10 18
From Knot Atlas
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 (KnotPlot image) |
See the full Rolfsen Knot Table. Visit 10 18's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 10_18's page at Knotilus! Visit 10 18's page at the original Knot Atlas! |
[edit] Knot presentations
| Planar diagram presentation | X1425 X3,12,4,13 X5,14,6,15 X15,20,16,1 X9,17,10,16 X7,19,8,18 X17,9,18,8 X19,7,20,6 X13,10,14,11 X11,2,12,3 |
| Gauss code | -1, 10, -2, 1, -3, 8, -6, 7, -5, 9, -10, 2, -9, 3, -4, 5, -7, 6, -8, 4 |
| Dowker-Thistlethwaite code | 4 12 14 18 16 2 10 20 8 6 |
| Conway Notation | [41122] |
| Minimum Braid Representative | A Morse Link Presentation | An Arc Presentation | |||||
Length is 12, width is 5, Braid index is 5 | 
|  [{12, 3}, {2, 10}, {9, 11}, {10, 12}, {11, 4}, {3, 5}, {4, 1}, {6, 2}, {5, 7}, {8, 6}, {7, 9}, {1, 8}] |
[edit Notes on presentations of 10 18]
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[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
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In[3]:=
| K = Knot["10 18"];
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In[4]:=
| PD[K]
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
| X1425 X3,12,4,13 X5,14,6,15 X15,20,16,1 X9,17,10,16 X7,19,8,18 X17,9,18,8 X19,7,20,6 X13,10,14,11 X11,2,12,3 |
In[5]:=
| GaussCode[K]
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Out[5]=
| -1, 10, -2, 1, -3, 8, -6, 7, -5, 9, -10, 2, -9, 3, -4, 5, -7, 6, -8, 4 |
In[6]:=
| DTCode[K]
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Out[6]=
| 4 12 14 18 16 2 10 20 8 6 |
(The path below may be different on your system)
In[7]:=
| AppendTo[$Path, "C:/bin/LinKnot/"];
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In[8]:=
| ConwayNotation[K]
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Out[8]=
| [41122] |
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,−1,−2,1,−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[{12, 3}, {2, 10}, {9, 11}, {10, 12}, {11, 4}, {3, 5}, {4, 1}, {6, 2}, {5, 7}, {8, 6}, {7, 9}, {1, 8}] |
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 + 14t−19 + 14t−1−4t−2 |
| Conway polynomial | −4z4−2z2 + 1 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 55, -2 } |
| Jones polynomial | q3−2q2 + 4q−6 + 8q−1−9q−2 + 9q−3−7q−4 + 5q−5−3q−6 + q−7 |
| HOMFLY-PT polynomial (db, data sources) | z2a6−z4a4 + a4−2z4a2−3z2a2−a2−z4−z2 + z2a−2 + a−2 |
| Kauffman polynomial (db, data sources) | a3z9 + az9 + 3a4z8 + 5a2z8 + 2z8 + 4a5z7 + 3a3z7 + az7 + 2z7a−1 + 4a6z6−5a4z6−15a2z6 + z6a−2−5z6 + 3a7z5−6a5z5−12a3z5−10az5−7z5a−1 + a8z4−5a6z4 + 6a4z4 + 17a2z4−4z4a−2 + z4−4a7z3 + 5a5z3 + 14a3z3 + 11az3 + 6z3a−1−a8z2 + a6z2−3a4z2−8a2z2 + 4z2a−2 + z2−2a5z−4a3z−4az−2za−1 + a4 + a2−a−2 |
| The A2 invariant | q22−q20−q18 + 2q16−q14 + q12 + q10−q8 + q6−2q4 + q2−q−2 + 2q−4 + q−10 |
| The G2 invariant | q114−2q112 + 4q110−6q108 + 4q106−q104−4q102 + 12q100−16q98 + 20q96−18q94 + 6q92 + 6q90−21q88 + 32q86−37q84 + 33q82−20q80 + 24q76−41q74 + 49q72−40q70 + 20q68 + 4q66−28q64 + 39q62−29q60 + 10q58 + 17q56−33q54 + 30q52−8q50−24q48 + 52q46−63q44 + 51q42−17q40−25q38 + 62q36−79q34 + 72q32−43q30 + 35q26−58q24 + 60q22−41q20 + 9q18 + 20q16−38q14 + 34q12−13q10−16q8 + 38q6−44q4 + 28q2 + 1−33q−2 + 58q−4−57q−6 + 40q−8−10q−10−21q−12 + 42q−14−47q−16 + 39q−18−21q−20 + 2q−22 + 13q−24−20q−26 + 20q−28−14q−30 + 9q−32−q−34−3q−36 + 4q−38−4q−40 + 3q−42−q−44 + q−46 |
Further Quantum Invariants
Further quantum knot invariants for 10_18.
