K11n114
From Knot Atlas
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 (Knotscape image) | See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.
Visit K11n114's page at Knotilus! Visit K11n114's page at the original Knot Atlas! |
[edit] Knot presentations
| Planar diagram presentation | X4251 X10,3,11,4 X14,6,15,5 X18,7,19,8 X16,9,17,10 X2,11,3,12 X13,21,14,20 X22,16,1,15 X8,17,9,18 X19,13,20,12 X6,21,7,22 |
| Gauss code | 1, -6, 2, -1, 3, -11, 4, -9, 5, -2, 6, 10, -7, -3, 8, -5, 9, -4, -10, 7, 11, -8 |
| Dowker-Thistlethwaite code | 4 10 14 18 16 2 -20 22 8 -12 6 |
| 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 | 3t2−13t + 21−13t−1 + 3t−2 |
| Conway polynomial | 3z4−z2 + 1 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 53, 0 } |
| Jones polynomial | −q5 + 3q4−5q3 + 8q2−9q + 9−8q−1 + 6q−2−3q−3 + q−4 |
| HOMFLY-PT polynomial (db, data sources) | a4−3z2a2−2a2 + 2z4 + 3z2 + 2 + z4a−2−z2a−4 |
| Kauffman polynomial (db, data sources) | az9 + z9a−1 + a2z8 + 3z8a−2 + 4z8−2az7 + 2z7a−1 + 4z7a−3−2a2z6−5z6a−2 + 3z6a−4−10z6 + 3a3z5 + 7az5−6z5a−1−9z5a−3 + z5a−5 + a4z4 + 8a2z4 + 2z4a−2−7z4a−4 + 16z4−4a3z3−5az3 + 6z3a−1 + 5z3a−3−2z3a−5−2a4z2−9a2z2 + 3z2a−4−10z2 + a3z−2za−1−za−3 + a4 + 2a2 + 2 |
| The A2 invariant | Data:K11n114/QuantumInvariant/A2/1,0 |
| The G2 invariant | Data:K11n114/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["K11n114"];
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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]=
| 3t2−13t + 21−13t−1 + 3t−2 |
In[5]:=
| Conway[K][z]
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Out[5]=
| 3z4−z2 + 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]=
| { 53, 0 } |
In[8]:=
| Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
| −q5 + 3q4−5q3 + 8q2−9q + 9−8q−1 + 6q−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]=
| a4−3z2a2−2a2 + 2z4 + 3z2 + 2 + z4a−2−z2a−4 |
In[10]:=
| Kauffman[K][a, z]
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KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
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Out[10]=
| az9 + z9a−1 + a2z8 + 3z8a−2 + 4z8−2az7 + 2z7a−1 + 4z7a−3−2a2z6−5z6a−2 + 3z6a−4−10z6 + 3a3z5 + 7az5−6z5a−1−9z5a−3 + z5a−5 + a4z4 + 8a2z4 + 2z4a−2−7z4a−4 + 16z4−4a3z3−5az3 + 6z3a−1 + 5z3a−3−2z3a−5−2a4z2−9a2z2 + 3z2a−4−10z2 + a3z−2za−1−za−3 + a4 + 2a2 + 2 |
[edit] "Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {K11a195,}
Same Jones Polynomial (up to mirroring, 
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{9_30,}
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["K11n114"];
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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]=
| { 3t2−13t + 21−13t−1 + 3t−2, −q5 + 3q4−5q3 + 8q2−9q + 9−8q−1 + 6q−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]=
| {K11a195,} |
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]=
| {9_30,} |
[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 K11n114. 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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