10 28
From Knot Atlas
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 (KnotPlot image) |
See the full Rolfsen Knot Table. Visit 10 28's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 10_28's page at Knotilus! Visit 10 28's page at the original Knot Atlas! |
[edit] Knot presentations
| Planar diagram presentation | X1425 X3,10,4,11 X13,19,14,18 X5,15,6,14 X17,7,18,6 X7,17,8,16 X15,9,16,8 X11,1,12,20 X19,13,20,12 X9,2,10,3 |
| Gauss code | -1, 10, -2, 1, -4, 5, -6, 7, -10, 2, -8, 9, -3, 4, -7, 6, -5, 3, -9, 8 |
| Dowker-Thistlethwaite code | 4 10 14 16 2 20 18 8 6 12 |
| Conway Notation | [31312] |
| Minimum Braid Representative | A Morse Link Presentation | An Arc Presentation | |||||
Length is 12, width is 5, Braid index is 5 | 
|  [{12, 2}, {1, 10}, {4, 11}, {10, 12}, {3, 5}, {6, 4}, {5, 7}, {2, 6}, {8, 3}, {7, 9}, {11, 8}, {9, 1}] |
[edit Notes on presentations of 10 28]
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[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
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In[3]:=
| K = Knot["10 28"];
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In[4]:=
| PD[K]
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
| X1425 X3,10,4,11 X13,19,14,18 X5,15,6,14 X17,7,18,6 X7,17,8,16 X15,9,16,8 X11,1,12,20 X19,13,20,12 X9,2,10,3 |
In[5]:=
| GaussCode[K]
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Out[5]=
| -1, 10, -2, 1, -4, 5, -6, 7, -10, 2, -8, 9, -3, 4, -7, 6, -5, 3, -9, 8 |
In[6]:=
| DTCode[K]
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Out[6]=
| 4 10 14 16 2 20 18 8 6 12 |
(The path below may be different on your system)
In[7]:=
| AppendTo[$Path, "C:/bin/LinKnot/"];
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In[8]:=
| ConwayNotation[K]
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Out[8]=
| [31312] |
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,−4,3,−4,−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, 2}, {1, 10}, {4, 11}, {10, 12}, {3, 5}, {6, 4}, {5, 7}, {2, 6}, {8, 3}, {7, 9}, {11, 8}, {9, 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 | 4t2−13t + 19−13t−1 + 4t−2 |
| Conway polynomial | 4z4 + 3z2 + 1 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 53, 0 } |
| Jones polynomial | −q7 + 2q6−4q5 + 6q4−7q3 + 9q2−8q + 7−5q−1 + 3q−2−q−3 |
| HOMFLY-PT polynomial (db, data sources) | 2z4a−2 + z4a−4 + z4−a2z2 + 4z2a−2 + z2a−4−z2a−6 + 3a−2−a−6−1 |
| Kauffman polynomial (db, data sources) | z9a−3 + z9a−5 + 3z8a−2 + 5z8a−4 + 2z8a−6 + 5z7a−1 + 4z7a−3 + z7a−7−z6a−2−16z6a−4−9z6a−6 + 6z6 + 5az5−6z5a−1−18z5a−3−12z5a−5−5z5a−7 + 3a2z4−12z4a−2 + 11z4a−4 + 12z4a−6−8z4 + a3z3−4az3−2z3a−1 + 13z3a−3 + 18z3a−5 + 8z3a−7−a2z2 + 10z2a−2−5z2a−6 + 4z2 + az + za−1−2za−3−6za−5−4za−7−3a−2 + a−6−1 |
| The A2 invariant | −q10 + q8 + q6−2q4 + q2−1 + 2q−4 + q−6 + 3q−8 + q−14−2q−16−q−22 |
| The G2 invariant | q52−2q50 + 3q48−4q46 + 2q44−q42−2q40 + 8q38−11q36 + 14q34−13q32 + 6q30−10q26 + 20q24−26q22 + 25q20−18q18 + 6q16 + 9q14−20q12 + 30q10−31q8 + 22q6−10q4−7q2 + 20−24q−2 + 21q−4−9q−6−5q−8 + 16q−10−21q−12 + 9q−14 + 10q−16−28q−18 + 38q−20−32q−22 + 12q−24 + 22q−26−45q−28 + 60q−30−53q−32 + 30q−34 + 7q−36−36q−38 + 56q−40−49q−42 + 34q−44−5q−46−18q−48 + 32q−50−29q−52 + 16q−54 + 4q−56−24q−58 + 29q−60−21q−62 + q−64 + 22q−66−40q−68 + 44q−70−34q−72 + 6q−74 + 20q−76−42q−78 + 47q−80−37q−82 + 15q−84 + 5q−86−22q−88 + 27q−90−23q−92 + 13q−94−2q−96−5q−98 + 6q−100−6q−102 + 4q−104−q−106 + q−108 |
Further Quantum Invariants
Further quantum knot invariants for 10_28.
