10 117
From Knot Atlas
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 (KnotPlot image) |
See the full Rolfsen Knot Table. Visit 10 117's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 10_117's page at Knotilus! Visit 10 117's page at the original Knot Atlas! |
[edit] Knot presentations
| Planar diagram presentation | X1627 X5,16,6,17 X13,1,14,20 X7,15,8,14 X19,9,20,8 X3,11,4,10 X11,5,12,4 X9,19,10,18 X17,13,18,12 X15,2,16,3 |
| Gauss code | -1, 10, -6, 7, -2, 1, -4, 5, -8, 6, -7, 9, -3, 4, -10, 2, -9, 8, -5, 3 |
| Dowker-Thistlethwaite code | 6 10 16 14 18 4 20 2 12 8 |
| Conway Notation | [8*2:20] |
| Minimum Braid Representative | A Morse Link Presentation | An Arc Presentation | ||||
Length is 11, width is 4, Braid index is 4 | 
|  [{12, 4}, {3, 10}, {5, 11}, {4, 6}, {2, 5}, {7, 3}, {6, 9}, {10, 8}, {9, 13}, {8, 12}, {1, 7}, {13, 2}, {11, 1}] |
[edit Notes on presentations of 10 117]
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[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
|
In[3]:=
| K = Knot["10 117"];
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In[4]:=
| PD[K]
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
| X1627 X5,16,6,17 X13,1,14,20 X7,15,8,14 X19,9,20,8 X3,11,4,10 X11,5,12,4 X9,19,10,18 X17,13,18,12 X15,2,16,3 |
In[5]:=
| GaussCode[K]
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Out[5]=
| -1, 10, -6, 7, -2, 1, -4, 5, -8, 6, -7, 9, -3, 4, -10, 2, -9, 8, -5, 3 |
In[6]:=
| DTCode[K]
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Out[6]=
| 6 10 16 14 18 4 20 2 12 8 |
(The path below may be different on your system)
In[7]:=
| AppendTo[$Path, "C:/bin/LinKnot/"];
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In[8]:=
| ConwayNotation[K]
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Out[8]=
| [8*2:20] |
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(4,{1,1,2,2,−3,2,−1,2,−3,2,−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]=
| { 4, 11, 4 } |
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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|
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Out[12]=
| -Graphics- |
In[13]:=
| ap = ArcPresentation[K]
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Out[13]=
| ArcPresentation[{12, 4}, {3, 10}, {5, 11}, {4, 6}, {2, 5}, {7, 3}, {6, 9}, {10, 8}, {9, 13}, {8, 12}, {1, 7}, {13, 2}, {11, 1}] |
In[14]:=
| Draw[ap]
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|
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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 | 2t3−10t2 + 24t−31 + 24t−1−10t−2 + 2t−3 |
| Conway polynomial | 2z6 + 2z4 + 2z2 + 1 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 103, 2 } |
| Jones polynomial | −q8 + 4q7−9q6 + 13q5−16q4 + 18q3−16q2 + 13q−8 + 4q−1−q−2 |
| HOMFLY-PT polynomial (db, data sources) | z6a−2 + z6a−4 + 2z4a−2 + 2z4a−4−z4a−6−z4 + 2z2a−2 + 2z2a−4−z2a−6−z2 + a−2 + a−4−a−6 |
| Kauffman polynomial (db, data sources) | 3z9a−3 + 3z9a−5 + 7z8a−2 + 15z8a−4 + 8z8a−6 + 7z7a−1 + 9z7a−3 + 10z7a−5 + 8z7a−7−8z6a−2−28z6a−4−12z6a−6 + 4z6a−8 + 4z6 + az5−11z5a−1−26z5a−3−29z5a−5−14z5a−7 + z5a−9 + 17z4a−4 + 6z4a−6−5z4a−8−6z4−az3 + 5z3a−1 + 18z3a−3 + 21z3a−5 + 8z3a−7−z3a−9 + 2z2a−2−4z2a−4−3z2a−6 + z2a−8 + 2z2−za−1−3za−3−5za−5−3za−7−a−2 + a−4 + a−6 |
| The A2 invariant | −q6 + 2q4−q2−1 + 4q−2−3q−4 + 3q−6 + 3q−12−3q−14 + 3q−16−2q−18−2q−20 + 2q−22−q−24 |
| The G2 invariant | q32−3q30 + 7q28−13q26 + 15q24−14q22 + 3q20 + 21q18−49q16 + 81q14−100q12 + 87q10−40q8−54q6 + 173q4−269q2 + 308−240q−2 + 63q−4 + 178q−6−398q−8 + 501q−10−428q−12 + 190q−14 + 119q−16−379q−18 + 476q−20−352q−22 + 81q−24 + 223q−26−405q−28 + 367q−30−133q−32−203q−34 + 486q−36−585q−38 + 465q−40−143q−42−252q−44 + 583q−46−727q−48 + 629q−50−331q−52−66q−54 + 415q−56−593q−58 + 555q−60−307q−62−27q−64 + 311q−66−430q−68 + 315q−70−41q−72−263q−74 + 450q−76−428q−78 + 211q−80 + 108q−82−391q−84 + 522q−86−465q−88 + 250q−90 + 19q−92−247q−94 + 349q−96−321q−98 + 213q−100−69q−102−46q−104 + 107q−106−122q−108 + 94q−110−52q−112 + 18q−114 + 7q−116−16q−118 + 16q−120−13q−122 + 7q−124−3q−126 + q−128 |
Further Quantum Invariants
Further quantum knot invariants for 10_117.
