K11n154
From Knot Atlas
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 (Knotscape image) | See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.
Visit K11n154's page at Knotilus! Visit K11n154's page at the original Knot Atlas! |
[edit] Knot presentations
| Planar diagram presentation | X6271 X8394 X12,6,13,5 X20,8,21,7 X9,17,10,16 X11,19,12,18 X22,13,1,14 X4,16,5,15 X17,11,18,10 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, 5, -9, 6, 10, -4, 11, -7 |
| Dowker-Thistlethwaite code | 6 8 12 20 -16 -18 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 | t3−7t2 + 19t−25 + 19t−1−7t−2 + t−3 |
| Conway polynomial | z6−z4 + 1 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 79, 2 } |
| Jones polynomial | −q8 + 4q7−8q6 + 11q5−13q4 + 14q3−12q2 + 9q−5 + 2q−1 |
| HOMFLY-PT polynomial (db, data sources) | z6a−4−3z4a−2 + 3z4a−4−z4a−6−6z2a−2 + 5z2a−4−z2a−6 + 2z2−3a−2 + 3a−4−a−6 + 2 |
| Kauffman polynomial (db, data sources) | 2z9a−3 + 2z9a−5 + 3z8a−2 + 9z8a−4 + 6z8a−6 + z7a−1 + 6z7a−5 + 7z7a−7−6z6a−2−19z6a−4−9z6a−6 + 4z6a−8 + 3z5a−1 + z5a−3−16z5a−5−13z5a−7 + z5a−9 + 14z4a−2 + 20z4a−4 + 3z4a−6−6z4a−8 + 3z4−5z3a−1−2z3a−3 + 9z3a−5 + 5z3a−7−z3a−9−13z2a−2−11z2a−4−2z2a−6 + z2a−8−5z2 + za−1 + za−3−za−5−za−7 + 3a−2 + 3a−4 + a−6 + 2 |
| The A2 invariant | 2q4−1 + 3q−2−3q−4 + q−6−q−10 + 3q−12−2q−14 + 3q−16−q−18−2q−20 + 2q−22−q−24 |
| The G2 invariant | Data:K11n154/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["K11n154"];
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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]=
| t3−7t2 + 19t−25 + 19t−1−7t−2 + t−3 |
In[5]:=
| Conway[K][z]
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Out[5]=
| z6−z4 + 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]=
| { 79, 2 } |
In[8]:=
| Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
| −q8 + 4q7−8q6 + 11q5−13q4 + 14q3−12q2 + 9q−5 + 2q−1 |
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−4−3z4a−2 + 3z4a−4−z4a−6−6z2a−2 + 5z2a−4−z2a−6 + 2z2−3a−2 + 3a−4−a−6 + 2 |
In[10]:=
| Kauffman[K][a, z]
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KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
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Out[10]=
| 2z9a−3 + 2z9a−5 + 3z8a−2 + 9z8a−4 + 6z8a−6 + z7a−1 + 6z7a−5 + 7z7a−7−6z6a−2−19z6a−4−9z6a−6 + 4z6a−8 + 3z5a−1 + z5a−3−16z5a−5−13z5a−7 + z5a−9 + 14z4a−2 + 20z4a−4 + 3z4a−6−6z4a−8 + 3z4−5z3a−1−2z3a−3 + 9z3a−5 + 5z3a−7−z3a−9−13z2a−2−11z2a−4−2z2a−6 + z2a−8−5z2 + za−1 + za−3−za−5−za−7 + 3a−2 + 3a−4 + a−6 + 2 |
[edit] "Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {10_44,}
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["K11n154"];
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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]=
| { t3−7t2 + 19t−25 + 19t−1−7t−2 + t−3, −q8 + 4q7−8q6 + 11q5−13q4 + 14q3−12q2 + 9q−5 + 2q−1 } |
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]=
| {10_44,} |
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 K11n154. 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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