K11a350
From Knot Atlas
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 (Knotscape image) | See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.
Visit K11a350's page at Knotilus! Visit K11a350's page at the original Knot Atlas! |
[edit] Knot presentations
| Planar diagram presentation | X6271 X18,4,19,3 X16,5,17,6 X14,8,15,7 X4,10,5,9 X2,11,3,12 X20,13,21,14 X22,16,1,15 X10,18,11,17 X12,19,13,20 X8,21,9,22 |
| Gauss code | 1, -6, 2, -5, 3, -1, 4, -11, 5, -9, 6, -10, 7, -4, 8, -3, 9, -2, 10, -7, 11, -8 |
| Dowker-Thistlethwaite code | 6 18 16 14 4 2 20 22 10 12 8 |
| 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 | t4−7t3 + 21t2−39t + 49−39t−1 + 21t−2−7t−3 + t−4 |
| Conway polynomial | z8 + z6−z4−2z2 + 1 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 185, 0 } |
| Jones polynomial | q6−5q5 + 12q4−19q3 + 26q2−30q + 30−26q−1 + 19q−2−11q−3 + 5q−4−q−5 |
| HOMFLY-PT polynomial (db, data sources) | z8−a2z6−2z6a−2 + 4z6−2a2z4−5z4a−2 + z4a−4 + 5z4−2z2a−2 + z2a−4−z2 + 2a2 + 2a−2−3 |
| Kauffman polynomial (db, data sources) | 5z10a−2 + 5z10 + 13az9 + 25z9a−1 + 12z9a−3 + 15a2z8 + 13z8a−2 + 11z8a−4 + 17z8 + 11a3z7−13az7−51z7a−1−22z7a−3 + 5z7a−5 + 5a4z6−20a2z6−49z6a−2−23z6a−4 + z6a−6−50z6 + a5z5−13a3z5−4az5 + 27z5a−1 + 9z5a−3−8z5a−5−4a4z4 + 5a2z4 + 35z4a−2 + 13z4a−4−z4a−6 + 30z4 + 3a3z3 + 7az3 + z3a−1−z3a−3 + 2z3a−5 + 3a2z2−3z2a−2−3z2a−4 + 3z2−2az−2za−1−2a2−2a−2−3 |
| The A2 invariant | −q14 + 3q12−3q10 + 4q8 + 2q6−5q4 + 5q2−7 + 3q−2−q−4−q−6 + 6q−8−4q−10 + 3q−12−2q−16 + q−18 |
| The G2 invariant | Data:K11a350/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["K11a350"];
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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]=
| t4−7t3 + 21t2−39t + 49−39t−1 + 21t−2−7t−3 + t−4 |
In[5]:=
| Conway[K][z]
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Out[5]=
| z8 + z6−z4−2z2 + 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]=
| { 185, 0 } |
In[8]:=
| Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
| q6−5q5 + 12q4−19q3 + 26q2−30q + 30−26q−1 + 19q−2−11q−3 + 5q−4−q−5 |
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]=
| z8−a2z6−2z6a−2 + 4z6−2a2z4−5z4a−2 + z4a−4 + 5z4−2z2a−2 + z2a−4−z2 + 2a2 + 2a−2−3 |
In[10]:=
| Kauffman[K][a, z]
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KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
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Out[10]=
| 5z10a−2 + 5z10 + 13az9 + 25z9a−1 + 12z9a−3 + 15a2z8 + 13z8a−2 + 11z8a−4 + 17z8 + 11a3z7−13az7−51z7a−1−22z7a−3 + 5z7a−5 + 5a4z6−20a2z6−49z6a−2−23z6a−4 + z6a−6−50z6 + a5z5−13a3z5−4az5 + 27z5a−1 + 9z5a−3−8z5a−5−4a4z4 + 5a2z4 + 35z4a−2 + 13z4a−4−z4a−6 + 30z4 + 3a3z3 + 7az3 + z3a−1−z3a−3 + 2z3a−5 + 3a2z2−3z2a−2−3z2a−4 + 3z2−2az−2za−1−2a2−2a−2−3 |
[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["K11a350"];
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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]=
| { t4−7t3 + 21t2−39t + 49−39t−1 + 21t−2−7t−3 + t−4, q6−5q5 + 12q4−19q3 + 26q2−30q + 30−26q−1 + 19q−2−11q−3 + 5q−4−q−5 } |
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 K11a350. 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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