K11a65
From Knot Atlas
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 (Knotscape image) | See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.
Visit K11a65's page at Knotilus! Visit K11a65's page at the original Knot Atlas! |
[edit] Knot presentations
| Planar diagram presentation | X4251 X8394 X16,5,17,6 X10,8,11,7 X2,9,3,10 X22,11,1,12 X20,13,21,14 X18,15,19,16 X6,17,7,18 X14,19,15,20 X12,21,13,22 |
| Gauss code | 1, -5, 2, -1, 3, -9, 4, -2, 5, -4, 6, -11, 7, -10, 8, -3, 9, -8, 10, -7, 11, -6 |
| Dowker-Thistlethwaite code | 4 8 16 10 2 22 20 18 6 14 12 |
| 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 + 15t−23 + 15t−1−3t−2 |
| Conway polynomial | −3z4 + 3z2 + 1 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 59, -2 } |
| Jones polynomial | q−2 + 4q−1−6q−2 + 8q−3−9q−4 + 9q−5−7q−6 + 6q−7−4q−8 + 2q−9−q−10 |
| HOMFLY-PT polynomial (db, data sources) | −a10 + 2z2a8 + a8−z4a6−z4a4 + z2a4 + a4−z4a2−z2a2−a2 + z2 + 1 |
| Kauffman polynomial (db, data sources) | z7a11−5z5a11 + 7z3a11−3za11 + 2z8a10−9z6a10 + 11z4a10−5z2a10 + a10 + 2z9a9−7z7a9 + 3z5a9 + 4z3a9−2za9 + z10a8−z8a8−7z6a8 + 9z4a8−3z2a8 + a8 + 4z9a7−15z7a7 + 17z5a7−8z3a7 + za7 + z10a6−z8a6−2z6a6 + z4a6 + z2a6 + 2z9a5−5z7a5 + 7z5a5−4z3a5 + 2z8a4−2z6a4 + 2z4a4−2z2a4 + a4 + 2z7a3−2z3a3 + za3 + 2z6a2−3z2a2 + a2 + 2z5a−3z3a + za + z4−2z2 + 1 |
| The A2 invariant | Data:K11a65/QuantumInvariant/A2/1,0 |
| The G2 invariant | Data:K11a65/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["K11a65"];
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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 + 15t−23 + 15t−1−3t−2 |
In[5]:=
| Conway[K][z]
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Out[5]=
| −3z4 + 3z2 + 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]=
| { 59, -2 } |
In[8]:=
| Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
| q−2 + 4q−1−6q−2 + 8q−3−9q−4 + 9q−5−7q−6 + 6q−7−4q−8 + 2q−9−q−10 |
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]=
| −a10 + 2z2a8 + a8−z4a6−z4a4 + z2a4 + a4−z4a2−z2a2−a2 + z2 + 1 |
In[10]:=
| Kauffman[K][a, z]
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KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
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Out[10]=
| z7a11−5z5a11 + 7z3a11−3za11 + 2z8a10−9z6a10 + 11z4a10−5z2a10 + a10 + 2z9a9−7z7a9 + 3z5a9 + 4z3a9−2za9 + z10a8−z8a8−7z6a8 + 9z4a8−3z2a8 + a8 + 4z9a7−15z7a7 + 17z5a7−8z3a7 + za7 + z10a6−z8a6−2z6a6 + z4a6 + z2a6 + 2z9a5−5z7a5 + 7z5a5−4z3a5 + 2z8a4−2z6a4 + 2z4a4−2z2a4 + a4 + 2z7a3−2z3a3 + za3 + 2z6a2−3z2a2 + a2 + 2z5a−3z3a + za + z4−2z2 + 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["K11a65"];
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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 + 15t−23 + 15t−1−3t−2, q−2 + 4q−1−6q−2 + 8q−3−9q−4 + 9q−5−7q−6 + 6q−7−4q−8 + 2q−9−q−10 } |
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 = -2 is the signature of K11a65. 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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