9 35
From Knot Atlas
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 (KnotPlot image) |
See the full Rolfsen Knot Table. Visit 9 35's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9_35's page at Knotilus! Visit 9 35's page at the original Knot Atlas! |
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9_35 is also known as the pretzel knot P(3,3,3). |
[edit] Knot presentations
| Planar diagram presentation | X1829 X7,14,8,15 X5,16,6,17 X9,18,10,1 X15,6,16,7 X17,10,18,11 X13,2,14,3 X3,12,4,13 X11,4,12,5 |
| Gauss code | -1, 7, -8, 9, -3, 5, -2, 1, -4, 6, -9, 8, -7, 2, -5, 3, -6, 4 |
| Dowker-Thistlethwaite code | 8 12 16 14 18 4 2 6 10 |
| Conway Notation | [3,3,3] |
| Minimum Braid Representative | A Morse Link Presentation | An Arc Presentation | |||||
Length is 14, width is 5, Braid index is 5 | 
|  [{8, 4}, {3, 7}, {4, 2}, {1, 3}, {9, 12}, {11, 8}, {12, 10}, {6, 9}, {7, 5}, {2, 6}, {5, 11}, {10, 1}] |
[edit Notes on presentations of 9 35]
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[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
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In[3]:=
| K = Knot["9 35"];
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In[4]:=
| PD[K]
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
| X1829 X7,14,8,15 X5,16,6,17 X9,18,10,1 X15,6,16,7 X17,10,18,11 X13,2,14,3 X3,12,4,13 X11,4,12,5 |
In[5]:=
| GaussCode[K]
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Out[5]=
| -1, 7, -8, 9, -3, 5, -2, 1, -4, 6, -9, 8, -7, 2, -5, 3, -6, 4 |
In[6]:=
| DTCode[K]
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Out[6]=
| 8 12 16 14 18 4 2 6 10 |
(The path below may be different on your system)
In[7]:=
| AppendTo[$Path, "C:/bin/LinKnot/"];
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In[8]:=
| ConwayNotation[K]
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Out[8]=
| [3,3,3] |
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,−3,2,2,−4,3,−2,−4,−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]=
| { 5, 14, 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[{8, 4}, {3, 7}, {4, 2}, {1, 3}, {9, 12}, {11, 8}, {12, 10}, {6, 9}, {7, 5}, {2, 6}, {5, 11}, {10, 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 | 7t−13 + 7t−1 |
| Conway polynomial | 7z2 + 1 |
| 2nd Alexander ideal (db, data sources) | {3,t + 1} |
| Determinant and Signature | { 27, -2 } |
| Jones polynomial | q−1−2q−2 + 3q−3−4q−4 + 5q−5−3q−6 + 4q−7−3q−8 + q−9−q−10 |
| HOMFLY-PT polynomial (db, data sources) | −a10 + z2a8−a8 + 3z2a6 + 3a6 + 2z2a4 + z2a2 |
| Kauffman polynomial (db, data sources) | z7a11−6z5a11 + 12z3a11−8za11 + z8a10−4z6a10 + 3z4a10 + z2a10 + a10 + 4z7a9−18z5a9 + 23z3a9−9za9 + z8a8 + z6a8−15z4a8 + 16z2a8−a8 + 3z7a7−8z5a7 + 3z3a7−za7 + 5z6a6−15z4a6 + 12z2a6−3a6 + 4z5a5−6z3a5 + 3z4a4−2z2a4 + 2z3a3 + z2a2 |
| The A2 invariant | −q32−q30−2q26−q24 + q22 + q20 + 3q18 + 2q16 + q14−q10 + q8−q4 + q2 |
| The G2 invariant | q156 + 3q152−3q150 + 2q148−q146−2q144 + 7q142−9q140 + 6q138−2q136−2q134 + 8q132−12q130 + 5q128−2q126−3q124 + 3q122−10q120−2q118 + 4q116−2q114 + q112−8q110−2q108 + 6q106−6q104 + 5q102−11q100 + 6q98 + 8q96−3q94 + 8q92−10q90 + 12q88 + 4q86−5q84 + 7q82−5q80 + 5q78 + 7q76−3q74 + 2q72 + q70−2q68 + 4q66−6q64 + 3q62−2q60−2q58 + 4q56−4q54 + 3q52−2q50 + q48−q46−q44 + 2q42−3q40 + 3q38 + q36 + q34−q30 + 2q28−2q26 + 2q24−q22−q16 + q14−q12 + q10 |
Further Quantum Invariants
Further quantum knot invariants for 9_35.
