5q−28 + 5q−32−8q−34−6q−36 + 9q−38 + 6q−40 + 14q−42−17q−44−24q−46 + q−48 + 19q−50 + 47q−52−10q−54−51q−56−33q−58 + 14q−60 + 83q−62 + 23q−64−54q−66−70q−68−17q−70 + 90q−72 + 55q−74−28q−76−73q−78−42q−80 + 56q−82 + 55q−84 + 6q−86−43q−88−44q−90 + 11q−92 + 35q−94 + 26q−96−10q−98−34q−100−28q−102 + 15q−104 + 44q−106 + 17q−108−28q−110−58q−112 + 60q−116 + 42q−118−19q−120−84q−122−19q−124 + 59q−126 + 62q−128 + 5q−130−86q−132−45q−134 + 27q−136 + 66q−138 + 43q−140−53q−142−52q−144−10q−146 + 38q−148 + 55q−150−8q−152−29q−154−28q−156 + 2q−158 + 32q−160 + 9q−162−4q−164−16q−166−9q−168 + 8q−170 + 3q−172 + 4q−174−3q−176−4q−178 + q−180−2q−182 + 2q−184−q−188 + q−190−q−192 + q−200 |
| 5 | q−15−q−17 + 2q−21 + q−27−q−29−q−31 + 3q−33 + 3q−35−2q−37−q−41 + 4q−45 + 6q−47−q−49−9q−51−11q−53−4q−55 + 14q−57 + 26q−59 + 22q−61−9q−63−49q−65−52q−67−8q−69 + 62q−71 + 99q−73 + 56q−75−60q−77−153q−79−118q−81 + 31q−83 + 186q−85 + 199q−87 + 30q−89−197q−91−273q−93−104q−95 + 171q−97 + 314q−99 + 183q−101−115q−103−323q−105−243q−107 + 53q−109 + 287q−111 + 266q−113 + 15q−115−227q−117−259q−119−71q−121 + 158q−123 + 220q−125 + 96q−127−80q−129−173q−131−115q−133 + 26q−135 + 121q−137 + 115q−139 + 26q−141−78q−143−122q−145−65q−147 + 45q−149 + 124q−151 + 100q−153−22q−155−140q−157−139q−159 + 5q−161 + 164q−163 + 177q−165 + 17q−167−179q−169−224q−171−48q−173 + 193q−175 + 264q−177 + 89q−179−176q−181−302q−183−140q−185 + 144q−187 + 300q−189 + 204q−191−76q−193−287q−195−239q−197 + 2q−199 + 221q−201 + 257q−203 + 86q−205−151q−207−235q−209−128q−211 + 58q−213 + 184q−215 + 157q−217 + 16q−219−118q−221−140q−223−62q−225 + 48q−227 + 102q−229 + 76q−231−3q−233−64q−235−64q−237−23q−239 + 22q−241 + 44q−243 + 25q−245−3q−247−20q−249−20q−251−7q−253 + 9q−255 + 11q−257 + 6q−259 + 2q−261−5q−263−6q−265 + q−269 + q−271 + 4q−273−2q−277−q−283 + q−285 + q−287−q−295 |
A2 Invariants.
| Weight | Invariant |
|---|---|
| 1,0 | q−6−q−8 + q−10 + 2q−16 + 2q−20 + q−22 + q−24 + q−26−2q−28−q−30−q−32−q−34 |
| 1,1 | q−12−2q−14 + 4q−16−8q−18 + 17q−20−22q−22 + 32q−24−42q−26 + 58q−28−56q−30 + 60q−32−54q−34 + 43q−36−16q−38−12q−40 + 44q−42−69q−44 + 100q−46−114q−48 + 118q−50−119q−52 + 94q−54−80q−56 + 46q−58−24q−60−8q−62 + 36q−64−46q−66 + 56q−68−56q−70 + 56q−72−44q−74 + 30q−76−28q−78 + 16q−80−12q−82 + 8q−84−4q−86 + 4q−88 + q−92 |
| 2,0 | q−12−q−14−q−16 + 3q−18 + 2q−20−3q−22−q−24 + 4q−26 + 3q−28−2q−30 + q−32 + 4q−34−q−36−2q−38 + 2q−40 + q−42 + 4q−46 + 4q−48 + 2q−54−6q−58−q−60−3q−64−6q−66−3q−68−q−72−q−74 + 2q−78 + q−80 + 2q−82 + q−84 + q−86 |
A3 Invariants.