A1 Invariants.
| Weight | Invariant |
|---|---|
| 1 | −q5 + 3q3−4q + 5q−1−3q−3 + 2q−5 + 2q−7−3q−9 + 4q−11−5q−13 + 3q−15−q−17 |
| 2 | q16−3q14 + 10q10−14q8−7q6 + 36q4−21q2−34 + 56q−2−3q−4−54q−6 + 40q−8 + 22q−10−40q−12 + q−14 + 31q−16−4q−18−38q−20 + 24q−22 + 33q−24−56q−26 + 3q−28 + 54q−30−40q−32−18q−34 + 41q−36−10q−38−17q−40 + 11q−42 + q−44−3q−46 + q−48 |
| 3 | −q33 + 3q31−6q27−q25 + 14q23 + 6q21−39q19−16q17 + 74q15 + 59q13−115q11−147q9 + 133q7 + 272q5−93q3−399q−20q−1 + 493q−3 + 173q−5−500q−7−332q−9 + 428q−11 + 451q−13−298q−15−495q−17 + 144q−19 + 475q−21 + 13q−23−414q−25−146q−27 + 323q−29 + 263q−31−219q−33−374q−35 + 107q−37 + 458q−39 + 28q−41−518q−43−170q−45 + 512q−47 + 319q−49−441q−51−431q−53 + 306q−55 + 478q−57−141q−59−437q−61−11q−63 + 335q−65 + 101q−67−210q−69−117q−71 + 93q−73 + 95q−75−28q−77−54q−79 + 5q−81 + 20q−83 + 2q−85−7q−87−q−89 + 3q−91−q−93 |
A2 Invariants.
| Weight | Invariant |
|---|---|
| 1,0 | −q6 + 2q4−q2−1 + 4q−2−3q−4 + 3q−6 + 3q−12−3q−14 + 3q−16−2q−18−2q−20 + 2q−22−q−24 |
| 1,1 | q20−6q18 + 20q16−50q14 + 107q12−206q10 + 360q8−594q6 + 914q4−1306q2 + 1748−2170q−2 + 2477q−4−2562q−6 + 2350q−8−1780q−10 + 871q−12 + 322q−14−1636q−16 + 2924q−18−4030q−20 + 4806q−22−5162q−24 + 5046q−26−4476q−28 + 3528q−30−2312q−32 + 982q−34 + 301q−36−1396q−38 + 2168q−40−2576q−42 + 2636q−44−2420q−46 + 2024q−48−1544q−50 + 1087q−52−708q−54 + 422q−56−230q−58 + 113q−60−50q−62 + 20q−64−6q−66 + q−68 |
| 2,0 | q18−2q16−2q14 + 6q12 + 2q10−11q8−6q6 + 17q4 + 10q2−22−8q−2 + 28q−4 + 5q−6−27q−8 + 2q−10 + 21q−12−2q−14−12q−16 + 10q−18 + 9q−20−15q−22 + 8q−24 + 8q−26−18q−28−7q−30 + 22q−32 + q−34−26q−36 + 3q−38 + 23q−40−q−42−24q−44 + 5q−46 + 17q−48−3q−50−9q−52−2q−54 + 6q−56−2q−60 + q−62 |
A3 Invariants.
| Weight | Invariant |
|---|---|
| 0,1,0 | q14−3q12 + q10 + 7q8−13q6 + 3q4 + 21q2−29 + 3q−2 + 36q−4−40q−6 + 2q−8 + 38q−10−29q−12−4q−14 + 23q−16−3q−18−9q−20−2q−22 + 21q−24−6q−26−30q−28 + 32q−30 + 2q−32−43q−34 + 34q−36 + 6q−38−32q−40 + 23q−42 + 3q−44−14q−46 + 8q−48 + q−50−3q−52 + q−54 |
| 1,0,0 | −q7 + 2q5−2q3 + 2q−2q−1 + 4q−3−3q−5 + 3q−7 + q−11 + q−13 + 3q−17−3q−19 + 3q−21−3q−23 + q−25−3q−27 + 2q−29−q−31 |
A4 Invariants.