A1 Invariants.
| Weight | Invariant |
|---|---|
| 1 | −q21−2q17 + q15 + q13 + 2q11 + q9−q7 + q5−q3 + q |
| 2 | q60−q56 + 2q54 + 2q52−3q50 + q46−4q44−3q42 + q40−q38−2q36 + 3q34 + 3q32 + 3q26−q24−2q22 + q20 + q18−2q16 + q14 + 3q12−q10 + q8−q4 + q2 |
| 3 | −q117 + q113 + q111−2q109−3q107 + 5q103 + 2q101−4q99−4q97 + 4q95 + 7q93 + 3q91−4q89−3q87 + 4q85 + 4q83−q81−8q79−3q77 + 2q75 + q73−7q71−4q69 + q67 + 6q65−2q63−4q61 + 4q59 + 9q57−q55−8q53 + 6q49 + 2q47−6q45−4q43−q41 + 6q39 + 4q37−3q35−6q33 + 4q31 + 8q29−5q25 + 3q21−q19−q17 + 2q13 + q11−q5 + q3 |
| 4 | q192−q188−q186−q184 + 3q182 + 3q180 + q178−2q176−8q174−2q172 + 4q170 + 9q168 + 6q166−8q164−11q162−10q160 + 3q158 + 15q156 + 8q154−q152−17q150−15q148 + 4q146 + 15q144 + 22q142 + 2q140−17q138−15q136−2q134 + 23q132 + 23q130−q128−17q126−21q124 + q122 + 19q120 + 14q118−7q116−25q114−15q112 + 9q110 + 17q108 + 3q106−15q104−12q102 + 8q100 + 17q98 + 6q96−14q94−12q92 + 10q90 + 18q88 + 2q86−18q84−21q82 + 6q80 + 20q78 + 13q76−6q74−25q72−15q70 + 6q68 + 21q66 + 23q64−4q62−27q60−20q58 + 6q56 + 35q54 + 21q52−14q50−31q48−15q46 + 20q44 + 22q42−15q38−11q36 + 8q34 + 10q32 + 2q30−3q28−5q26 + 5q24 + q22−2q20−q18−q16 + 4q14−q6 + q4 |
| 5 | −q285 + q281 + q279 + q277−3q273−4q271−q269 + 2q267 + 5q265 + 8q263 + 3q261−6q259−12q257−11q255−2q253 + 12q251 + 20q249 + 16q247−18q243−27q241−18q239 + 4q237 + 27q235 + 37q233 + 20q231−11q229−38q227−44q225−23q223 + 20q221 + 50q219 + 46q217 + 10q215−36q213−66q211−51q209 + 3q207 + 57q205 + 69q203 + 39q201−26q199−74q197−67q195−10q193 + 55q191 + 87q189 + 58q187−16q185−76q183−76q181−19q179 + 57q177 + 92q175 + 49q173−28q171−83q169−76q167−9q165 + 65q163 + 76q161 + 27q159−46q157−76q155−44q153 + 29q151 + 66q149 + 43q147−11q145−49q143−37q141 + 15q139 + 42q137 + 23q135−19q133−40q131−15q129 + 30q127 + 48q125 + 13q123−44q121−64q119−24q117 + 36q115 + 73q113 + 47q111−23q109−76q107−69q105−16q103 + 53q101 + 85q99 + 61q97−7q95−75q93−95q91−44q89 + 46q87 + 106q85 + 93q83−3q81−100q79−114q77−40q75 + 66q73 + 115q71 + 63q69−34q67−94q65−66q63 + 11q61 + 68q59 + 60q57 + 4q55−44q53−42q51−3q49 + 24q47 + 25q45 + 3q43−15q41−14q39 + 2q37 + 9q35 + 5q33−q31−q29 + q25 + q23−2q21−2q19 + q17 + 3q15−q7 + q5 |
A2 Invariants.
| Weight | Invariant |
|---|---|
| 1,0 | −q32−q30−2q26−q24 + q22 + q20 + 3q18 + 2q16 + q14−q10 + q8−q4 + q2 |
| 1,1 | q84 + 6q80−6q78 + 12q76−18q74 + 18q72−22q70 + 14q68−16q66 + 4q64 + 8q62−7q60 + 20q58−24q56 + 30q54−35q52 + 22q50−32q48 + 16q46−14q44 + 2q42 + 8q40−2q38 + 15q36−4q34 + 12q32−10q30 + 7q28−4q26 + 4q24−6q22 + 5q20 + 2q18 + 4q16−4q14 + 3q12−2q10 + 2q8−2q6 + q4 |
| 2,0 | q82 + q80 + q78−q76 + q74 + 3q72 + 3q70−q68−3q66−q64−q62−5q60−7q58−4q56−2q54−2q52−3q50 + 2q48 + 5q46 + 6q44 + 4q42 + 3q40 + 2q38 + q36−q34−q32−q30 + q28 + 3q26−q24−3q22 + 2q20 + 4q18 + q16−2q14 + 2q10−q8−q6 + q4 |
A3 Invariants.
