K11a74
From Knot Atlas
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 (Knotscape image) | See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.
Visit K11a74's page at Knotilus! Visit K11a74's page at the original Knot Atlas! |
[edit] Knot presentations
| Planar diagram presentation | X4251 X10,3,11,4 X12,6,13,5 X14,8,15,7 X18,9,19,10 X2,11,3,12 X6,14,7,13 X20,16,21,15 X22,18,1,17 X8,19,9,20 X16,22,17,21 |
| Gauss code | 1, -6, 2, -1, 3, -7, 4, -10, 5, -2, 6, -3, 7, -4, 8, -11, 9, -5, 10, -8, 11, -9 |
| Dowker-Thistlethwaite code | 4 10 12 14 18 2 6 20 22 8 16 |
| A Braid Representative |
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| A Morse Link Presentation | 
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[edit] Three dimensional invariants
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[edit] Four dimensional invariants
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[edit] Polynomial invariants
| Alexander polynomial | t4−5t3 + 10t2−13t + 15−13t−1 + 10t−2−5t−3 + t−4 |
| Conway polynomial | z8 + 3z6−2z2 + 1 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 73, 4 } |
| Jones polynomial | −q9 + 3q8−5q7 + 8q6−10q5 + 11q4−11q3 + 9q2−7q + 5−2q−1 + q−2 |
| HOMFLY-PT polynomial (db, data sources) | z8a−4−2z6a−2 + 6z6a−4−z6a−6−10z4a−2 + 13z4a−4−4z4a−6 + z4−15z2a−2 + 13z2a−4−4z2a−6 + 4z2−7a−2 + 5a−4−a−6 + 4 |
| Kauffman polynomial (db, data sources) | z10a−2 + z10a−4 + 2z9a−1 + 6z9a−3 + 4z9a−5 + z8a−2 + 7z8a−4 + 7z8a−6 + z8−10z7a−1−23z7a−3−5z7a−5 + 8z7a−7−23z6a−2−40z6a−4−16z6a−6 + 7z6a−8−6z6 + 15z5a−1 + 18z5a−3−18z5a−5−16z5a−7 + 5z5a−9 + 47z4a−2 + 51z4a−4 + 5z4a−6−9z4a−8 + 3z4a−10 + 13z4−6z3a−1 + 8z3a−3 + 26z3a−5 + 8z3a−7−3z3a−9 + z3a−11−32z2a−2−23z2a−4 + 2z2a−8−z2a−10−12z2−za−1−7za−3−9za−5−3za−7 + 7a−2 + 5a−4 + a−6 + 4 |
| The A2 invariant | q6 + q4 + q2 + 2−q−2−2q−6−2q−8 + q−10−2q−12 + 3q−14 + q−18 + q−20−q−22 + q−24−q−26 |
| The G2 invariant | q26−q24 + 5q22−7q20 + 10q18−9q16 + 2q14 + 15q12−31q10 + 46q8−42q6 + 23q4 + 15q2−53 + 84q−2−81q−4 + 53q−6−2q−8−52q−10 + 82q−12−81q−14 + 50q−16−4q−18−40q−20 + 57q−22−50q−24 + 11q−26 + 25q−28−58q−30 + 60q−32−40q−34−7q−36 + 51q−38−87q−40 + 94q−42−70q−44 + 20q−46 + 40q−48−85q−50 + 103q−52−82q−54 + 41q−56 + 18q−58−54q−60 + 67q−62−48q−64 + 15q−66 + 24q−68−40q−70 + 35q−72−9q−74−21q−76 + 43q−78−46q−80 + 32q−82−8q−84−19q−86 + 35q−88−43q−90 + 38q−92−25q−94 + 9q−96 + 5q−98−20q−100 + 27q−102−31q−104 + 28q−106−18q−108 + 7q−110 + 7q−112−19q−114 + 23q−116−23q−118 + 18q−120−8q−122 + 7q−126−12q−128 + 13q−130−10q−132 + 7q−134−2q−136−q−138 + 2q−140−4q−142 + 3q−144−2q−146 + q−148 |
Further Quantum Invariants
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["K11a74"];
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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]=
| t4−5t3 + 10t2−13t + 15−13t−1 + 10t−2−5t−3 + t−4 |
In[5]:=
| Conway[K][z]
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Out[5]=
| z8 + 3z6−2z2 + 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]=
| { 73, 4 } |
In[8]:=
| Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
| −q9 + 3q8−5q7 + 8q6−10q5 + 11q4−11q3 + 9q2−7q + 5−2q−1 + q−2 |
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]=
| z8a−4−2z6a−2 + 6z6a−4−z6a−6−10z4a−2 + 13z4a−4−4z4a−6 + z4−15z2a−2 + 13z2a−4−4z2a−6 + 4z2−7a−2 + 5a−4−a−6 + 4 |
In[10]:=
| Kauffman[K][a, z]
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KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
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Out[10]=
| z10a−2 + z10a−4 + 2z9a−1 + 6z9a−3 + 4z9a−5 + z8a−2 + 7z8a−4 + 7z8a−6 + z8−10z7a−1−23z7a−3−5z7a−5 + 8z7a−7−23z6a−2−40z6a−4−16z6a−6 + 7z6a−8−6z6 + 15z5a−1 + 18z5a−3−18z5a−5−16z5a−7 + 5z5a−9 + 47z4a−2 + 51z4a−4 + 5z4a−6−9z4a−8 + 3z4a−10 + 13z4−6z3a−1 + 8z3a−3 + 26z3a−5 + 8z3a−7−3z3a−9 + z3a−11−32z2a−2−23z2a−4 + 2z2a−8−z2a−10−12z2−za−1−7za−3−9za−5−3za−7 + 7a−2 + 5a−4 + a−6 + 4 |
[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["K11a74"];
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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]=
| { t4−5t3 + 10t2−13t + 15−13t−1 + 10t−2−5t−3 + t−4, −q9 + 3q8−5q7 + 8q6−10q5 + 11q4−11q3 + 9q2−7q + 5−2q−1 + q−2 } |
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 = 4 is the signature of K11a74. 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] Computer Talk
Much of the above data can be recomputed by Mathematica using the packageKnotTheory`. See A Sample KnotTheory` Session.
[edit] Modifying This Page
| Read me first: Modifying Knot Pages.
See/edit the Hoste-Thistlethwaite Knot Page master template (intermediate). See/edit the Hoste-Thistlethwaite_Splice_Base (expert). Back to the top. |
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