10 34
From Knot Atlas
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 (KnotPlot image) |
See the full Rolfsen Knot Table. Visit 10 34's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 10_34's page at Knotilus! Visit 10 34's page at the original Knot Atlas! |
[edit] Knot presentations
| Planar diagram presentation | X1425 X3849 X13,17,14,16 X5,15,6,14 X15,7,16,6 X9,1,10,20 X11,19,12,18 X17,13,18,12 X19,11,20,10 X7283 |
| Gauss code | -1, 10, -2, 1, -4, 5, -10, 2, -6, 9, -7, 8, -3, 4, -5, 3, -8, 7, -9, 6 |
| Dowker-Thistlethwaite code | 4 8 14 2 20 18 16 6 12 10 |
| Conway Notation | [2512] |
| 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}, {2, 4}, {6, 3}, {5, 7}, {8, 6}, {7, 9}, {11, 8}, {9, 1}] |
[edit Notes on presentations of 10 34]
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[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
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In[3]:=
| K = Knot["10 34"];
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In[4]:=
| PD[K]
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
| X1425 X3849 X13,17,14,16 X5,15,6,14 X15,7,16,6 X9,1,10,20 X11,19,12,18 X17,13,18,12 X19,11,20,10 X7283 |
In[5]:=
| GaussCode[K]
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Out[5]=
| -1, 10, -2, 1, -4, 5, -10, 2, -6, 9, -7, 8, -3, 4, -5, 3, -8, 7, -9, 6 |
In[6]:=
| DTCode[K]
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Out[6]=
| 4 8 14 2 20 18 16 6 12 10 |
(The path below may be different on your system)
In[7]:=
| AppendTo[$Path, "C:/bin/LinKnot/"];
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In[8]:=
| ConwayNotation[K]
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Out[8]=
| [2512] |
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,−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}, {2, 4}, {6, 3}, {5, 7}, {8, 6}, {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 | 3t2−9t + 13−9t−1 + 3t−2 |
| Conway polynomial | 3z4 + 3z2 + 1 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 37, 0 } |
| Jones polynomial | −q7 + 2q6−3q5 + 4q4−5q3 + 6q2−5q + 5−3q−1 + 2q−2−q−3 |
| HOMFLY-PT polynomial (db, data sources) | z4a−2 + z4a−4 + z4−a2z2 + z2a−2 + 2z2a−4−z2a−6 + 2z2−a2 + a−4−a−6 + 2 |
| Kauffman polynomial (db, data sources) | z9a−3 + z9a−5 + 2z8a−2 + 4z8a−4 + 2z8a−6 + 2z7a−1−z7a−3−2z7a−5 + z7a−7−5z6a−2−17z6a−4−10z6a−6 + 2z6 + 2az5−2z5a−1−5z5a−3−6z5a−5−5z5a−7 + 2a2z4 + 4z4a−2 + 20z4a−4 + 14z4a−6 + a3z3−z3a−1 + 5z3a−3 + 12z3a−5 + 7z3a−7−2a2z2−3z2a−2−8z2a−4−6z2a−6−3z2−a3z−az−za−3−4za−5−3za−7 + a2 + a−4 + a−6 + 2 |
| The A2 invariant | −q10−q4 + 2q2 + 1 + q−2 + q−4 + q−8 + q−14−q−16−q−22 |
| The G2 invariant | q52−q50 + 2q48−2q46−3q40 + 4q38−5q36 + 4q34−4q32 + 3q28−6q26 + 7q24−7q22 + 4q20−2q18−2q16 + 5q14−5q12 + 6q10−2q8 + 2q6 + q2 + 2−q−2 + 5q−4−4q−6 + 4q−8−q−10−2q−12 + 4q−14−3q−16 + 2q−18 + 2q−20−4q−22 + 3q−24 + q−26−6q−28 + 11q−30−11q−32 + 7q−34 + q−36−7q−38 + 15q−40−15q−42 + 12q−44−6q−46−2q−48 + 8q−50−10q−52 + 11q−54−6q−56 + 2q−58 + 3q−60−6q−62 + 5q−64−q−66−6q−68 + 9q−70−9q−72 + 4q−74 + 4q−76−12q−78 + 16q−80−15q−82 + 7q−84−10q−88 + 12q−90−12q−92 + 8q−94−2q−96−2q−98 + 3q−100−4q−102 + 3q−104−q−106 + q−108 |
Further Quantum Invariants
Further quantum knot invariants for 10_34.
