K11n126
From Knot Atlas
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 (Knotscape image) | See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.
Visit K11n126's page at Knotilus! Visit K11n126's page at the original Knot Atlas! |
[edit] Knot presentations
| Planar diagram presentation | X4251 X10,4,11,3 X5,19,6,18 X7,13,8,12 X2,10,3,9 X11,17,12,16 X13,21,14,20 X15,9,16,8 X17,1,18,22 X19,15,20,14 X21,7,22,6 |
| Gauss code | 1, -5, 2, -1, -3, 11, -4, 8, 5, -2, -6, 4, -7, 10, -8, 6, -9, 3, -10, 7, -11, 9 |
| Dowker-Thistlethwaite code | 4 10 -18 -12 2 -16 -20 -8 -22 -14 -6 |
| 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 | 3t3−6t2 + 4t−1 + 4t−1−6t−2 + 3t−3 |
| Conway polynomial | 3z6 + 12z4 + 7z2 + 1 |
| 2nd Alexander ideal (db, data sources) | {3,t + 1} |
| Determinant and Signature | { 27, 6 } |
| Jones polynomial | q11−3q10 + 3q9−5q8 + 5q7−3q6 + 4q5−2q4 + q3 |
| HOMFLY-PT polynomial (db, data sources) | z6a−6 + 2z6a−8 + 4z4a−6 + 10z4a−8−2z4a−10 + 3z2a−6 + 13z2a−8−9z2a−10 + a−6 + 5a−8−7a−10 + 2a−12 |
| Kauffman polynomial (db, data sources) | z9a−9 + z9a−11 + 3z8a−8 + 4z8a−10 + z8a−12 + 2z7a−7−z7a−9−3z7a−11 + z6a−6−14z6a−8−20z6a−10−5z6a−12−7z5a−7−11z5a−9−4z5a−11−4z4a−6 + 19z4a−8 + 31z4a−10 + 8z4a−12 + 3z3a−7 + 20z3a−9 + 19z3a−11 + 2z3a−13 + 3z2a−6−13z2a−8−19z2a−10−3z2a−12−10za−9−13za−11−3za−13−a−6 + 5a−8 + 7a−10 + 2a−12 |
| The A2 invariant | Data:K11n126/QuantumInvariant/A2/1,0 |
| The G2 invariant | Data:K11n126/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["K11n126"];
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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]=
| 3t3−6t2 + 4t−1 + 4t−1−6t−2 + 3t−3 |
In[5]:=
| Conway[K][z]
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Out[5]=
| 3z6 + 12z4 + 7z2 + 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]=
| {3,t + 1} |
In[7]:=
| {KnotDet[K], KnotSignature[K]}
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Out[7]=
| { 27, 6 } |
In[8]:=
| Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
| q11−3q10 + 3q9−5q8 + 5q7−3q6 + 4q5−2q4 + q3 |
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]=
| z6a−6 + 2z6a−8 + 4z4a−6 + 10z4a−8−2z4a−10 + 3z2a−6 + 13z2a−8−9z2a−10 + a−6 + 5a−8−7a−10 + 2a−12 |
In[10]:=
| Kauffman[K][a, z]
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KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
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Out[10]=
| z9a−9 + z9a−11 + 3z8a−8 + 4z8a−10 + z8a−12 + 2z7a−7−z7a−9−3z7a−11 + z6a−6−14z6a−8−20z6a−10−5z6a−12−7z5a−7−11z5a−9−4z5a−11−4z4a−6 + 19z4a−8 + 31z4a−10 + 8z4a−12 + 3z3a−7 + 20z3a−9 + 19z3a−11 + 2z3a−13 + 3z2a−6−13z2a−8−19z2a−10−3z2a−12−10za−9−13za−11−3za−13−a−6 + 5a−8 + 7a−10 + 2a−12 |
[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["K11n126"];
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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]=
| { 3t3−6t2 + 4t−1 + 4t−1−6t−2 + 3t−3, q11−3q10 + 3q9−5q8 + 5q7−3q6 + 4q5−2q4 + q3 } |
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 = 6 is the signature of K11n126. 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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