10 33
From Knot Atlas
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 (KnotPlot image) |
See the full Rolfsen Knot Table. Visit 10 33's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 10_33's page at Knotilus! Visit 10 33's page at the original Knot Atlas! |
[edit] Knot presentations
| Planar diagram presentation | X6271 X14,6,15,5 X20,15,1,16 X16,7,17,8 X8,19,9,20 X18,9,19,10 X10,17,11,18 X2,14,3,13 X12,4,13,3 X4,12,5,11 |
| Gauss code | 1, -8, 9, -10, 2, -1, 4, -5, 6, -7, 10, -9, 8, -2, 3, -4, 7, -6, 5, -3 |
| Dowker-Thistlethwaite code | 6 12 14 16 18 4 2 20 10 8 |
| Conway Notation | [311113] |
| Minimum Braid Representative | A Morse Link Presentation | An Arc Presentation | |||||
Length is 12, width is 5, Braid index is 5 | 
|  [{3, 9}, {2, 7}, {6, 8}, {7, 10}, {9, 5}, {4, 6}, {5, 11}, {10, 4}, {12, 3}, {11, 13}, {1, 12}, {13, 2}, {8, 1}] |
[edit Notes on presentations of 10 33]
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[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
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In[3]:=
| K = Knot["10 33"];
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In[4]:=
| PD[K]
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
| X6271 X14,6,15,5 X20,15,1,16 X16,7,17,8 X8,19,9,20 X18,9,19,10 X10,17,11,18 X2,14,3,13 X12,4,13,3 X4,12,5,11 |
In[5]:=
| GaussCode[K]
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Out[5]=
| 1, -8, 9, -10, 2, -1, 4, -5, 6, -7, 10, -9, 8, -2, 3, -4, 7, -6, 5, -3 |
In[6]:=
| DTCode[K]
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Out[6]=
| 6 12 14 16 18 4 2 20 10 8 |
(The path below may be different on your system)
In[7]:=
| AppendTo[$Path, "C:/bin/LinKnot/"];
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In[8]:=
| ConwayNotation[K]
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Out[8]=
| [311113] |
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,−2,3,−2,3,3,4,−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, 9}, {2, 7}, {6, 8}, {7, 10}, {9, 5}, {4, 6}, {5, 11}, {10, 4}, {12, 3}, {11, 13}, {1, 12}, {13, 2}, {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 | 4t2−16t + 25−16t−1 + 4t−2 |
| Conway polynomial | 4z4 + 1 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 65, 0 } |
| Jones polynomial | −q5 + 3q4−5q3 + 8q2−10q + 11−10q−1 + 8q−2−5q−3 + 3q−4−q−5 |
| HOMFLY-PT polynomial (db, data sources) | −z2a4 + z4a2 + 2z4 + 2z2 + 1 + z4a−2−z2a−4 |
| Kauffman polynomial (db, data sources) | az9 + z9a−1 + 3a2z8 + 3z8a−2 + 6z8 + 4a3z7 + 5az7 + 5z7a−1 + 4z7a−3 + 3a4z6−4a2z6−4z6a−2 + 3z6a−4−14z6 + a5z5−9a3z5−16az5−16z5a−1−9z5a−3 + z5a−5−7a4z4 + a2z4 + z4a−2−7z4a−4 + 16z4−2a5z3 + 6a3z3 + 18az3 + 18z3a−1 + 6z3a−3−2z3a−5 + 3a4z2 + 3z2a−4−6z2−2a3z−6az−6za−1−2za−3 + 1 |
| The A2 invariant | −q16 + q14 + q12−2q10 + 2q8−q4 + 2q2−1 + 2q−2−q−4 + 2q−8−2q−10 + q−12 + q−14−q−16 |
| The G2 invariant | q80−2q78 + 4q76−7q74 + 6q72−5q70−2q68 + 14q66−23q64 + 32q62−32q60 + 19q58 + 3q56−33q54 + 60q52−73q50 + 67q48−37q46−9q44 + 57q42−88q40 + 96q38−70q36 + 20q34 + 31q32−69q30 + 73q28−46q26 + 45q22−65q20 + 47q18−3q16−55q14 + 102q12−111q10 + 80q8−14q6−61q4 + 124q2−145 + 124q−2−61q−4−14q−6 + 80q−8−111q−10 + 102q−12−55q−14−3q−16 + 47q−18−65q−20 + 45q−22−46q−26 + 73q−28−69q−30 + 31q−32 + 20q−34−70q−36 + 96q−38−88q−40 + 57q−42−9q−44−37q−46 + 67q−48−73q−50 + 60q−52−33q−54 + 3q−56 + 19q−58−32q−60 + 32q−62−23q−64 + 14q−66−2q−68−5q−70 + 6q−72−7q−74 + 4q−76−2q−78 + q−80 |
Further Quantum Invariants
Further quantum knot invariants for 10_33.