A1 Invariants.
| Weight | Invariant |
|---|---|
| 1 | −q7 + 2q5−2q3 + 2q−q−1 + q−3 + 2q−5−q−7 + 2q−9−2q−11 + q−13−q−15 |
| 2 | q20−2q18 + 4q14−6q12 + q10 + 6q8−9q6 + 3q4 + 8q2−8 + 8q−4−2q−6−5q−8 + 4q−10 + 5q−12−5q−14−3q−16 + 9q−18−3q−20−8q−22 + 9q−24 + q−26−10q−28 + 5q−30 + 4q−32−7q−34 + q−36 + 4q−38−2q−40−q−42 + q−44 |
| 3 | −q39 + 2q37−2q33 + 3q29 + q27−6q25 + 2q23 + 4q21−5q19−5q17 + 10q15 + 5q13−16q11−5q9 + 21q7 + 8q5−23q3−14q + 23q−1 + 19q−3−12q−5−23q−7 + 2q−9 + 25q−11 + 14q−13−21q−15−21q−17 + 14q−19 + 29q−21−7q−23−32q−25−q−27 + 28q−29 + 8q−31−28q−33−13q−35 + 22q−37 + 23q−39−19q−41−26q−43 + 10q−45 + 30q−47−2q−49−30q−51−9q−53 + 27q−55 + 16q−57−18q−59−21q−61 + 8q−63 + 21q−65−15q−69−6q−71 + 9q−73 + 7q−75−3q−77−5q−79 + 2q−83 + q−85−q−87 |
| 4 | q64−2q62 + 2q58−2q56 + 3q54−5q52 + 2q50 + 4q48−6q46 + 11q44−8q42−q40−q38−13q36 + 28q34 + 2q32−7q30−25q28−28q26 + 59q24 + 36q22−13q20−72q18−65q16 + 86q14 + 97q12 + 12q10−110q8−126q6 + 64q4 + 136q2 + 81−73q−2−161q−4−26q−6 + 86q−8 + 129q−10 + 32q−12−103q−14−100q−16−34q−18 + 93q−20 + 121q−22 + 11q−24−101q−26−116q−28 + 17q−30 + 129q−32 + 85q−34−66q−36−130q−38−31q−40 + 103q−42 + 109q−44−45q−46−118q−48−54q−50 + 79q−52 + 128q−54−16q−56−105q−58−90q−60 + 37q−62 + 141q−64 + 41q−66−59q−68−124q−70−43q−72 + 108q−74 + 94q−76 + 28q−78−102q−80−111q−82 + 17q−84 + 78q−86 + 103q−88−13q−90−99q−92−60q−94−q−96 + 92q−98 + 57q−100−21q−102−55q−104−58q−106 + 23q−108 + 48q−110 + 29q−112−4q−114−43q−116−16q−118 + 6q−120 + 20q−122 + 17q−124−9q−126−9q−128−7q−130 + q−132 + 7q−134 + q−136−2q−140−q−142 + q−144 |
| 5 | −q95 + 2q93−2q89 + 2q87−q85−q83 + 2q81−3q77−q73 + 6q69 + 8q67−18q63−17q61 + 3q59 + 27q57 + 42q55 + 2q53−64q51−72q49 + 5q47 + 100q45 + 122q43 + 9q41−167q39−205q37−17q35 + 244q33 + 303q31 + 56q29−318q27−446q25−132q23 + 379q21 + 595q19 + 251q17−382q15−720q13−422q11 + 304q9 + 801q7 + 602q5−146q3−774q−739q−1−92q−3 + 627q−5 + 812q−7 + 332q−9−376q−11−743q−13−531q−15 + 64q−17 + 577q−19 + 630q−21 + 232q−23−322q−25−624q−27−447q−29 + 61q−31 + 525q−33 + 571q−35 + 150q−37−399q−39−599q−41−276q−43 + 276q−45 + 571q−47 + 335q−49−212q−51−536q−53−336q−55 + 185q−57 + 517q−59 + 337q−61−181q−63−538q−65−365q−67 + 185q−69 + 564q−71 + 418q−73−130q−75−590q−77−519q−79 + 46q−81 + 573q−83 + 600q−85 + 114q−87−487q−89−677q−91−285q−93 + 337q−95 + 670q−97 + 458q−99−116q−101−588q−103−571q−105−121q−107 + 405q−109 + 604q−111 + 328q−113−170q−115−515q−117−466q−119−81q−121 + 343q−123 + 487q−125 + 269q−127−113q−129−390q−131−375q−133−96q−135 + 226q−137 + 358q−139 + 231q−141−32q−143−252q−145−277q−147−111q−149 + 110q−151 + 224q−153 + 177q−155 + 25q−157−127q−159−171q−161−95q−163 + 28q−165 + 110q−167 + 107q−169 + 38q−171−43q−173−79q−175−58q−177−3q−179 + 38q−181 + 44q−183 + 25q−185−7q−187−26q−189−21q−191−4q−193 + 6q−195 + 11q−197 + 9q−199−q−201−5q−203−3q−205−q−207 + 2q−211 + q−213−q−215 |
A2 Invariants.