A1 Invariants.
| Weight | Invariant |
|---|---|
| 1 | −q7 + q5−q3 + 2q + q−3 + q−5−q−7 + q−9−q−11 + q−13−q−15 |
| 2 | q20−q18−q16 + 2q14−2q12−q10 + 2q8−2q6 + q4 + 2q2−2 + 2q−2 + 3q−4−q−6−q−8 + 3q−10 + q−12−2q−14 + 2q−18−2q−20−2q−22 + 3q−24−q−26−4q−28 + 3q−30 + 2q−32−4q−34 + q−36 + 3q−38−2q−40−q−42 + q−44 |
| 3 | −q39 + q37 + q35−2q31 + q29 + 3q27−q25−4q23−2q21 + 5q19 + 2q17−4q15−6q13 + 3q11 + 8q9 + q7−11q5−2q3 + 8q + 8q−1−5q−3−7q−5 + 2q−7 + 7q−9 + 6q−11−2q−13−6q−15−q−17 + 9q−19 + 3q−21−8q−23−6q−25 + 7q−27 + 3q−29−7q−31−5q−33 + 6q−35 + 5q−37−3q−39−6q−41 + q−43 + 5q−45 + 2q−47−4q−49−6q−51 + q−53 + 8q−55 + 3q−57−6q−59−7q−61 + 4q−63 + 9q−65−q−67−8q−69−3q−71 + 6q−73 + 5q−75−3q−77−4q−79 + 2q−83 + q−85−q−87 |
| 4 | q64−q62−q60 + 3q54−3q52−q50 + q48 + 2q46 + 7q44−7q42−6q40−q38 + 7q36 + 15q34−10q32−16q30−8q28 + 14q26 + 29q24−9q22−29q20−21q18 + 19q16 + 46q14−39q10−39q8 + 15q6 + 57q4 + 19q2−30−48q−2−7q−4 + 43q−6 + 36q−8−33q−12−28q−14 + 6q−16 + 23q−18 + 25q−20 + 3q−22−25q−24−22q−26 + q−28 + 25q−30 + 21q−32−14q−34−27q−36−7q−38 + 21q−40 + 24q−42−13q−44−32q−46−6q−48 + 25q−50 + 29q−52−10q−54−38q−56−13q−58 + 22q−60 + 35q−62 + 3q−64−34q−66−22q−68 + 6q−70 + 29q−72 + 19q−74−14q−76−18q−78−12q−80 + 7q−82 + 17q−84 + 4q−86 + 3q−88−11q−90−13q−92−q−94 + 2q−96 + 19q−98 + 8q−100−7q−102−13q−104−18q−106 + 10q−108 + 17q−110 + 10q−112−2q−114−21q−116−7q−118 + 4q−120 + 12q−122 + 11q−124−7q−126−7q−128−5q−130 + q−132 + 6q−134 + q−136−2q−140−q−142 + q−144 |
| 5 | −q95 + q93 + q91−q85−q83 + q81 + q79−2q77−q75−2q73 + 2q71 + 6q69 + 3q67−5q65−9q63−5q61 + 6q59 + 16q57 + 12q55−9q53−25q51−16q49 + 13q47 + 30q45 + 22q43−12q41−44q39−26q37 + 24q35 + 52q33 + 23q31−34q29−68q27−28q25 + 56q23 + 86q21 + 27q19−73q17−115q15−36q13 + 87q11 + 139q9 + 59q7−81q5−162q3−91q + 67q−1 + 160q−3 + 122q−5−14q−7−138q−9−140q−11−30q−13 + 93q−15 + 132q−17 + 75q−19−28q−21−102q−23−99q−25−25q−27 + 60q−29 + 93q−31 + 66q−33−15q−35−85q−37−80q−39−9q−41 + 65q−43 + 81q−45 + 18q−47−60q−49−76q−51−14q−53 + 61q−55 + 75q−57 + 8q−59−72q−61−83q−63−10q−65 + 78q−67 + 100q−69 + 19q−71−84q−73−115q−75−42q−77 + 76q−79 + 134q−81 + 71q−83−60q−85−138q−87−98q−89 + 26q−91 + 134q−93 + 128q−95 + 9q−97−114q−99−138q−101−50q−103 + 77q−105 + 135q−107 + 81q−109−33q−111−114q−113−95q−115−9q−117 + 73q−119 + 91q−121 + 42q−123−30q−125−67q−127−49q−129−6q−131 + 29q−133 + 37q−135 + 25q−137 + 3q−139−14q−141−18q−143−21q−145−15q−147−q−149 + 19q−151 + 29q−153 + 22q−155 + 3q−157−25q−159−37q−161−23q−163 + 9q−165 + 32q−167 + 33q−169 + 14q−171−17q−173−33q−175−25q−177 + 19q−181 + 24q−183 + 15q−185−6q−187−17q−189−14q−191−3q−193 + 5q−195 + 9q−197 + 7q−199−q−201−4q−203−3q−205−q−207 + 2q−211 + q−213−q−215 |
A2 Invariants.