A1 Invariants.
| Weight | Invariant |
|---|---|
| 1 | q15−2q13 + 2q11−2q9 + 2q7−q3 + 2q−2q−1 + 2q−3−q−5 + q−7 |
| 2 | q42−2q40−q38 + 5q36−4q34−2q32 + 10q30−7q28−6q26 + 13q24−6q22−9q20 + 10q18 + q16−6q14 + 7q10−q8−9q6 + 9q4 + 5q2−12 + 6q−2 + 8q−4−11q−6 + 8q−10−5q−12−2q−14 + 4q−16−q−18−q−20 + q−22 |
| 3 | q81−2q79−q77 + 2q75 + 3q73−q71−5q69 + 3q67 + 3q65−5q63−4q61 + 10q59 + 6q57−17q55−11q53 + 25q51 + 21q49−30q47−31q45 + 27q43 + 41q41−20q39−43q37 + 6q35 + 41q33 + 5q31−31q29−17q27 + 18q25 + 28q23−8q21−32q19−3q17 + 37q15 + 13q13−39q11−22q9 + 37q7 + 31q5−31q3−38q + 22q−1 + 44q−3−7q−5−43q−7−7q−9 + 37q−11 + 17q−13−25q−15−23q−17 + 14q−19 + 22q−21−5q−23−16q−25−q−27 + 11q−29 + 2q−31−6q−33−2q−35 + 3q−37 + q−39−q−41−q−43 + q−45 |
| 4 | q132−2q130−q128 + 2q126 + 6q122−4q120−4q118−2q116−8q114 + 17q112 + 2q110 + 3q108−2q106−30q104 + 8q102 + 3q100 + 33q98 + 28q96−48q94−35q92−41q90 + 57q88 + 109q86−4q84−79q82−149q80 + 12q78 + 182q76 + 112q74−39q72−235q70−104q68 + 153q66 + 195q64 + 72q62−197q60−180q58 + 29q56 + 157q54 + 150q52−66q50−150q48−77q46 + 50q44 + 139q42 + 55q40−70q38−130q36−37q34 + 104q32 + 132q30−14q28−162q26−89q24 + 81q22 + 195q20 + 36q18−184q16−147q14 + 33q12 + 231q10 + 111q8−143q6−190q4−72q2 + 187 + 180q−2−21q−4−150q−6−168q−8 + 55q−10 + 156q−12 + 96q−14−24q−16−164q−18−64q−20 + 47q−22 + 107q−24 + 78q−26−71q−28−79q−30−41q−32 + 40q−34 + 83q−36 + 2q−38−29q−40−46q−42−8q−44 + 38q−46 + 13q−48 + 2q−50−19q−52−11q−54 + 11q−56 + 3q−58 + 4q−60−4q−62−4q−64 + 3q−66 + q−70−q−72−q−74 + q−76 |
| 5 | q195−2q193−q191 + 2q189 + 3q185 + 3q183−3q181−9q179−5q177−q175 + 9q173 + 21q171 + 12q169−13q167−35q165−28q163 + q161 + 44q159 + 67q157 + 26q155−56q153−105q151−75q149 + 30q147 + 153q145 + 171q143 + 15q141−192q139−280q137−132q135 + 190q133 + 425q131 + 316q129−128q127−554q125−555q123−37q121 + 620q119 + 833q117 + 299q115−593q113−1063q111−626q109 + 424q107 + 1194q105 + 955q103−148q101−1168q99−1200q97−190q95 + 985q93 + 1288q91 + 502q89−664q87−1218q85−723q83 + 321q81 + 994q79 + 802q77 + 17q75−698q73−778q71−254q69 + 394q67 + 663q65 + 401q63−128q61−532q59−497q57−49q55 + 431q53 + 547q51 + 181q49−374q47−625q45−281q43 + 375q41 + 718q39 + 383q37−372q35−842q33−524q31 + 349q29 + 963q27 + 693q25−260q23−1030q21−884q19 + 85q17 + 1016q15 + 1055q13 + 154q11−882q9−1144q7−430q5 + 627q3 + 1126q + 676q−1−294q−3−965q−5−818q−7−61q−9 + 675q−11 + 833q−13 + 357q−15−326q−17−692q−19−521q−21−20q−23 + 438q−25 + 549q−27 + 267q−29−160q−31−431q−33−382q−35−87q−37 + 243q−39 + 375q−41 + 229q−43−56q−45−269q−47−265q−49−82q−51 + 139q−53 + 229q−55 + 136q−57−33q−59−147q−61−135q−63−30q−65 + 74q−67 + 100q−69 + 48q−71−25q−73−60q−75−38q−77 + 27q−81 + 28q−83 + 5q−85−13q−87−12q−89−3q−91 + q−93 + 6q−95 + 4q−97−3q−99−2q−101 + q−103 + q−109−q−111−q−113 + q−115 |
A2 Invariants.