| Weight | Invariant |
|---|---|
| 1,0 | −q10 + q8 + q6−2q4 + q2−1 + 2q−4 + q−6 + 3q−8 + q−14−2q−16−q−22 |
| 1,1 | q28−4q26 + 8q24−12q22 + 20q20−32q18 + 42q16−50q14 + 62q12−78q10 + 82q8−86q6 + 92q4−94q2 + 86−70q−2 + 55q−4−24q−6−14q−8 + 66q−10−111q−12 + 172q−14−212q−16 + 246q−18−262q−20 + 258q−22−234q−24 + 184q−26−131q−28 + 70q−30−4q−32−60q−34 + 104q−36−136q−38 + 154q−40−156q−42 + 138q−44−112q−46 + 86q−48−58q−50 + 33q−52−18q−54 + 8q−56−2q−58 + q−60 |
| 2,0 | q26−q24−2q22 + 2q20 + 3q18−2q16−5q14 + 3q12 + 4q10−7q8−4q6 + 8q4 + 4q2−5−q−2 + 5q−4 + q−6−4q−8 + q−10 + 2q−12−q−14 + 6q−16 + 5q−18−q−20−q−22 + 5q−24 + q−26−6q−28−4q−30 + 4q−32 + q−34−7q−36−3q−38 + 3q−40 + 2q−42−3q−44−2q−46 + 3q−48 + 2q−50−q−52−q−54 + q−58 |
A3 Invariants.
| Weight | Invariant |
|---|---|
| 0,1,0 | q22−2q20−q18 + 5q16−3q14−4q12 + 8q10−2q8−8q6 + 7q4−8 + 5q−2 + 3q−4−4q−6 + 2q−8 + 5q−10 + 4q−12−3q−14 + 4q−16 + 8q−18−5q−20 + 7q−24−7q−26−3q−28 + 3q−30−7q−32−2q−34 + 3q−36−3q−38 + q−40 + 2q−42−q−44 + q−46 |
| 1,0,0 | −q13 + q11 + q7−2q5 + q3−2q + 2q−5 + 2q−7 + 2q−9 + 3q−11 + q−15−q−17 + q−19−2q−21−q−25−q−29 |
B2 Invariants.
| Weight | Invariant |
|---|---|
| 0,1 | −q22 + 2q20−3q18 + 5q16−7q14 + 8q12−10q10 + 10q8−10q6 + 9q4−6q2 + 2 + 3q−2−7q−4 + 14q−6−16q−8 + 21q−10−20q−12 + 21q−14−18q−16 + 14q−18−9q−20 + 4q−22 + q−24−5q−26 + 9q−28−11q−30 + 11q−32−10q−34 + 9q−36−7q−38 + 5q−40−4q−42 + q−44−q−46 |
| 1,0 | q36−2q32−2q30 + q28 + 5q26 + 2q24−5q22−6q20 + q18 + 9q16 + 4q14−7q12−9q10 + q8 + 10q6 + 3q4−8q2−7 + 5q−2 + 9q−4−q−6−9q−8−q−10 + 8q−12 + 5q−14−5q−16−3q−18 + 5q−20 + 6q−22−3q−24−4q−26 + 5q−28 + 9q−30−q−32−10q−34−3q−36 + 10q−38 + 8q−40−6q−42−13q−44−q−46 + 10q−48 + 4q−50−8q−52−9q−54 + 2q−56 + 7q−58 + q−60−5q−62−3q−64 + 3q−66 + 3q−68−q−70−q−72 + q−76 |
G2 Invariants.
| Weight | Invariant |
|---|---|
| 1,0 | q52−2q50 + 3q48−4q46 + 2q44−q42−2q40 + 8q38−11q36 + 14q34−13q32 + 6q30−10q26 + 20q24−26q22 + 25q20−18q18 + 6q16 + 9q14−20q12 + 30q10−31q8 + 22q6−10q4−7q2 + 20−24q−2 + 21q−4−9q−6−5q−8 + 16q−10−21q−12 + 9q−14 + 10q−16−28q−18 + 38q−20−32q−22 + 12q−24 + 22q−26−45q−28 + 60q−30−53q−32 + 30q−34 + 7q−36−36q−38 + 56q−40−49q−42 + 34q−44−5q−46−18q−48 + 32q−50−29q−52 + 16q−54 + 4q−56−24q−58 + 29q−60−21q−62 + q−64 + 22q−66−40q−68 + 44q−70−34q−72 + 6q−74 + 20q−76−42q−78 + 47q−80−37q−82 + 15q−84 + 5q−86−22q−88 + 27q−90−23q−92 + 13q−94−2q−96−5q−98 + 6q−100−6q−102 + 4q−104−q−106 + q−108 |
.