| Weight | Invariant |
|---|---|
| 1,0 | −q10−q4 + 2q2 + 1 + q−2 + q−4 + q−8 + q−14−q−16−q−22 |
| 1,1 | q28−2q26 + 4q24−6q22 + 9q20−12q18 + 14q16−18q14 + 17q12−20q10 + 16q8−16q6 + 12q4−8q2 + 8−2q−2 + 10q−4−4q−6 + 10q−8−2q−10 + q−12 + 12q−14−24q−16 + 38q−18−53q−20 + 62q−22−70q−24 + 68q−26−65q−28 + 52q−30−38q−32 + 18q−34 + q−36−20q−38 + 36q−40−46q−42 + 49q−44−46q−46 + 40q−48−30q−50 + 19q−52−12q−54 + 6q−56−2q−58 + q−60 |
| 2,0 | q26−q22 + q18−q16−2q14 + q12 + q10−4q8−3q6 + 3q4−1 + 2q−2 + 5q−4 + 3q−6 + q−8 + 3q−10 + 2q−12 + 2q−16−2q−20−q−22 + q−24−q−26−2q−28−q−30 + q−32−3q−36−q−38 + q−40 + q−42−q−44−q−46 + 2q−48 + q−50−q−52−q−54 + q−58 |
A3 Invariants.
| Weight | Invariant |
|---|---|
| 0,1,0 | q22−q20 + 2q16−3q14−3q12 + 2q10−2q8−3q6 + 4q4 + 2q2 + 3q−2 + 3q−4−q−6 + q−8 + 2q−10 + 2q−12−q−14 + q−16 + 3q−18−2q−20 + 2q−24−2q−26−q−28−3q−32−q−34 + q−36−2q−38 + q−40 + q−42−q−44 + q−46 |
| 1,0,0 | −q13−q9−q5 + 2q3 + q + 2q−1 + q−3 + q−5 + q−11 + q−15 + q−19−q−21−q−25−q−29 |
B2 Invariants.
| Weight | Invariant |
|---|---|
| 0,1 | −q22 + q20−2q18 + 2q16−3q14 + 3q12−4q10 + 4q8−3q6 + 4q4−2q2 + 2 + q−2−q−4 + 5q−6−5q−8 + 8q−10−8q−12 + 9q−14−9q−16 + 7q−18−6q−20 + 4q−22−2q−24 + 3q−28−4q−30 + 5q−32−5q−34 + 5q−36−4q−38 + 3q−40−3q−42 + q−44−q−46 |
| 1,0 | q36−q32−q30 + q28 + 2q26−3q22−3q20 + 3q16 + q14−3q12−3q10 + 4q6 + 2q4−q2−2 + 2q−2 + 4q−4 + 2q−6−3q−8−q−10 + 3q−12 + 4q−14−q−16−2q−18 + q−20 + 3q−22−2q−26 + q−28 + 3q−30−4q−34−2q−36 + 3q−38 + 4q−40−q−42−6q−44−2q−46 + 4q−48 + 3q−50−3q−52−5q−54 + 4q−58 + q−60−3q−62−2q−64 + 2q−66 + 2q−68−q−70−q−72 + q−76 |
G2 Invariants.
