K11n17
From Knot Atlas
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 (Knotscape image) | See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.
Visit K11n17's page at Knotilus! Visit K11n17's page at the original Knot Atlas! |
[edit] Knot presentations
| Planar diagram presentation | X4251 X8394 X10,6,11,5 X7,16,8,17 X2,9,3,10 X18,11,19,12 X20,13,21,14 X15,6,16,7 X22,17,1,18 X14,19,15,20 X12,21,13,22 |
| Gauss code | 1, -5, 2, -1, 3, 8, -4, -2, 5, -3, 6, -11, 7, -10, -8, 4, 9, -6, 10, -7, 11, -9 |
| Dowker-Thistlethwaite code | 4 8 10 -16 2 18 20 -6 22 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 | −2t2 + 12t−19 + 12t−1−2t−2 |
| Conway polynomial | −2z4 + 4z2 + 1 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 47, -2 } |
| Jones polynomial | 2q−1−4q−2 + 6q−3−7q−4 + 8q−5−7q−6 + 6q−7−4q−8 + 2q−9−q−10 |
| HOMFLY-PT polynomial (db, data sources) | −a10 + 2z2a8 + a8−z4a6−z4a4 + 2z2a2 + a2 |
| Kauffman polynomial (db, data sources) | z7a11−5z5a11 + 8z3a11−4za11 + 2z8a10−9z6a10 + 12z4a10−5z2a10 + a10 + z9a9−12z5a9 + 17z3a9−5za9 + 5z8a8−17z6a8 + 14z4a8−4z2a8 + a8 + z9a7 + 3z7a7−16z5a7 + 12z3a7−3za7 + 3z8a6−5z6a6−z4a6 + 4z7a5−8z5a5 + 6z3a5−3za5 + 3z6a4−3z4a4 + 2z2a4 + z5a3 + 3z3a3−za3 + 3z2a2−a2 |
| The A2 invariant | Data:K11n17/QuantumInvariant/A2/1,0 |
| The G2 invariant | Data:K11n17/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["K11n17"];
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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]=
| −2t2 + 12t−19 + 12t−1−2t−2 |
In[5]:=
| Conway[K][z]
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Out[5]=
| −2z4 + 4z2 + 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]=
| { 47, -2 } |
In[8]:=
| Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
| 2q−1−4q−2 + 6q−3−7q−4 + 8q−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 + 2z2a2 + a2 |
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 + 8z3a11−4za11 + 2z8a10−9z6a10 + 12z4a10−5z2a10 + a10 + z9a9−12z5a9 + 17z3a9−5za9 + 5z8a8−17z6a8 + 14z4a8−4z2a8 + a8 + z9a7 + 3z7a7−16z5a7 + 12z3a7−3za7 + 3z8a6−5z6a6−z4a6 + 4z7a5−8z5a5 + 6z3a5−3za5 + 3z6a4−3z4a4 + 2z2a4 + z5a3 + 3z3a3−za3 + 3z2a2−a2 |
[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["K11n17"];
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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]=
| { −2t2 + 12t−19 + 12t−1−2t−2, 2q−1−4q−2 + 6q−3−7q−4 + 8q−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 K11n17. 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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