| Weight | Invariant |
|---|---|
| 1,0 | q22−q20−q18 + 2q16−q14 + q12 + q10−q8 + q6−2q4 + q2−q−2 + 2q−4 + q−10 |
| 1,1 | q60−4q58 + 10q56−20q54 + 32q52−48q50 + 70q48−90q46 + 108q44−126q42 + 144q40−156q38 + 156q36−148q34 + 128q32−88q30 + 28q28 + 48q26−128q24 + 212q22−288q20 + 340q18−376q16 + 378q14−351q12 + 302q10−226q8 + 144q6−44q4−44q2 + 118−172q−2 + 204q−4−210q−6 + 192q−8−164q−10 + 130q−12−96q−14 + 64q−16−40q−18 + 25q−20−12q−22 + 6q−24−2q−26 + q−28 |
| 2,0 | q56−q54−2q52 + q50 + 3q48−5q44 + 2q42 + 7q40−3q38−7q36 + 3q34 + 8q32−4q30−8q28 + 4q26 + 4q24−5q22−q20 + 3q18−2q16 + 4q12−q10−4q8 + 4q6 + 9q4−3q2−5 + 6q−2 + 4q−4−6q−6−6q−8 + 2q−10 + 4q−12−2q−14−3q−16 + 2q−18 + 3q−20−q−24 + q−28 |
A3 Invariants.
| Weight | Invariant |
|---|---|
| 0,1,0 | q48−2q46 + 4q42−5q40 + 8q36−8q34−4q32 + 11q30−6q28−5q26 + 11q24−q22−3q20 + 3q18 + 3q16−2q14−6q12 + 5q10 + q8−11q6 + 5q4 + 6q2−9 + 5q−2 + 6q−4−6q−6 + 3q−8 + 2q−10−3q−12 + 2q−14 + q−16−q−18 + q−20 |
| 1,0,0 | q29−q27−q23 + 2q21−q19 + 2q17 + q13−q11−2q5 + q3−q + q−1−q−3 + 2q−5 + q−9 + q−13 |
B2 Invariants.
| Weight | Invariant |
|---|---|
| 0,1 | q48−2q46 + 4q44−6q42 + 7q40−10q38 + 12q36−12q34 + 12q32−9q30 + 6q28−q26−5q24 + 11q22−17q20 + 21q18−23q16 + 24q14−22q12 + 19q10−13q8 + 7q6−q4−4q2 + 7−11q−2 + 12q−4−12q−6 + 11q−8−8q−10 + 7q−12−4q−14 + 3q−16−q−18 + q−20 |
| 1,0 | q78−2q74−2q72 + 2q70 + 5q68−6q64−5q62 + 5q60 + 10q58−11q54−8q52 + 7q50 + 12q48−q46−12q44−4q42 + 10q40 + 8q38−6q36−8q34 + 4q32 + 9q30−q28−8q26−q24 + 7q22 + 2q20−7q18−4q16 + 7q14 + 6q12−7q10−11q8 + 3q6 + 13q4 + 4q2−11−10q−2 + 7q−4 + 13q−6 + q−8−10q−10−5q−12 + 6q−14 + 6q−16−q−18−5q−20−q−22 + 3q−24 + 2q−26−q−28−q−30 + q−34 |
G2 Invariants.
| Weight | Invariant |
|---|---|
| 1,0 | q114−2q112 + 4q110−6q108 + 4q106−q104−4q102 + 12q100−16q98 + 20q96−18q94 + 6q92 + 6q90−21q88 + 32q86−37q84 + 33q82−20q80 + 24q76−41q74 + 49q72−40q70 + 20q68 + 4q66−28q64 + 39q62−29q60 + 10q58 + 17q56−33q54 + 30q52−8q50−24q48 + 52q46−63q44 + 51q42−17q40−25q38 + 62q36−79q34 + 72q32−43q30 + 35q26−58q24 + 60q22−41q20 + 9q18 + 20q16−38q14 + 34q12−13q10−16q8 + 38q6−44q4 + 28q2 + 1−33q−2 + 58q−4−57q−6 + 40q−8−10q−10−21q−12 + 42q−14−47q−16 + 39q−18−21q−20 + 2q−22 + 13q−24−20q−26 + 20q−28−14q−30 + 9q−32−q−34−3q−36 + 4q−38−4q−40 + 3q−42−q−44 + q−46 |
.
