9 23
From Knot Atlas
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 (KnotPlot image) |
See the full Rolfsen Knot Table. Visit 9 23's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9_23's page at Knotilus! Visit 9 23's page at the original Knot Atlas! |
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[edit] Knot presentations
| Planar diagram presentation | X1425 X3,10,4,11 X5,12,6,13 X7,16,8,17 X13,18,14,1 X17,14,18,15 X15,6,16,7 X11,8,12,9 X9,2,10,3 |
| Gauss code | -1, 9, -2, 1, -3, 7, -4, 8, -9, 2, -8, 3, -5, 6, -7, 4, -6, 5 |
| Dowker-Thistlethwaite code | 4 10 12 16 2 8 18 6 14 |
| Conway Notation | [22122] |
| Minimum Braid Representative | A Morse Link Presentation | An Arc Presentation | ||||
Length is 11, width is 4, Braid index is 4 | 
|  [{11, 4}, {3, 9}, {8, 10}, {9, 11}, {10, 5}, {4, 6}, {5, 2}, {1, 3}, {2, 7}, {6, 8}, {7, 1}] |
[edit Notes on presentations of 9 23]
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[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
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In[3]:=
| K = Knot["9 23"];
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In[4]:=
| PD[K]
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
| X1425 X3,10,4,11 X5,12,6,13 X7,16,8,17 X13,18,14,1 X17,14,18,15 X15,6,16,7 X11,8,12,9 X9,2,10,3 |
In[5]:=
| GaussCode[K]
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Out[5]=
| -1, 9, -2, 1, -3, 7, -4, 8, -9, 2, -8, 3, -5, 6, -7, 4, -6, 5 |
In[6]:=
| DTCode[K]
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Out[6]=
| 4 10 12 16 2 8 18 6 14 |
(The path below may be different on your system)
In[7]:=
| AppendTo[$Path, "C:/bin/LinKnot/"];
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In[8]:=
| ConwayNotation[K]
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Out[8]=
| [22122] |
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,−2,−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[{11, 4}, {3, 9}, {8, 10}, {9, 11}, {10, 5}, {4, 6}, {5, 2}, {1, 3}, {2, 7}, {6, 8}, {7, 1}] |
In[14]:=
| Draw[ap]
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|
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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 | 4t2−11t + 15−11t−1 + 4t−2 |
| Conway polynomial | 4z4 + 5z2 + 1 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 45, -4 } |
| Jones polynomial | q−2−2q−3 + 5q−4−6q−5 + 8q−6−8q−7 + 6q−8−5q−9 + 3q−10−q−11 |
| HOMFLY-PT polynomial (db, data sources) | −z2a10 + z4a8−2a8 + 2z4a6 + 4z2a6 + 2a6 + z4a4 + 2z2a4 + a4 |
| Kauffman polynomial (db, data sources) | z5a13−2z3a13 + za13 + 3z6a12−7z4a12 + 3z2a12 + 3z7a11−5z5a11 + za11 + z8a10 + 4z6a10−10z4a10 + 3z2a10 + 5z7a9−6z5a9−2z3a9 + 4za9 + z8a8 + 4z6a8−8z4a8 + 6z2a8−2a8 + 2z7a7 + 2z5a7−6z3a7 + 4za7 + 3z6a6−4z4a6 + 4z2a6−2a6 + 2z5a5−2z3a5 + z4a4−2z2a4 + a4 |
| The A2 invariant | −q34 + q32 + q30−2q28−2q24−q22 + q20 + 3q16 + q12 + 2q10−q8 + q6 |
| The G2 invariant | q176−2q174 + 5q172−8q170 + 7q168−3q166−6q164 + 20q162−27q160 + 30q158−24q156 + q154 + 23q152−47q150 + 55q148−43q146 + 18q144 + 14q142−38q140 + 48q138−38q136 + 13q134 + 12q132−30q130 + 30q128−10q126−16q124 + 40q122−43q120 + 36q118−13q116−28q114 + 56q112−73q110 + 66q108−39q106−6q104 + 43q102−64q100 + 61q98−44q96 + 7q94 + 20q92−36q90 + 30q88−7q86−16q84 + 35q82−29q80 + 11q78 + 13q76−34q74 + 47q72−41q70 + 28q68−2q66−19q64 + 35q62−35q60 + 31q58−17q56 + 4q54 + 7q52−15q50 + 16q48−12q46 + 9q44−2q42−q40 + 3q38−3q36 + 3q34−q32 + q30 |
Further Quantum Invariants
Further quantum knot invariants for 9_23.