| Weight | Invariant |
|---|---|
| 0,1,0 | q−12−q−14 + 2q−18−3q−20 + q−22 + 6q−24−4q−26 + q−28 + 7q−30−2q−32 + q−34 + 6q−36 + q−38 + 2q−44−2q−46−4q−48 + 3q−50−8q−54 + 2q−56−q−58−7q−60 + q−62−q−66 + 2q−68 + 2q−70 + q−74 |
| 1,0,0 | q−9−q−11 + q−13−q−15 + q−17 + 2q−21 + q−23 + q−25 + 2q−27 + q−29 + 2q−31 + q−35−2q−37−q−39−2q−41−q−43−q−45 |
A4 Invariants.
| Weight | Invariant |
|---|---|
| 0,1,0,0 | q−18−q−20 + q−24−q−26−q−28 + 3q−30 + 2q−32−2q−34 + q−36 + 5q−38 + 2q−40−2q−42 + 4q−44 + 8q−46 + q−50 + 6q−52 + 3q−54−q−56 + 4q−58 + 5q−60−q−64 + 2q−66−4q−68−11q−70−7q−72−5q−74−10q−76−8q−78 + 2q−82 + q−84 + 2q−86 + 5q−88 + 3q−90 + q−92 + q−94 + q−96 |
| 1,0,0,0 | q−12−q−14 + q−16−q−18 + q−22 + 2q−26 + q−28 + 2q−30 + q−32 + 2q−34 + q−36 + 2q−38 + q−40 + q−44−2q−46−q−48−2q−50−2q−52−q−54−q−56 |
B2 Invariants.
| Weight | Invariant |
|---|---|
| 0,1 | q−12−q−14 + 2q−16−4q−18 + 5q−20−5q−22 + 6q−24−4q−26 + 5q−28−q−30 + 5q−34−6q−36 + 9q−38−10q−40 + 10q−42−10q−44 + 8q−46−6q−48 + 3q−50−2q−54 + 4q−56−5q−58 + 5q−60−5q−62 + 4q−64−3q−66 + 2q−68−2q−70−q−74 |
| 1,0 | q−18−q−22−q−24 + q−26 + 3q−28−4q−32−2q−34 + 4q−36 + 6q−38−q−40−5q−42−2q−44 + 6q−46 + 5q−48−2q−50−4q−52 + 3q−54 + 5q−56 + q−58−3q−60 + 4q−64 + 2q−66−3q−68−3q−70 + 2q−72 + 3q−74−2q−76−4q−78 + q−80 + 5q−82−6q−86−5q−88 + 3q−90 + 4q−92−3q−94−7q−96−3q−98 + 3q−100 + 2q−102−q−104−2q−106 + 2q−110 + 2q−112 + 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 + 6q−34−3q−36 + 4q−38−q−40 + 5q−42 + 2q−44−q−46 + 5q−48−2q−50 + 9q−52−5q−54 + 8q−56−7q−58 + 9q−60−7q−62 + 5q−64−8q−66 + 2q−68−3q−70−q−72−2q−74−4q−76 + 2q−78−5q−80 + 2q−82−6q−84 + 3q−86−4q−88 + 2q−90−2q−92 + 3q−94 + 2q−98 + q−102 |
G2 Invariants.