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["10 18"];
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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 + 14t−19 + 14t−1−4t−2 |
In[5]:=
| Conway[K][z]
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Out[5]=
| −4z4−2z2 + 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]=
| { 55, -2 } |
In[8]:=
| Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
| q3−2q2 + 4q−6 + 8q−1−9q−2 + 9q−3−7q−4 + 5q−5−3q−6 + q−7 |
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]=
| z2a6−z4a4 + a4−2z4a2−3z2a2−a2−z4−z2 + z2a−2 + a−2 |
In[10]:=
| Kauffman[K][a, z]
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KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
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Out[10]=
| a3z9 + az9 + 3a4z8 + 5a2z8 + 2z8 + 4a5z7 + 3a3z7 + az7 + 2z7a−1 + 4a6z6−5a4z6−15a2z6 + z6a−2−5z6 + 3a7z5−6a5z5−12a3z5−10az5−7z5a−1 + a8z4−5a6z4 + 6a4z4 + 17a2z4−4z4a−2 + z4−4a7z3 + 5a5z3 + 14a3z3 + 11az3 + 6z3a−1−a8z2 + a6z2−3a4z2−8a2z2 + 4z2a−2 + z2−2a5z−4a3z−4az−2za−1 + a4 + a2−a−2 |
[edit] "Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {10_24,}
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["10 18"];
|
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`.
|
Out[4]=
| { −4t2 + 14t−19 + 14t−1−4t−2, q3−2q2 + 4q−6 + 8q−1−9q−2 + 9q−3−7q−4 + 5q−5−3q−6 + q−7 } |
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]=
| {10_24,} |
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: | (-2, 1) |
| 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 = -2 is the signature of 10 18. 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 | q10−2q9 + 6q7−8q6−3q5 + 19q4−16q3−14q2 + 38q−18−32q−1 + 55q−2−14q−3−50q−4 + 63q−5−6q−6−57q−7 + 57q−8 + q−9−48q−10 + 38q−11 + 4q−12−29q−13 + 19q−14 + 3q−15−12q−16 + 7q−17 + q−18−3q−19 + q−20 |
| 3 | q21−2q20 + 2q18 + 3q17−7q16−4q15 + 10q14 + 12q13−19q12−19q11 + 21q10 + 39q9−27q8−56q7 + 19q6 + 81q5−7q4−100q3−17q2 + 117q + 44−122q−1−77q−2 + 124q−3 + 106q−4−116q−5−136q−6 + 107q−7 + 158q−8−92q−9−176q−10 + 78q−11 + 182q−12−56q−13−186q−14 + 43q−15 + 168q−16−20q−17−150q−18 + 8q−19 + 119q−20 + 3q−21−89q−22−6q−23 + 61q−24 + 4q−25−38q−26−2q−27 + 25q−28−2q−29−15q−30 + 2q−31 + 11q−32−3q−33−7q−34 + 2q−35 + 3q−36 + q−37−3q−38 + q−39 |
| 4 | q36−2q35 + 2q33−q32 + 4q31−9q30 + 10q28−2q27 + 12q26−31q25−8q24 + 31q23 + 9q22 + 37q21−77q20−46q19 + 48q18 + 40q17 + 118q16−120q15−127q14 + 10q13 + 48q12 + 267q11−91q10−187q9−101q8−52q7 + 407q6 + 29q5−127q4−202q3−275q2 + 425q + 158 + 74q−1−195q−2−534q−3 + 307q−4 + 205q−5 + 328q−6−75q−7−732q−8 + 127q−9 + 168q−10 + 548q−11 + 84q−12−846q−13−43q−14 + 95q−15 + 696q−16 + 230q−17−874q−18−184q−19 + 2q−20 + 756q−21 + 355q−22−790q−23−273q−24−125q−25 + 683q−26 + 439q−27−574q−28−266q−29−253q−30 + 474q−31 + 422q−32−305q−33−143q−34−295q−35 + 217q−36 + 291q−37−109q−38 + 8q−39−225q−40 + 47q−41 + 130q−42−39q−43 + 83q−44−112q−45−5q−46 + 32q−47−33q−48 + 70q−49−36q−50 + 2q−52−28q−53 + 32q−54−8q−55 + 5q−56 + q−57−13q−58 + 7q−59−2q−60 + 3q−61 + q−62−3q−63 + q−64 |
[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. |
|