A1 Invariants.
| Weight | Invariant |
|---|---|
| 1 | −q11 + 2q9−2q7 + 3q5−2q3 + q + q−1−2q−3 + 3q−5−2q−7 + 2q−9−q−11 |
| 2 | q32−2q30−q28 + 6q26−5q24−6q22 + 13q20−4q18−13q16 + 18q14 + q12−18q10 + 13q8 + 6q6−13q4 + q2 + 9 + q−2−13q−4 + 6q−6 + 13q−8−18q−10 + q−12 + 18q−14−13q−16−4q−18 + 13q−20−6q−22−5q−24 + 6q−26−q−28−2q−30 + q−32 |
| 3 | −q63 + 2q61 + q59−3q57−3q55 + 5q53 + 8q51−9q49−14q47 + 10q45 + 23q43−10q41−35q39 + 4q37 + 51q35 + 6q33−61q31−25q29 + 69q27 + 45q25−66q23−63q21 + 56q19 + 76q17−42q15−78q13 + 20q11 + 73q9 + q7−61q5−21q3 + 44q + 44q−1−21q−3−61q−5 + q−7 + 73q−9 + 20q−11−78q−13−42q−15 + 76q−17 + 56q−19−63q−21−66q−23 + 45q−25 + 69q−27−25q−29−61q−31 + 6q−33 + 51q−35 + 4q−37−35q−39−10q−41 + 23q−43 + 10q−45−14q−47−9q−49 + 8q−51 + 5q−53−3q−55−3q−57 + q−59 + 2q−61−q−63 |
| 4 | q104−2q102−q100 + 3q98 + 3q94−8q92−3q90 + 11q88 + q86 + 9q84−24q82−15q80 + 28q78 + 16q76 + 24q74−58q72−54q70 + 38q68 + 61q66 + 91q64−82q62−151q60−31q58 + 98q56 + 245q54−q52−237q50−218q48 + 6q46 + 393q44 + 216q42−179q40−398q38−220q36 + 375q34 + 411q32 + 21q30−408q28−406q26 + 200q24 + 430q22 + 202q20−256q18−418q16−3q14 + 297q12 + 276q10−65q8−310q6−165q4 + 118q2 + 283 + 118q−2−165q−4−310q−6−65q−8 + 276q−10 + 297q−12−3q−14−418q−16−256q−18 + 202q−20 + 430q−22 + 200q−24−406q−26−408q−28 + 21q−30 + 411q−32 + 375q−34−220q−36−398q−38−179q−40 + 216q−42 + 393q−44 + 6q−46−218q−48−237q−50−q−52 + 245q−54 + 98q−56−31q−58−151q−60−82q−62 + 91q−64 + 61q−66 + 38q−68−54q−70−58q−72 + 24q−74 + 16q−76 + 28q−78−15q−80−24q−82 + 9q−84 + q−86 + 11q−88−3q−90−8q−92 + 3q−94 + 3q−98−q−100−2q−102 + q−104 |
| 5 | −q155 + 2q153 + q151−3q149 + 3q141 + 2q139−7q137−5q135 + 7q133 + 9q131 + 5q129−7q127−22q125−20q123 + 17q121 + 55q119 + 38q117−21q115−89q113−98q111−4q109 + 142q107 + 197q105 + 68q103−161q101−321q99−234q97 + 111q95 + 454q93 + 475q91 + 74q89−504q87−780q85−421q83 + 393q81 + 1046q79 + 927q77−58q75−1179q73−1467q71−522q69 + 1052q67 + 1955q65 + 1250q63−658q61−2208q59−1983q57 + 6q55 + 2170q53 + 2578q51 + 734q49−1836q47−2882q45−1428q43 + 1275q41 + 2871q39 + 1948q37−653q35−2575q33−2178q31 + 55q29 + 2088q27 + 2182q25 + 404q23−1548q21−1986q19−711q17 + 1020q15 + 1708q13 + 910q11−563q9−1444q7−1048q5 + 181q3 + 1215q + 1215q−1 + 181q−3−1048q−5−1444q−7−563q−9 + 910q−11 + 1708q−13 + 1020q−15−711q−17−1986q−19−1548q−21 + 404q−23 + 2182q−25 + 2088q−27 + 55q−29−2178q−31−2575q−33−653q−35 + 1948q−37 + 2871q−39 + 1275q−41−1428q−43−2882q−45−1836q−47 + 734q−49 + 2578q−51 + 2170q−53 + 6q−55−1983q−57−2208q−59−658q−61 + 1250q−63 + 1955q−65 + 1052q−67−522q−69−1467q−71−1179q−73−58q−75 + 927q−77 + 1046q−79 + 393q−81−421q−83−780q−85−504q−87 + 74q−89 + 475q−91 + 454q−93 + 111q−95−234q−97−321q−99−161q−101 + 68q−103 + 197q−105 + 142q−107−4q−109−98q−111−89q−113−21q−115 + 38q−117 + 55q−119 + 17q−121−20q−123−22q−125−7q−127 + 5q−129 + 9q−131 + 7q−133−5q−135−7q−137 + 2q−139 + 3q−141−3q−149 + q−151 + 2q−153−q−155 |
A2 Invariants.