A1 Invariants.
| Weight | Invariant |
|---|---|
| 1 | −q23 + 2q21−2q19 + q17−2q15 + 2q11−q9 + 3q7−q5 + q3 |
| 2 | q64−2q62−2q60 + 7q58−q56−9q54 + 9q52 + 3q50−13q48 + 6q46 + 7q44−10q42 + 2q40 + 7q38−3q36−6q34 + q32 + 7q30−10q28−4q26 + 14q24−6q22−6q20 + 11q18−2q16−3q14 + 5q12−q8 + q6 |
| 3 | −q123 + 2q121 + 2q119−3q117−7q115 + q113 + 16q111 + 4q109−21q107−15q105 + 22q103 + 29q101−20q99−41q97 + 10q95 + 49q93 + 3q91−50q89−16q87 + 50q85 + 24q83−41q81−30q79 + 32q77 + 32q75−23q73−32q71 + 8q69 + 33q67 + 5q65−24q63−25q61 + 21q59 + 38q57−10q55−51q53−3q51 + 53q49 + 14q47−52q45−23q43 + 40q41 + 24q39−28q37−23q35 + 20q33 + 16q31−9q29−10q27 + 8q25 + 7q23−3q21−3q19 + 3q17 + 2q15−q11 + q9 |
| 4 | q200−2q198−2q196 + 3q194 + 3q192 + 7q190−8q188−16q186−4q184 + 8q182 + 42q180 + 8q178−38q176−47q174−26q172 + 80q170 + 74q168−2q166−91q164−129q162 + 48q160 + 141q158 + 111q156−53q154−222q152−72q150 + 115q148 + 222q146 + 68q144−223q142−187q140 + 12q138 + 241q136 + 171q134−143q132−218q130−80q128 + 186q126 + 200q124−53q122−179q120−116q118 + 108q116 + 175q114 + 23q112−124q110−133q108 + 16q106 + 128q104 + 114q102−40q100−145q98−109q96 + 55q94 + 212q92 + 80q90−120q88−226q86−64q84 + 234q82 + 193q80−21q78−254q76−175q74 + 153q72 + 208q70 + 81q68−167q66−188q64 + 43q62 + 122q60 + 103q58−58q56−115q54−3q52 + 36q50 + 61q48−8q46−44q44 + 4q40 + 22q38−2q36−13q34 + 6q32 + 7q28−q26−4q24 + 3q22 + 2q18−q14 + q12 |
| 5 | −q295 + 2q293 + 2q291−3q289−3q287−3q285 + 8q281 + 16q279 + 4q277−17q275−29q273−26q271 + 8q269 + 54q267 + 74q265 + 19q263−70q261−124q259−94q257 + 38q255 + 183q253 + 209q251 + 46q249−195q247−327q245−217q243 + 118q241 + 429q239 + 433q237 + 53q235−435q233−637q231−334q229 + 320q227 + 792q225 + 648q223−82q221−822q219−937q217−253q215 + 725q213 + 1141q211 + 595q209−506q207−1226q205−894q203 + 234q201 + 1175q199 + 1097q197 + 55q195−1039q193−1188q191−273q189 + 841q187 + 1168q185 + 438q183−642q181−1082q179−522q177 + 464q175 + 946q173 + 545q171−302q169−814q167−563q165 + 174q163 + 686q161 + 565q159−13q157−553q155−629q153−153q151 + 432q149 + 669q147 + 394q145−249q143−752q141−648q139 + 33q137 + 765q135 + 919q133 + 263q131−717q129−1145q127−588q125 + 552q123 + 1285q121 + 894q119−303q117−1273q115−1131q113−4q111 + 1127q109 + 1242q107 + 280q105−859q103−1192q101−493q99 + 546q97 + 1019q95 + 593q93−269q91−766q89−564q87 + 43q85 + 502q83 + 475q81 + 66q79−295q77−329q75−112q73 + 136q71 + 212q69 + 96q67−57q65−112q63−66q61 + 16q59 + 55q57 + 34q55−2q53−19q51−16q49 + 3q47 + 10q45 + 3q43−2q41 + 2q39−2q37 + 2q35 + 5q33−q31−2q29 + 2q27 + 2q21−q17 + q15 |
A2 Invariants.