| Weight | Invariant |
|---|---|
| 0,1,0,0 | q16−2q14−q12 + 5q10−2q8−8q6 + 6q4 + 12q2−11−14q−2 + 20q−4 + 15q−6−31q−8−6q−10 + 39q−12−5q−14−35q−16 + 20q−18 + 28q−20−25q−22−11q−24 + 36q−26 + q−28−30q−30 + 24q−32 + 24q−34−39q−36−11q−38 + 34q−40−13q−42−37q−44 + 14q−46 + 26q−48−17q−50−15q−52 + 20q−54 + 9q−56−15q−58−q−60 + 8q−62−2q−64−2q−66 + q−68 |
| 1,0,0,0 | −q8 + 2q6−2q4 + q2 + 1−2q−2 + 4q−4−3q−6 + 3q−8 + q−12 + q−14 + q−16 + q−18 + 3q−22−3q−24 + 3q−26−3q−28−3q−34 + 2q−36−q−38 |
B2 Invariants.
| Weight | Invariant |
|---|---|
| 0,1 | −q14 + 3q12−7q10 + 13q8−21q6 + 31q4−41q2 + 49−51q−2 + 48q−4−36q−6 + 20q−8 + 4q−10−29q−12 + 56q−14−77q−16 + 93q−18−99q−20 + 96q−22−83q−24 + 62q−26−36q−28 + 10q−30 + 14q−32−33q−34 + 46q−36−52q−38 + 50q−40−45q−42 + 35q−44−24q−46 + 14q−48−7q−50 + 3q−52−q−54 |
| 1,0 | q24−3q20−3q18 + 4q16 + 10q14−17q10−13q8 + 17q6 + 31q4−3q2−43−21q−2 + 39q−4 + 45q−6−19q−8−55q−10−6q−12 + 52q−14 + 26q−16−36q−18−34q−20 + 23q−22 + 37q−24−10q−26−36q−28 + 3q−30 + 36q−32 + 6q−34−34q−36−14q−38 + 33q−40 + 25q−42−31q−44−39q−46 + 20q−48 + 50q−50−2q−52−55q−54−25q−56 + 43q−58 + 44q−60−20q−62−46q−64−5q−66 + 34q−68 + 19q−70−15q−72−19q−74 + q−76 + 11q−78 + 4q−80−3q−82−3q−84 + q−88 |
D4 Invariants.
| Weight | Invariant |
|---|---|
| 1,0,0,0 | q18−3q16 + 4q14−6q12 + 11q10−17q8 + 20q6−24q4 + 34q2−39 + 38q−2−38q−4 + 41q−6−33q−8 + 20q−10−12q−12 + 4q−14 + 19q−16−34q−18 + 43q−20−53q−22 + 73q−24−73q−26 + 74q−28−74q−30 + 75q−32−58q−34 + 47q−36−42q−38 + 21q−40−4q−42−10q−44 + 13q−46−31q−48 + 39q−50−39q−52 + 40q−54−41q−56 + 37q−58−27q−60 + 23q−62−19q−64 + 12q−66−6q−68 + 4q−70−3q−72 + q−74 |
G2 Invariants.
| Weight | Invariant |
|---|---|
| 1,0 | q32−3q30 + 7q28−13q26 + 15q24−14q22 + 3q20 + 21q18−49q16 + 81q14−100q12 + 87q10−40q8−54q6 + 173q4−269q2 + 308−240q−2 + 63q−4 + 178q−6−398q−8 + 501q−10−428q−12 + 190q−14 + 119q−16−379q−18 + 476q−20−352q−22 + 81q−24 + 223q−26−405q−28 + 367q−30−133q−32−203q−34 + 486q−36−585q−38 + 465q−40−143q−42−252q−44 + 583q−46−727q−48 + 629q−50−331q−52−66q−54 + 415q−56−593q−58 + 555q−60−307q−62−27q−64 + 311q−66−430q−68 + 315q−70−41q−72−263q−74 + 450q−76−428q−78 + 211q−80 + 108q−82−391q−84 + 522q−86−465q−88 + 250q−90 + 19q−92−247q−94 + 349q−96−321q−98 + 213q−100−69q−102−46q−104 + 107q−106−122q−108 + 94q−110−52q−112 + 18q−114 + 7q−116−16q−118 + 16q−120−13q−122 + 7q−124−3q−126 + q−128 |
.