| Weight | Invariant |
|---|---|
| 1,0 | −q16 + q14 + q12−2q10 + 2q8−q4 + 2q2−1 + 2q−2−q−4 + 2q−8−2q−10 + q−12 + q−14−q−16 |
| 1,1 | q44−4q42 + 10q40−22q38 + 44q36−74q34 + 112q32−166q30 + 222q28−278q26 + 330q24−362q22 + 371q20−340q18 + 270q16−158q14 + 8q12 + 164q10−340q8 + 510q6−645q4 + 732q2−762 + 732q−2−645q−4 + 510q−6−340q−8 + 164q−10 + 8q−12−158q−14 + 270q−16−340q−18 + 371q−20−362q−22 + 330q−24−278q−26 + 222q−28−166q−30 + 112q−32−74q−34 + 44q−36−22q−38 + 10q−40−4q−42 + q−44 |
| 2,0 | q42−q40−2q38 + q36 + 4q34−8q30−q28 + 9q26 + q24−11q22−q20 + 14q18 + 4q16−13q14 + 12q10−2q8−7q6 + 2q4 + 2q2−2 + 2q−2 + 2q−4−7q−6−2q−8 + 12q−10−13q−14 + 4q−16 + 14q−18−q−20−11q−22 + q−24 + 9q−26−q−28−8q−30 + 4q−34 + q−36−2q−38−q−40 + q−42 |
A3 Invariants.
| Weight | Invariant |
|---|---|
| 0,1,0 | q34−2q32 + 3q28−6q26 + 3q24 + 8q22−11q20 + 3q18 + 12q16−16q14−q12 + 13q10−11q8−3q6 + 10q4 + q2−2 + q−2 + 10q−4−3q−6−11q−8 + 13q−10−q−12−16q−14 + 12q−16 + 3q−18−11q−20 + 8q−22 + 3q−24−6q−26 + 3q−28−2q−32 + q−34 |
| 1,0,0 | −q21 + q19 + q15−2q13 + 2q11−q9 + q7−q5 + 2q3 + 2q−3−q−5 + q−7−q−9 + 2q−11−2q−13 + q−15 + q−19−q−21 |
B2 Invariants.
| Weight | Invariant |
|---|---|
| 0,1 | −q34 + 2q32−4q30 + 7q28−10q26 + 13q24−16q22 + 17q20−17q18 + 16q16−10q14 + 5q12 + 5q10−13q8 + 21q6−28q4 + 33q2−36 + 33q−2−28q−4 + 21q−6−13q−8 + 5q−10 + 5q−12−10q−14 + 16q−16−17q−18 + 17q−20−16q−22 + 13q−24−10q−26 + 7q−28−4q−30 + 2q−32−q−34 |
| 1,0 | q56−2q52−2q50 + 2q48 + 5q46−q44−8q42−4q40 + 9q38 + 12q36−4q34−16q32−5q30 + 15q28 + 15q26−9q24−19q22−2q20 + 17q18 + 8q16−12q14−11q12 + 7q10 + 12q8−3q6−11q4 + 2q2 + 13 + 2q−2−11q−4−3q−6 + 12q−8 + 7q−10−11q−12−12q−14 + 8q−16 + 17q−18−2q−20−19q−22−9q−24 + 15q−26 + 15q−28−5q−30−16q−32−4q−34 + 12q−36 + 9q−38−4q−40−8q−42−q−44 + 5q−46 + 2q−48−2q−50−2q−52 + q−56 |
G2 Invariants.
| Weight | Invariant |
|---|---|
| 1,0 | q80−2q78 + 4q76−7q74 + 6q72−5q70−2q68 + 14q66−23q64 + 32q62−32q60 + 19q58 + 3q56−33q54 + 60q52−73q50 + 67q48−37q46−9q44 + 57q42−88q40 + 96q38−70q36 + 20q34 + 31q32−69q30 + 73q28−46q26 + 45q22−65q20 + 47q18−3q16−55q14 + 102q12−111q10 + 80q8−14q6−61q4 + 124q2−145 + 124q−2−61q−4−14q−6 + 80q−8−111q−10 + 102q−12−55q−14−3q−16 + 47q−18−65q−20 + 45q−22−46q−26 + 73q−28−69q−30 + 31q−32 + 20q−34−70q−36 + 96q−38−88q−40 + 57q−42−9q−44−37q−46 + 67q−48−73q−50 + 60q−52−33q−54 + 3q−56 + 19q−58−32q−60 + 32q−62−23q−64 + 14q−66−2q−68−5q−70 + 6q−72−7q−74 + 4q−76−2q−78 + q−80 |
.