| Weight | Invariant |
|---|---|
| 1,0 | q−30−q−32 + 2q−34−3q−36 + 2q−38−q−40−2q−42 + 7q−44−9q−46 + 11q−48−8q−50 + 3q−52 + 5q−54−13q−56 + 21q−58−19q−60 + 12q−62−2q−64−10q−66 + 18q−68−17q−70 + 14q−72−2q−74−7q−76 + 13q−78−9q−80−q−82 + 12q−84−19q−86 + 18q−88−9q−90−4q−92 + 21q−94−28q−96 + 31q−98−20q−100 + 6q−102 + 11q−104−21q−106 + 26q−108−18q−110 + 12q−112 + 2q−114−10q−116 + 15q−118−10q−120−2q−122 + 9q−124−16q−126 + 10q−128−3q−130−12q−132 + 18q−134−20q−136 + 15q−138−10q−140−9q−142 + 13q−144−16q−146 + 13q−148−9q−150 + 2q−152 + 3q−154−4q−156 + 6q−158−6q−160 + 4q−162−q−164 + q−168−2q−170 + 2q−172 + 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.
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In[3]:=
| K = Knot["9 10"];
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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]=
| 4t2−8t + 9−8t−1 + 4t−2 |
In[5]:=
| Conway[K][z]
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Out[5]=
| 4z4 + 8z2 + 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]=
| { 33, 4 } |
In[8]:=
| Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
| −q11 + q10−3q9 + 5q8−5q7 + 6q6−5q5 + 4q4−2q3 + q2 |
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]=
| z4a−4 + 2z4a−6 + z4a−8 + 2z2a−4 + 5z2a−6 + 2z2a−8−z2a−10 + 2a−6 + a−8−2a−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−z6a−8−3z6a−10 + z6a−12 + 2z5a−5−3z5a−7−7z5a−9−z5a−11 + z5a−13 + z4a−4−7z4a−6 + 3z4a−8 + 9z4a−10−2z4a−12−3z3a−5 + 3z3a−7 + 9z3a−9−z3a−11−4z3a−13−2z2a−4 + 7z2a−6−2z2a−8−11z2a−10−4za−9 + 4za−13−2a−6 + a−8 + 2a−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 10"];
|
In[4]:=
| {A = Alexander[K][t], J = Jones[K][q]}
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
|
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
|
Out[4]=
| { 4t2−8t + 9−8t−1 + 4t−2, −q11 + q10−3q9 + 5q8−5q7 + 6q6−5q5 + 4q4−2q3 + q2 } |
In[5]:=
| DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]
|
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.
|
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 10. 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−q30 + 3q28−4q27−2q26 + 10q25−10q24−7q23 + 22q22−14q21−15q20 + 32q19−13q18−22q17 + 35q16−11q15−22q14 + 28q13−5q12−16q11 + 15q10−8q8 + 5q7 + q6−2q5 + q4 |
| 3 | −q60 + q59−2q56 + 3q55−q53−5q52 + 8q51 + 5q50−7q49−16q48 + 16q47 + 21q46−13q45−37q44 + 13q43 + 50q42−10q41−62q40−q39 + 76q38 + 7q37−81q36−22q35 + 94q34 + 24q33−90q32−37q31 + 94q30 + 36q29−84q28−43q27 + 77q26 + 41q25−60q24−41q23 + 46q22 + 35q21−28q20−32q19 + 19q18 + 21q17−6q16−17q15 + 4q14 + 9q13−6q11 + q10 + 2q9 + q8−2q7 + q6 |
| 4 | q98−q97−q94 + 3q93−3q92 + q91 + 2q90−5q89 + 6q88−8q87 + 2q86 + 9q85−6q84 + 11q83−25q82−5q81 + 21q80 + 7q79 + 34q78−55q77−35q76 + 20q75 + 28q74 + 97q73−72q72−83q71−22q70 + 27q69 + 193q68−49q67−122q66−94q65−14q64 + 284q63 + 8q62−125q61−172q60−79q59 + 349q58 + 69q57−107q56−232q55−137q54 + 379q53 + 114q52−80q51−261q50−180q49 + 373q48 + 138q47−44q46−252q45−204q44 + 318q43 + 139q42 + 5q41−203q40−203q39 + 220q38 + 108q37 + 50q36−120q35−168q34 + 113q33 + 55q32 + 66q31−43q30−108q29 + 44q28 + 8q27 + 48q26−2q25−51q24 + 16q23−10q22 + 23q21 + 5q20−20q19 + 8q18−7q17 + 8q16 + 3q15−7q14 + 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. |
|
