K11n123
From Knot Atlas
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 (Knotscape image) | See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.
Visit K11n123's page at Knotilus! Visit K11n123's page at the original Knot Atlas! |
[edit] Knot presentations
| Planar diagram presentation | X4251 X10,3,11,4 X5,16,6,17 X7,15,8,14 X12,10,13,9 X2,11,3,12 X13,18,14,19 X15,21,16,20 X17,22,18,1 X19,8,20,9 X21,7,22,6 |
| Gauss code | 1, -6, 2, -1, -3, 11, -4, 10, 5, -2, 6, -5, -7, 4, -8, 3, -9, 7, -10, 8, -11, 9 |
| Dowker-Thistlethwaite code | 4 10 -16 -14 12 2 -18 -20 -22 -8 -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 | 3t2−14t + 23−14t−1 + 3t−2 |
| Conway polynomial | 3z4−2z2 + 1 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 57, 0 } |
| Jones polynomial | −q3 + 4q2−6q + 9−10q−1 + 9q−2−8q−3 + 6q−4−3q−5 + q−6 |
| HOMFLY-PT polynomial (db, data sources) | a6−3z2a4−2a4 + 2z4a2 + 3z2a2 + 2a2 + z4−z2−1−z2a−2 + a−2 |
| Kauffman polynomial (db, data sources) | a3z9 + az9 + 3a4z8 + 5a2z8 + 2z8 + 3a5z7 + 5a3z7 + 3az7 + z7a−1 + a6z6−5a4z6−8a2z6−2z6−9a5z5−20a3z5−9az5 + 2z5a−1−3a6z4−6a4z4−4a2z4 + 4z4a−2 + 3z4 + 7a5z3 + 14a3z3 + 6az3 + z3a−3 + 3a6z2 + 8a4z2 + 7a2z2−2z2a−2−2a5z−2a3z−a6−2a4−2a2−a−2−1 |
| The A2 invariant | Data:K11n123/QuantumInvariant/A2/1,0 |
| The G2 invariant | Data:K11n123/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["K11n123"];
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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]=
| 3t2−14t + 23−14t−1 + 3t−2 |
In[5]:=
| Conway[K][z]
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Out[5]=
| 3z4−2z2 + 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]=
| { 57, 0 } |
In[8]:=
| Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
| −q3 + 4q2−6q + 9−10q−1 + 9q−2−8q−3 + 6q−4−3q−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]=
| a6−3z2a4−2a4 + 2z4a2 + 3z2a2 + 2a2 + z4−z2−1−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]=
| a3z9 + az9 + 3a4z8 + 5a2z8 + 2z8 + 3a5z7 + 5a3z7 + 3az7 + z7a−1 + a6z6−5a4z6−8a2z6−2z6−9a5z5−20a3z5−9az5 + 2z5a−1−3a6z4−6a4z4−4a2z4 + 4z4a−2 + 3z4 + 7a5z3 + 14a3z3 + 6az3 + z3a−3 + 3a6z2 + 8a4z2 + 7a2z2−2z2a−2−2a5z−2a3z−a6−2a4−2a2−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["K11n123"];
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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]=
| { 3t2−14t + 23−14t−1 + 3t−2, −q3 + 4q2−6q + 9−10q−1 + 9q−2−8q−3 + 6q−4−3q−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 K11n123. 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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