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["10 33"];
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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]=
| 4t2−16t + 25−16t−1 + 4t−2 |
In[5]:=
| Conway[K][z]
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Out[5]=
| 4z4 + 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]=
| { 65, 0 } |
In[8]:=
| Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
| −q5 + 3q4−5q3 + 8q2−10q + 11−10q−1 + 8q−2−5q−3 + 3q−4−q−5 |
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]=
| −z2a4 + z4a2 + 2z4 + 2z2 + 1 + z4a−2−z2a−4 |
In[10]:=
| Kauffman[K][a, z]
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KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
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Out[10]=
| az9 + z9a−1 + 3a2z8 + 3z8a−2 + 6z8 + 4a3z7 + 5az7 + 5z7a−1 + 4z7a−3 + 3a4z6−4a2z6−4z6a−2 + 3z6a−4−14z6 + a5z5−9a3z5−16az5−16z5a−1−9z5a−3 + z5a−5−7a4z4 + a2z4 + z4a−2−7z4a−4 + 16z4−2a5z3 + 6a3z3 + 18az3 + 18z3a−1 + 6z3a−3−2z3a−5 + 3a4z2 + 3z2a−4−6z2−2a3z−6az−6za−1−2za−3 + 1 |
[edit] "Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {K11a333,}
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 33"];
|
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]=
| { 4t2−16t + 25−16t−1 + 4t−2, −q5 + 3q4−5q3 + 8q2−10q + 11−10q−1 + 8q−2−5q−3 + 3q−4−q−5 } |
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]=
| {K11a333,} |
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 = 0 is the signature of 10 33. 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 | q15−3q14 + q13 + 8q12−14q11 + 27q9−31q8−9q7 + 58q6−48q5−28q4 + 89q3−55q2−47q + 103−47q−1−55q−2 + 89q−3−28q−4−48q−5 + 58q−6−9q−7−31q−8 + 27q−9−14q−11 + 8q−12 + q−13−3q−14 + q−15 |
| 3 | −q30 + 3q29−q28−4q27−q26 + 11q25 + 2q24−21q23−6q22 + 35q21 + 15q20−54q19−31q18 + 74q17 + 62q16−99q15−98q14 + 110q13 + 156q12−123q11−209q10 + 113q9 + 275q8−103q7−327q6 + 77q5 + 373q4−50q3−399q2 + 15q + 413 + 15q−1−399q−2−50q−3 + 373q−4 + 77q−5−327q−6−103q−7 + 275q−8 + 113q−9−209q−10−123q−11 + 156q−12 + 110q−13−98q−14−99q−15 + 62q−16 + 74q−17−31q−18−54q−19 + 15q−20 + 35q−21−6q−22−21q−23 + 2q−24 + 11q−25−q−26−4q−27−q−28 + 3q−29−q−30 |
| 4 | q50−3q49 + q48 + 4q47−3q46 + 4q45−14q44 + 6q43 + 18q42−13q41 + 12q40−47q39 + 15q38 + 61q37−25q36 + 20q35−129q34 + 19q33 + 153q32−2q31 + 50q30−302q29−50q28 + 273q27 + 127q26 + 197q25−548q24−286q23 + 292q22 + 351q21 + 584q20−725q19−681q18 + 73q17 + 529q16 + 1179q15−689q14−1071q13−356q12 + 531q11 + 1785q10−459q9−1299q8−814q7 + 369q6 + 2200q5−159q4−1320q3−1155q2 + 124q + 2345 + 124q−1−1155q−2−1320q−3−159q−4 + 2200q−5 + 369q−6−814q−7−1299q−8−459q−9 + 1785q−10 + 531q−11−356q−12−1071q−13−689q−14 + 1179q−15 + 529q−16 + 73q−17−681q−18−725q−19 + 584q−20 + 351q−21 + 292q−22−286q−23−548q−24 + 197q−25 + 127q−26 + 273q−27−50q−28−302q−29 + 50q−30−2q−31 + 153q−32 + 19q−33−129q−34 + 20q−35−25q−36 + 61q−37 + 15q−38−47q−39 + 12q−40−13q−41 + 18q−42 + 6q−43−14q−44 + 4q−45−3q−46 + 4q−47 + q−48−3q−49 + q−50 |
[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. |
|
