K11a115
From Knot Atlas
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 (Knotscape image) | See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.
Visit K11a115's page at Knotilus! Visit K11a115's page at the original Knot Atlas! |
[edit] Knot presentations
| Planar diagram presentation | X4251 X10,4,11,3 X14,6,15,5 X18,7,19,8 X2,10,3,9 X20,11,21,12 X6,14,7,13 X22,16,1,15 X12,17,13,18 X8,19,9,20 X16,22,17,21 |
| Gauss code | 1, -5, 2, -1, 3, -7, 4, -10, 5, -2, 6, -9, 7, -3, 8, -11, 9, -4, 10, -6, 11, -8 |
| Dowker-Thistlethwaite code | 4 10 14 18 2 20 6 22 12 8 16 |
| 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 | −3t3 + 13t2−27t + 35−27t−1 + 13t−2−3t−3 |
| Conway polynomial | −3z6−5z4−2z2 + 1 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 121, 0 } |
| Jones polynomial | −q7 + 4q6−8q5 + 13q4−17q3 + 19q2−19q + 17−12q−1 + 7q−2−3q−3 + q−4 |
| HOMFLY-PT polynomial (db, data sources) | −2z6a−2−z6 + a2z4−7z4a−2 + 3z4a−4−2z4 + 2a2z2−9z2a−2 + 7z2a−4−z2a−6−z2 + a2−4a−2 + 4a−4−a−6 + 1 |
| Kauffman polynomial (db, data sources) | 2z10a−2 + 2z10a−4 + 7z9a−1 + 12z9a−3 + 5z9a−5 + 14z8a−2 + 8z8a−4 + 4z8a−6 + 10z8 + 9az7−6z7a−1−29z7a−3−13z7a−5 + z7a−7 + 6a2z6−53z6a−2−44z6a−4−14z6a−6−17z6 + 3a3z5−13az5−12z5a−1 + 10z5a−3 + 3z5a−5−3z5a−7 + a4z4−6a2z4 + 54z4a−2 + 51z4a−4 + 15z4a−6 + 11z4−2a3z3 + 10az3 + 13z3a−1 + 6z3a−3 + 8z3a−5 + 3z3a−7−a4z2 + 4a2z2−25z2a−2−23z2a−4−5z2a−6−2z2−2az−3za−1−3za−3−3za−5−za−7−a2 + 4a−2 + 4a−4 + a−6 + 1 |
| The A2 invariant | Data:K11a115/QuantumInvariant/A2/1,0 |
| The G2 invariant | Data:K11a115/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["K11a115"];
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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]=
| −3t3 + 13t2−27t + 35−27t−1 + 13t−2−3t−3 |
In[5]:=
| Conway[K][z]
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Out[5]=
| −3z6−5z4−2z2 + 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]=
| { 121, 0 } |
In[8]:=
| Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
| −q7 + 4q6−8q5 + 13q4−17q3 + 19q2−19q + 17−12q−1 + 7q−2−3q−3 + q−4 |
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]=
| −2z6a−2−z6 + a2z4−7z4a−2 + 3z4a−4−2z4 + 2a2z2−9z2a−2 + 7z2a−4−z2a−6−z2 + a2−4a−2 + 4a−4−a−6 + 1 |
In[10]:=
| Kauffman[K][a, z]
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KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
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Out[10]=
| 2z10a−2 + 2z10a−4 + 7z9a−1 + 12z9a−3 + 5z9a−5 + 14z8a−2 + 8z8a−4 + 4z8a−6 + 10z8 + 9az7−6z7a−1−29z7a−3−13z7a−5 + z7a−7 + 6a2z6−53z6a−2−44z6a−4−14z6a−6−17z6 + 3a3z5−13az5−12z5a−1 + 10z5a−3 + 3z5a−5−3z5a−7 + a4z4−6a2z4 + 54z4a−2 + 51z4a−4 + 15z4a−6 + 11z4−2a3z3 + 10az3 + 13z3a−1 + 6z3a−3 + 8z3a−5 + 3z3a−7−a4z2 + 4a2z2−25z2a−2−23z2a−4−5z2a−6−2z2−2az−3za−1−3za−3−3za−5−za−7−a2 + 4a−2 + 4a−4 + a−6 + 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["K11a115"];
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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]=
| { −3t3 + 13t2−27t + 35−27t−1 + 13t−2−3t−3, −q7 + 4q6−8q5 + 13q4−17q3 + 19q2−19q + 17−12q−1 + 7q−2−3q−3 + q−4 } |
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 K11a115. 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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