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 28"];
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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]=
| 4t2−13t + 19−13t−1 + 4t−2 |
In[5]:=
| Conway[K][z]
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Out[5]=
| 4z4 + 3z2 + 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]=
| { 53, 0 } |
In[8]:=
| Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
| −q7 + 2q6−4q5 + 6q4−7q3 + 9q2−8q + 7−5q−1 + 3q−2−q−3 |
In[9]:=
| HOMFLYPT[K][a, z]
|
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
|
Out[9]=
| 2z4a−2 + z4a−4 + z4−a2z2 + 4z2a−2 + z2a−4−z2a−6 + 3a−2−a−6−1 |
In[10]:=
| Kauffman[K][a, z]
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KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
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Out[10]=
| z9a−3 + z9a−5 + 3z8a−2 + 5z8a−4 + 2z8a−6 + 5z7a−1 + 4z7a−3 + z7a−7−z6a−2−16z6a−4−9z6a−6 + 6z6 + 5az5−6z5a−1−18z5a−3−12z5a−5−5z5a−7 + 3a2z4−12z4a−2 + 11z4a−4 + 12z4a−6−8z4 + a3z3−4az3−2z3a−1 + 13z3a−3 + 18z3a−5 + 8z3a−7−a2z2 + 10z2a−2−5z2a−6 + 4z2 + az + za−1−2za−3−6za−5−4za−7−3a−2 + a−6−1 |
[edit] "Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {10_37,}
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 28"];
|
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]=
| { 4t2−13t + 19−13t−1 + 4t−2, −q7 + 2q6−4q5 + 6q4−7q3 + 9q2−8q + 7−5q−1 + 3q−2−q−3 } |
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]=
| {10_37,} |
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 28. 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 | q21−2q20−q19 + 7q18−5q17−9q16 + 18q15−4q14−24q13 + 29q12 + 4q11−41q10 + 34q9 + 16q8−53q7 + 32q6 + 26q5−54q4 + 23q3 + 29q2−44q + 15 + 21q−1−28q−2 + 10q−3 + 9q−4−13q−5 + 5q−6 + 2q−7−3q−8 + q−9 |
| 3 | −q42 + 2q41 + q40−2q39−6q38 + 4q37 + 11q36−21q34−5q33 + 26q32 + 21q31−34q30−34q29 + 29q28 + 55q27−23q26−70q25 + 8q24 + 83q23 + 9q22−90q21−28q20 + 90q19 + 51q18−91q17−63q16 + 75q15 + 87q14−71q13−92q12 + 44q11 + 112q10−35q9−107q8 + 9q7 + 112q6−96q4−14q3 + 87q2 + 11q−65−10q−1 + 50q−2 + 2q−3−34q−4 + 3q−5 + 24q−6−9q−7−13q−8 + 8q−9 + 9q−10−9q−11−4q−12 + 6q−13 + q−14−2q−15−2q−16 + 3q−17−q−18 |
| 4 | q70−2q69−q68 + 2q67 + q66 + 7q65−8q64−9q63 + q61 + 33q60−5q59−23q58−22q57−26q56 + 72q55 + 28q54−4q53−47q52−107q51 + 75q50 + 62q49 + 74q48−12q47−200q46 + 16q45 + 23q44 + 160q43 + 104q42−225q41−45q40−105q39 + 169q38 + 234q37−159q36−31q35−256q34 + 88q33 + 299q32−59q31 + 69q30−360q29−39q28 + 284q27 + 30q26 + 213q25−409q24−172q23 + 220q22 + 103q21 + 367q20−415q19−306q18 + 121q17 + 167q16 + 518q15−371q14−418q13−12q12 + 182q11 + 630q10−261q9−446q8−139q7 + 116q6 + 627q5−126q4−349q3−182q2 + 4q + 492−38q−1−195q−2−127q−3−68q−4 + 302q−5−22q−6−73q−7−42q−8−79q−9 + 151q−10−29q−11−14q−12 + 7q−13−56q−14 + 64q−15−26q−16 + 4q−17 + 16q−18−30q−19 + 23q−20−14q−21 + 4q−22 + 9q−23−11q−24 + 6q−25−4q−26 + 2q−27 + 2q−28−3q−29 + q−30 |
[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. |
|