| Weight | Invariant |
|---|---|
| 1,0 | q52−q50 + 2q48−2q46−3q40 + 4q38−5q36 + 4q34−4q32 + 3q28−6q26 + 7q24−7q22 + 4q20−2q18−2q16 + 5q14−5q12 + 6q10−2q8 + 2q6 + q2 + 2−q−2 + 5q−4−4q−6 + 4q−8−q−10−2q−12 + 4q−14−3q−16 + 2q−18 + 2q−20−4q−22 + 3q−24 + q−26−6q−28 + 11q−30−11q−32 + 7q−34 + q−36−7q−38 + 15q−40−15q−42 + 12q−44−6q−46−2q−48 + 8q−50−10q−52 + 11q−54−6q−56 + 2q−58 + 3q−60−6q−62 + 5q−64−q−66−6q−68 + 9q−70−9q−72 + 4q−74 + 4q−76−12q−78 + 16q−80−15q−82 + 7q−84−10q−88 + 12q−90−12q−92 + 8q−94−2q−96−2q−98 + 3q−100−4q−102 + 3q−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.
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In[3]:=
| K = Knot["10 34"];
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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]=
| 3t2−9t + 13−9t−1 + 3t−2 |
In[5]:=
| Conway[K][z]
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Out[5]=
| 3z4 + 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]=
| { 37, 0 } |
In[8]:=
| Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
| −q7 + 2q6−3q5 + 4q4−5q3 + 6q2−5q + 5−3q−1 + 2q−2−q−3 |
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]=
| z4a−2 + z4a−4 + z4−a2z2 + z2a−2 + 2z2a−4−z2a−6 + 2z2−a2 + a−4−a−6 + 2 |
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 + 2z8a−2 + 4z8a−4 + 2z8a−6 + 2z7a−1−z7a−3−2z7a−5 + z7a−7−5z6a−2−17z6a−4−10z6a−6 + 2z6 + 2az5−2z5a−1−5z5a−3−6z5a−5−5z5a−7 + 2a2z4 + 4z4a−2 + 20z4a−4 + 14z4a−6 + a3z3−z3a−1 + 5z3a−3 + 12z3a−5 + 7z3a−7−2a2z2−3z2a−2−8z2a−4−6z2a−6−3z2−a3z−az−za−3−4za−5−3za−7 + a2 + a−4 + a−6 + 2 |
[edit] "Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {10_135,}
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 34"];
|
In[4]:=
| {A = Alexander[K][t], J = Jones[K][q]}
|
KnotTheory::loading: Loading precomputed data in PD4Knots`.
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
|
Out[4]=
| { 3t2−9t + 13−9t−1 + 3t−2, −q7 + 2q6−3q5 + 4q4−5q3 + 6q2−5q + 5−3q−1 + 2q−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_135,} |
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 34. 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 + 6q18−4q17−6q16 + 12q15−3q14−13q13 + 15q12 + q11−18q10 + 15q9 + 5q8−20q7 + 13q6 + 8q5−18q4 + 9q3 + 8q2−14q + 8 + 4q−1−10q−2 + 7q−3 + q−4−6q−5 + 4q−6−2q−8 + q−9 |
| 3 | −q42 + 2q41 + q40−2q39−5q38 + 3q37 + 9q36−q35−14q34−2q33 + 16q32 + 9q31−19q30−13q29 + 17q28 + 18q27−14q26−20q25 + 10q24 + 20q23−8q22−17q21 + 6q20 + 13q19−5q18−9q17 + 7q16 + 2q15−7q14 + q13 + 11q12−11q11−9q10 + 12q9 + 17q8−21q7−14q6 + 16q5 + 25q4−20q3−19q2 + 7q + 27−7q−1−19q−2−3q−3 + 18q−4 + 5q−5−12q−6−8q−7 + 9q−8 + 7q−9−6q−10−5q−11 + 2q−12 + 5q−13−3q−14−q−15 + 2q−17−q−18 |
[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. |
|