| Weight | Invariant |
|---|---|
| 1,0 | −q34 + q32 + q30−2q28−2q24−q22 + q20 + 3q16 + q12 + 2q10−q8 + q6 |
| 1,1 | q92−4q90 + 12q88−28q86 + 50q84−80q82 + 118q80−154q78 + 184q76−194q74 + 186q72−156q70 + 93q68−18q66−74q64 + 166q62−244q60 + 316q58−352q56 + 364q54−340q52 + 284q50−212q48 + 120q46−36q44−54q42 + 108q40−152q38 + 165q36−166q34 + 154q32−124q30 + 105q28−72q26 + 56q24−32q22 + 23q20−10q18 + 6q16−2q14 + q12 |
| 2,0 | q86−q84−2q82 + 4q78 + 3q76−5q74−2q72 + 4q70 + 2q68−5q66−3q64 + 7q62 + 2q60−4q58 + 5q54−2q52−4q50−q48−4q46−6q44−q42 + 3q40−6q38−q36 + 10q34 + 4q32−5q30 + 2q28 + 8q26 + 2q24−4q22 + 2q20 + 4q18−q16−q14 + q12 |
A3 Invariants.
| Weight | Invariant |
|---|---|
| 0,1,0 | q74−2q72 + q70 + 3q68−6q66 + 4q64 + 4q62−10q60 + 5q58 + 5q56−9q54 + 3q52 + 8q50−5q48−2q46−2q42−7q40−6q38 + 8q36−4q34−4q32 + 13q30−5q26 + 10q24 + q22−3q20 + 4q18 + q16−q14 + q12 |
| 1,0,0 | −q45 + q43 + q39−2q37−3q33−q31−q29 + q27 + q25 + q23 + 3q21 + 2q17 + 2q13−q11 + q9 |
A4 Invariants.
| Weight | Invariant |
|---|---|
| 0,1,0,0 | q96−q94−2q92 + 3q90 + 3q88−4q86−2q84 + 6q82 + q80−8q78−q76 + 9q74−6q70 + 7q68 + 6q66−6q64−3q62 + 2q60−8q58−11q56−2q54−2q52−10q50−2q48 + 11q46 + 2q44−3q42 + 8q40 + 10q38−q34 + 5q32 + 5q30−q28 + 3q24 + q22−q20 + q18 |
| 1,0,0,0 | −q56 + q54 + q48−2q46−3q42−2q40−q38−q36 + q34 + q32 + 2q30 + q28 + 3q26 + 2q22 + q20 + 2q16−q14 + q12 |
B2 Invariants.
| Weight | Invariant |
|---|---|
| 0,1 | −q74 + 2q72−5q70 + 7q68−8q66 + 10q64−10q62 + 10q60−7q58 + 3q56 + q54−7q52 + 10q50−17q48 + 18q46−20q44 + 18q42−15q40 + 12q38−6q36 + 2q34 + 4q32−7q30 + 10q28−9q26 + 10q24−7q22 + 7q20−4q18 + 3q16−q14 + q12 |
| 1,0 | q120−2q116−2q114 + 3q112 + 5q110−2q108−7q106−2q104 + 9q102 + 7q100−7q98−11q96 + q94 + 11q92 + 4q90−8q88−7q86 + 5q84 + 8q82−7q78 + 6q74 + q72−9q70−5q68 + 5q66 + 3q64−7q62−8q60 + 4q58 + 8q56−q54−10q52−q50 + 11q48 + 8q46−5q44−8q42 + q40 + 10q38 + 5q36−3q34−5q32 + q30 + 4q28 + 2q26−q24−q22 + q18 |
D4 Invariants.
| Weight | Invariant |
|---|---|
| 1,0,0,0 | q102−2q100 + 3q98−4q96 + 6q94−7q92 + 7q90−8q88 + 9q86−8q84 + 5q82−4q80 + 3q78−4q74 + 7q72−6q70 + 12q68−14q66 + 13q64−16q62 + 13q60−18q58 + 6q56−13q54 + 6q52−4q50 + q48 + 2q46−q44 + 11q42−4q40 + 8q38−6q36 + 10q34−4q32 + 6q30−4q28 + 5q26−q24 + 2q22−q20 + q18 |
G2 Invariants.
| Weight | Invariant |
|---|---|
| 1,0 | q176−2q174 + 5q172−8q170 + 7q168−3q166−6q164 + 20q162−27q160 + 30q158−24q156 + q154 + 23q152−47q150 + 55q148−43q146 + 18q144 + 14q142−38q140 + 48q138−38q136 + 13q134 + 12q132−30q130 + 30q128−10q126−16q124 + 40q122−43q120 + 36q118−13q116−28q114 + 56q112−73q110 + 66q108−39q106−6q104 + 43q102−64q100 + 61q98−44q96 + 7q94 + 20q92−36q90 + 30q88−7q86−16q84 + 35q82−29q80 + 11q78 + 13q76−34q74 + 47q72−41q70 + 28q68−2q66−19q64 + 35q62−35q60 + 31q58−17q56 + 4q54 + 7q52−15q50 + 16q48−12q46 + 9q44−2q42−q40 + 3q38−3q36 + 3q34−q32 + q30 |
.
