K11a319
From Knot Atlas
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 (Knotscape image) | See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.
Visit K11a319's page at Knotilus! Visit K11a319's page at the original Knot Atlas! |
[edit] Knot presentations
| Planar diagram presentation | X6271 X12,4,13,3 X16,6,17,5 X22,8,1,7 X18,10,19,9 X4,12,5,11 X20,14,21,13 X2,16,3,15 X8,18,9,17 X10,20,11,19 X14,22,15,21 |
| Gauss code | 1, -8, 2, -6, 3, -1, 4, -9, 5, -10, 6, -2, 7, -11, 8, -3, 9, -5, 10, -7, 11, -4 |
| Dowker-Thistlethwaite code | 6 12 16 22 18 4 20 2 8 10 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 | 5t3−15t2 + 25t−29 + 25t−1−15t−2 + 5t−3 |
| Conway polynomial | 5z6 + 15z4 + 10z2 + 1 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 119, 6 } |
| Jones polynomial | −q14 + 3q13−7q12 + 12q11−17q10 + 19q9−19q8 + 17q7−12q6 + 8q5−3q4 + q3 |
| HOMFLY-PT polynomial (db, data sources) | z6a−6 + 3z6a−8 + z6a−10 + 3z4a−6 + 12z4a−8 + z4a−10−z4a−12 + 2z2a−6 + 14z2a−8−4z2a−10−2z2a−12 + 5a−8−4a−10 |
| Kauffman polynomial (db, data sources) | 2z10a−10 + 2z10a−12 + 5z9a−9 + 10z9a−11 + 5z9a−13 + 6z8a−8 + 6z8a−10 + 6z8a−12 + 6z8a−14 + 3z7a−7−8z7a−9−21z7a−11−5z7a−13 + 5z7a−15 + z6a−6−18z6a−8−24z6a−10−16z6a−12−8z6a−14 + 3z6a−16−7z5a−7−2z5a−9 + 16z5a−11 + 3z5a−13−7z5a−15 + z5a−17−3z4a−6 + 22z4a−8 + 27z4a−10 + 12z4a−12 + 5z4a−14−5z4a−16 + 3z3a−7 + 7z3a−9−3z3a−11−2z3a−13 + 3z3a−15−2z3a−17 + 2z2a−6−17z2a−8−16z2a−10−z2a−12−2z2a−14 + 2z2a−16−4za−9−za−11 + za−13−za−15 + za−17 + 5a−8 + 4a−10 |
| The A2 invariant | Data:K11a319/QuantumInvariant/A2/1,0 |
| The G2 invariant | Data:K11a319/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["K11a319"];
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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]=
| 5t3−15t2 + 25t−29 + 25t−1−15t−2 + 5t−3 |
In[5]:=
| Conway[K][z]
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Out[5]=
| 5z6 + 15z4 + 10z2 + 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]=
| { 119, 6 } |
In[8]:=
| Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
| −q14 + 3q13−7q12 + 12q11−17q10 + 19q9−19q8 + 17q7−12q6 + 8q5−3q4 + 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 + 3z6a−8 + z6a−10 + 3z4a−6 + 12z4a−8 + z4a−10−z4a−12 + 2z2a−6 + 14z2a−8−4z2a−10−2z2a−12 + 5a−8−4a−10 |
In[10]:=
| Kauffman[K][a, z]
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KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
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Out[10]=
| 2z10a−10 + 2z10a−12 + 5z9a−9 + 10z9a−11 + 5z9a−13 + 6z8a−8 + 6z8a−10 + 6z8a−12 + 6z8a−14 + 3z7a−7−8z7a−9−21z7a−11−5z7a−13 + 5z7a−15 + z6a−6−18z6a−8−24z6a−10−16z6a−12−8z6a−14 + 3z6a−16−7z5a−7−2z5a−9 + 16z5a−11 + 3z5a−13−7z5a−15 + z5a−17−3z4a−6 + 22z4a−8 + 27z4a−10 + 12z4a−12 + 5z4a−14−5z4a−16 + 3z3a−7 + 7z3a−9−3z3a−11−2z3a−13 + 3z3a−15−2z3a−17 + 2z2a−6−17z2a−8−16z2a−10−z2a−12−2z2a−14 + 2z2a−16−4za−9−za−11 + za−13−za−15 + za−17 + 5a−8 + 4a−10 |
[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["K11a319"];
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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]=
| { 5t3−15t2 + 25t−29 + 25t−1−15t−2 + 5t−3, −q14 + 3q13−7q12 + 12q11−17q10 + 19q9−19q8 + 17q7−12q6 + 8q5−3q4 + 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 K11a319. 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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