K11a266
From Knot Atlas
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 (Knotscape image) | See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.
Visit K11a266's page at Knotilus! Visit K11a266's page at the original Knot Atlas! |
[edit] Knot presentations
| Planar diagram presentation | X6271 X10,3,11,4 X12,6,13,5 X14,7,15,8 X16,10,17,9 X18,11,19,12 X22,13,1,14 X20,16,21,15 X4,18,5,17 X2,19,3,20 X8,21,9,22 |
| Gauss code | 1, -10, 2, -9, 3, -1, 4, -11, 5, -2, 6, -3, 7, -4, 8, -5, 9, -6, 10, -8, 11, -7 |
| Dowker-Thistlethwaite code | 6 10 12 14 16 18 22 20 4 2 8 |
| 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 | t4−7t3 + 23t2−45t + 57−45t−1 + 23t−2−7t−3 + t−4 |
| Conway polynomial | z8 + z6 + z4 + 1 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 209, 0 } |
| Jones polynomial | −q5 + 5q4−12q3 + 21q2−29q + 34−34q−1 + 30q−2−22q−3 + 14q−4−6q−5 + q−6 |
| HOMFLY-PT polynomial (db, data sources) | z8−2a2z6−z6a−2 + 4z6 + a4z4−4a2z4−2z4a−2 + 6z4−z2a−2 + z2−a4 + 3a2 + a−2−2 |
| Kauffman polynomial (db, data sources) | 6a2z10 + 6z10 + 15a3z9 + 31az9 + 16z9a−1 + 14a4z8 + 20a2z8 + 18z8a−2 + 24z8 + 6a5z7−23a3z7−57az7−16z7a−1 + 12z7a−3 + a6z6−26a4z6−65a2z6−26z6a−2 + 5z6a−4−69z6−7a5z5 + 2a3z5 + 19az5−4z5a−1−13z5a−3 + z5a−5 + 10a4z4 + 37a2z4 + 14z4a−2−3z4a−4 + 44z4 + 2a3z3 + 5az3 + 7z3a−1 + 4z3a−3 + 2a4z2 + a2z2−2z2a−2−3z2 + a5z + a3z−az−za−1−a4−3a2−a−2−2 |
| The A2 invariant | q18−3q16 + 3q12−5q10 + 7q8−q6 + 4q2−7 + 6q−2−6q−4 + 2q−6 + 4q−8−4q−10 + 3q−12−q−14 |
| The G2 invariant | Data:K11a266/QuantumInvariant/G2/1,0 |
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["K11a266"];
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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]=
| t4−7t3 + 23t2−45t + 57−45t−1 + 23t−2−7t−3 + t−4 |
In[5]:=
| Conway[K][z]
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Out[5]=
| z8 + z6 + z4 + 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]=
| { 209, 0 } |
In[8]:=
| Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
| −q5 + 5q4−12q3 + 21q2−29q + 34−34q−1 + 30q−2−22q−3 + 14q−4−6q−5 + q−6 |
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]=
| z8−2a2z6−z6a−2 + 4z6 + a4z4−4a2z4−2z4a−2 + 6z4−z2a−2 + z2−a4 + 3a2 + a−2−2 |
In[10]:=
| Kauffman[K][a, z]
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KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
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Out[10]=
| 6a2z10 + 6z10 + 15a3z9 + 31az9 + 16z9a−1 + 14a4z8 + 20a2z8 + 18z8a−2 + 24z8 + 6a5z7−23a3z7−57az7−16z7a−1 + 12z7a−3 + a6z6−26a4z6−65a2z6−26z6a−2 + 5z6a−4−69z6−7a5z5 + 2a3z5 + 19az5−4z5a−1−13z5a−3 + z5a−5 + 10a4z4 + 37a2z4 + 14z4a−2−3z4a−4 + 44z4 + 2a3z3 + 5az3 + 7z3a−1 + 4z3a−3 + 2a4z2 + a2z2−2z2a−2−3z2 + a5z + a3z−az−za−1−a4−3a2−a−2−2 |
[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["K11a266"];
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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]=
| { t4−7t3 + 23t2−45t + 57−45t−1 + 23t−2−7t−3 + t−4, −q5 + 5q4−12q3 + 21q2−29q + 34−34q−1 + 30q−2−22q−3 + 14q−4−6q−5 + q−6 } |
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 = 0 is the signature of K11a266. 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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