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 23"];
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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]=
| 4t2−11t + 15−11t−1 + 4t−2 |
In[5]:=
| Conway[K][z]
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Out[5]=
| 4z4 + 5z2 + 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]=
| { 45, -4 } |
In[8]:=
| Jones[K][q]
|
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
|
Out[8]=
| q−2−2q−3 + 5q−4−6q−5 + 8q−6−8q−7 + 6q−8−5q−9 + 3q−10−q−11 |
In[9]:=
| HOMFLYPT[K][a, z]
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KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
|
Out[9]=
| −z2a10 + z4a8−2a8 + 2z4a6 + 4z2a6 + 2a6 + z4a4 + 2z2a4 + a4 |
In[10]:=
| Kauffman[K][a, z]
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KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
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Out[10]=
| z5a13−2z3a13 + za13 + 3z6a12−7z4a12 + 3z2a12 + 3z7a11−5z5a11 + za11 + z8a10 + 4z6a10−10z4a10 + 3z2a10 + 5z7a9−6z5a9−2z3a9 + 4za9 + z8a8 + 4z6a8−8z4a8 + 6z2a8−2a8 + 2z7a7 + 2z5a7−6z3a7 + 4za7 + 3z6a6−4z4a6 + 4z2a6−2a6 + 2z5a5−2z3a5 + z4a4−2z2a4 + a4 |
[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 23"];
|
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]=
| { 4t2−11t + 15−11t−1 + 4t−2, q−2−2q−3 + 5q−4−6q−5 + 8q−6−8q−7 + 6q−8−5q−9 + 3q−10−q−11 } |
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] Vassiliev invariants
| V2 and V3: | (5, -11) |
| V2,1 through V6,9: |
|
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 = -4 is the signature of 9 23. 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 | q−4−2q−5 + q−6 + 6q−7−10q−8 + 2q−9 + 19q−10−27q−11 + 2q−12 + 39q−13−45q−14−4q−15 + 56q−16−51q−17−11q−18 + 59q−19−41q−20−16q−21 + 47q−22−24q−23−17q−24 + 28q−25−8q−26−11q−27 + 10q−28−3q−30 + q−31 |
| 3 | q−6−2q−7 + q−8 + 2q−9 + 2q−10−8q−11 + q−12 + 12q−13 + 3q−14−26q−15 + 2q−16 + 37q−17 + 7q−18−69q−19−3q−20 + 89q−21 + 23q−22−132q−23−32q−24 + 155q−25 + 62q−26−188q−27−80q−28 + 196q−29 + 110q−30−205q−31−126q−32 + 197q−33 + 139q−34−177q−35−151q−36 + 157q−37 + 148q−38−122q−39−151q−40 + 95q−41 + 137q−42−57q−43−125q−44 + 29q−45 + 103q−46−4q−47−79q−48−10q−49 + 52q−50 + 17q−51−30q−52−17q−53 + 15q−54 + 11q−55−5q−56−5q−57 + 3q−59−q−60 |
| 4 | q−8−2q−9 + q−10 + 2q−11−2q−12 + 4q−13−9q−14 + 4q−15 + 10q−16−9q−17 + 10q−18−28q−19 + 15q−20 + 34q−21−27q−22 + 6q−23−72q−24 + 51q−25 + 103q−26−52q−27−33q−28−184q−29 + 108q−30 + 264q−31−33q−32−112q−33−415q−34 + 129q−35 + 512q−36 + 94q−37−167q−38−743q−39 + 50q−40 + 745q−41 + 308q−42−126q−43−1041q−44−112q−45 + 851q−46 + 508q−47 + 6q−48−1198q−49−276q−50 + 815q−51 + 613q−52 + 160q−53−1184q−54−388q−55 + 666q−56 + 622q−57 + 307q−58−1032q−59−455q−60 + 442q−61 + 559q−62 + 433q−63−779q−64−469q−65 + 176q−66 + 421q−67 + 508q−68−465q−69−399q−70−53q−71 + 222q−72 + 472q−73−174q−74−245q−75−160q−76 + 35q−77 + 322q−78−5q−79−81q−80−130q−81−58q−82 + 145q−83 + 33q−84 + 8q−85−54q−86−52q−87 + 39q−88 + 12q−89 + 17q−90−8q−91−18q−92 + 5q−93 + 5q−95−3q−97 + q−98 |
[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. |
|