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["10 117"];
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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]=
| 2t3−10t2 + 24t−31 + 24t−1−10t−2 + 2t−3 |
In[5]:=
| Conway[K][z]
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Out[5]=
| 2z6 + 2z4 + 2z2 + 1 |
In[6]:=
| Alexander[K, 2][t]
|
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.
|
Out[6]=
| {1} |
In[7]:=
| {KnotDet[K], KnotSignature[K]}
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Out[7]=
| { 103, 2 } |
In[8]:=
| Jones[K][q]
|
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
|
Out[8]=
| −q8 + 4q7−9q6 + 13q5−16q4 + 18q3−16q2 + 13q−8 + 4q−1−q−2 |
In[9]:=
| HOMFLYPT[K][a, z]
|
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
|
Out[9]=
| z6a−2 + z6a−4 + 2z4a−2 + 2z4a−4−z4a−6−z4 + 2z2a−2 + 2z2a−4−z2a−6−z2 + a−2 + a−4−a−6 |
In[10]:=
| Kauffman[K][a, z]
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KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
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Out[10]=
| 3z9a−3 + 3z9a−5 + 7z8a−2 + 15z8a−4 + 8z8a−6 + 7z7a−1 + 9z7a−3 + 10z7a−5 + 8z7a−7−8z6a−2−28z6a−4−12z6a−6 + 4z6a−8 + 4z6 + az5−11z5a−1−26z5a−3−29z5a−5−14z5a−7 + z5a−9 + 17z4a−4 + 6z4a−6−5z4a−8−6z4−az3 + 5z3a−1 + 18z3a−3 + 21z3a−5 + 8z3a−7−z3a−9 + 2z2a−2−4z2a−4−3z2a−6 + z2a−8 + 2z2−za−1−3za−3−5za−5−3za−7−a−2 + a−4 + a−6 |
[edit] "Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {K11a23, K11a111,}
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["10 117"];
|
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]=
| { 2t3−10t2 + 24t−31 + 24t−1−10t−2 + 2t−3, −q8 + 4q7−9q6 + 13q5−16q4 + 18q3−16q2 + 13q−8 + 4q−1−q−2 } |
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]=
| {K11a23, K11a111,} |
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 = 2 is the signature of 10 117. 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 | q23−4q22 + 4q21 + 11q20−32q19 + 11q18 + 62q17−91q16−11q15 + 156q14−142q13−70q12 + 245q11−151q10−132q9 + 279q8−116q7−162q6 + 238q5−54q4−144q3 + 144q2−3q−85 + 54q−1 + 10q−2−28q−3 + 11q−4 + 3q−5−4q−6 + q−7 |
| 3 | −q45 + 4q44−4q43−6q42 + 8q41 + 22q40−19q39−65q38 + 34q37 + 145q36−21q35−275q34−59q33 + 456q32 + 213q31−621q30−485q29 + 752q28 + 832q27−793q26−1222q25 + 742q24 + 1592q23−600q22−1904q21 + 394q20 + 2138q19−170q18−2255q17−87q16 + 2293q15 + 312q14−2195q13−556q12 + 2025q11 + 739q10−1733q9−887q8 + 1386q7 + 936q6−984q5−910q4 + 626q3 + 768q2−311q−590 + 113q−1 + 389q−2−5q−3−225q−4−26q−5 + 109q−6 + 27q−7−51q−8−11q−9 + 19q−10 + 4q−11−6q−12−3q−13 + 4q−14−q−15 |
| 4 | q74−4q73 + 4q72 + 6q71−13q70 + 2q69−14q68 + 38q67 + 46q66−88q65−53q64−87q63 + 214q62 + 349q61−203q60−415q59−686q58 + 456q57 + 1495q56 + 419q55−901q54−2805q53−534q52 + 3266q51 + 3182q50 + 346q49−6096q48−4545q47 + 3246q46 + 7568q45 + 5377q44−7709q43−10783q42−763q41 + 10486q40 + 13037q39−5450q38−15867q37−7514q36 + 9848q35 + 19812q34−491q33−17645q32−13824q31 + 6586q30 + 23528q29 + 4657q28−16540q27−17947q26 + 2447q25 + 24171q24 + 8872q23−13455q22−19780q21−2028q20 + 21961q19 + 12073q18−8455q17−19075q16−6616q15 + 16639q14 + 13394q13−2078q12−15051q11−9723q10 + 8999q9 + 11381q8 + 3259q7−8428q6−9269q5 + 2126q4 + 6525q3 + 4878q2−2399q−5685−1045q−1 + 1945q−2 + 3200q−3 + 355q−4−2079q−5−1000q−6−61q−7 + 1128q−8 + 521q−9−418q−10−272q−11−228q−12 + 223q−13 + 156q−14−65q−15−11q−16−65q−17 + 34q−18 + 24q−19−18q−20 + 5q−21−9q−22 + 6q−23 + 3q−24−4q−25 + q−26 |
[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. |
|