| Weight | Invariant |
|---|---|
| 0,1,0 | q66 + 3q62 + 2q60 + q58 + q56−3q54−7q52−6q50−7q48−5q46 + 2q44 + 2q42 + 5q40 + 6q38 + 4q36 + q28 + 3q24 + 3q22−q20 + q18 + q16−2q14 + 2q10−q8−q6 + q4 |
| 1,0,0 | −q43−q41−q39−2q35−q33−q31 + q29 + q27 + 3q25 + 3q23 + 2q21 + q19−q13 + q11−q5 + q3 |
B2 Invariants.
| Weight | Invariant |
|---|---|
| 0,1 | −q66−3q62 + 2q60−3q58 + 3q56−3q54 + 3q52−2q50 + q48 + q46−2q44 + 4q42−5q40 + 6q38−6q36 + 6q34−4q32 + 4q30−q28 + 2q26 + q24−q22 + 3q20−3q18 + 3q16−2q14 + 2q12−2q10 + q8−q6 + q4 |
| 1,0 | q108 + 3q100 + 2q98−q96−q94 + 2q92 + 2q90−q88−5q86−4q84−q82−q80−5q78−7q76−2q74 + 2q72 + 2q70−q68 + q66 + 4q64 + 5q62 + 2q60 + q58 + 2q56 + 3q54−q52−3q50−q48 + 2q46 + q44−q42−2q40 + 2q38 + 4q36 + q34−2q32 + 2q28 + 2q26−q24−2q22−q20 + q18 + 2q16−q12−q10 + q6 |
G2 Invariants.
| Weight | Invariant |
|---|---|
| 1,0 | q156 + 3q152−3q150 + 2q148−q146−2q144 + 7q142−9q140 + 6q138−2q136−2q134 + 8q132−12q130 + 5q128−2q126−3q124 + 3q122−10q120−2q118 + 4q116−2q114 + q112−8q110−2q108 + 6q106−6q104 + 5q102−11q100 + 6q98 + 8q96−3q94 + 8q92−10q90 + 12q88 + 4q86−5q84 + 7q82−5q80 + 5q78 + 7q76−3q74 + 2q72 + q70−2q68 + 4q66−6q64 + 3q62−2q60−2q58 + 4q56−4q54 + 3q52−2q50 + q48−q46−q44 + 2q42−3q40 + 3q38 + q36 + q34−q30 + 2q28−2q26 + 2q24−q22−q16 + q14−q12 + q10 |
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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 35"];
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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]=
| 7t−13 + 7t−1 |
In[5]:=
| Conway[K][z]
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Out[5]=
| 7z2 + 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]=
| {3,t + 1} |
In[7]:=
| {KnotDet[K], KnotSignature[K]}
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Out[7]=
| { 27, -2 } |
In[8]:=
| Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
| q−1−2q−2 + 3q−3−4q−4 + 5q−5−3q−6 + 4q−7−3q−8 + q−9−q−10 |
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]=
| −a10 + z2a8−a8 + 3z2a6 + 3a6 + 2z2a4 + z2a2 |
In[10]:=
| Kauffman[K][a, z]
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KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
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Out[10]=
| z7a11−6z5a11 + 12z3a11−8za11 + z8a10−4z6a10 + 3z4a10 + z2a10 + a10 + 4z7a9−18z5a9 + 23z3a9−9za9 + z8a8 + z6a8−15z4a8 + 16z2a8−a8 + 3z7a7−8z5a7 + 3z3a7−za7 + 5z6a6−15z4a6 + 12z2a6−3a6 + 4z5a5−6z3a5 + 3z4a4−2z2a4 + 2z3a3 + z2a2 |
[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 35"];
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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]=
| { 7t−13 + 7t−1, q−1−2q−2 + 3q−3−4q−4 + 5q−5−3q−6 + 4q−7−3q−8 + q−9−q−10 } |
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 35. 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 | q−2−2q−3 + q−4 + 2q−5−4q−6 + 5q−7−7q−9 + 8q−10−10q−12 + 9q−13 + 4q−14−13q−15 + 9q−16 + 7q−17−13q−18 + 4q−19 + 8q−20−11q−21 + 7q−23−6q−24−q−25 + 4q−26−q−27−q−28 + q−29 |
[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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