K11a256
From Knot Atlas
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 (Knotscape image) | See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.
Visit K11a256's page at Knotilus! Visit K11a256's page at the original Knot Atlas! |
[edit] Knot presentations
| Planar diagram presentation | X6271 X8394 X12,6,13,5 X20,8,21,7 X18,9,19,10 X16,11,17,12 X22,13,1,14 X4,16,5,15 X10,17,11,18 X2,19,3,20 X14,21,15,22 |
| Gauss code | 1, -10, 2, -8, 3, -1, 4, -2, 5, -9, 6, -3, 7, -11, 8, -6, 9, -5, 10, -4, 11, -7 |
| Dowker-Thistlethwaite code | 6 8 12 20 18 16 22 4 10 2 14 |
| 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 | −2t3 + 12t2−31t + 43−31t−1 + 12t−2−2t−3 |
| Conway polynomial | −2z6−z2 + 1 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 133, 0 } |
| Jones polynomial | q4−4q3 + 9q2−14q + 19−21q−1 + 21q−2−18q−3 + 13q−4−8q−5 + 4q−6−q−7 |
| HOMFLY-PT polynomial (db, data sources) | −z2a6 + 2z4a4 + 2z2a4−z6a2−z4a2 + a2−z6−2z4−3z2−1 + z4a−2 + z2a−2 + a−2 |
| Kauffman polynomial (db, data sources) | 3a4z10 + 3a2z10 + 6a5z9 + 15a3z9 + 9az9 + 4a6z8 + 2a4z8 + 11a2z8 + 13z8 + a7z7−20a5z7−43a3z7−9az7 + 13z7a−1−14a6z6−30a4z6−43a2z6 + 9z6a−2−18z6−3a7z5 + 19a5z5 + 34a3z5−9az5−17z5a−1 + 4z5a−3 + 14a6z4 + 35a4z4 + 32a2z4−8z4a−2 + z4a−4 + 2z4 + 2a7z3−5a5z3−8a3z3 + 7az3 + 7z3a−1−z3a−3−4a6z2−10a4z2−5a2z2 + 3z2a−2 + 4z2−az−za−1−a2−a−2−1 |
| The A2 invariant | Data:K11a256/QuantumInvariant/A2/1,0 |
| The G2 invariant | Data:K11a256/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["K11a256"];
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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 + 12t2−31t + 43−31t−1 + 12t−2−2t−3 |
In[5]:=
| Conway[K][z]
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Out[5]=
| −2z6−z2 + 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]=
| { 133, 0 } |
In[8]:=
| Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
| q4−4q3 + 9q2−14q + 19−21q−1 + 21q−2−18q−3 + 13q−4−8q−5 + 4q−6−q−7 |
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]=
| −z2a6 + 2z4a4 + 2z2a4−z6a2−z4a2 + a2−z6−2z4−3z2−1 + z4a−2 + z2a−2 + a−2 |
In[10]:=
| Kauffman[K][a, z]
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KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
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Out[10]=
| 3a4z10 + 3a2z10 + 6a5z9 + 15a3z9 + 9az9 + 4a6z8 + 2a4z8 + 11a2z8 + 13z8 + a7z7−20a5z7−43a3z7−9az7 + 13z7a−1−14a6z6−30a4z6−43a2z6 + 9z6a−2−18z6−3a7z5 + 19a5z5 + 34a3z5−9az5−17z5a−1 + 4z5a−3 + 14a6z4 + 35a4z4 + 32a2z4−8z4a−2 + z4a−4 + 2z4 + 2a7z3−5a5z3−8a3z3 + 7az3 + 7z3a−1−z3a−3−4a6z2−10a4z2−5a2z2 + 3z2a−2 + 4z2−az−za−1−a2−a−2−1 |
[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["K11a256"];
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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]=
| { −2t3 + 12t2−31t + 43−31t−1 + 12t−2−2t−3, q4−4q3 + 9q2−14q + 19−21q−1 + 21q−2−18q−3 + 13q−4−8q−5 + 4q−6−q−7 } |
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 K11a256. 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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