K11n102
From Knot Atlas
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 (Knotscape image) | See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.
Visit K11n102's page at Knotilus! Visit K11n102's page at the original Knot Atlas! |
[edit] Knot presentations
| Planar diagram presentation | X4251 X10,3,11,4 X5,14,6,15 X7,12,8,13 X9,19,10,18 X2,11,3,12 X13,6,14,7 X15,22,16,1 X17,20,18,21 X19,9,20,8 X21,16,22,17 |
| Gauss code | 1, -6, 2, -1, -3, 7, -4, 10, -5, -2, 6, 4, -7, 3, -8, 11, -9, 5, -10, 9, -11, 8 |
| Dowker-Thistlethwaite code | 4 10 -14 -12 -18 2 -6 -22 -20 -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 | −t2 + t + 1 + t−1−t−2 |
| Conway polynomial | −z4−3z2 + 1 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 3, -2 } |
| Jones polynomial | q−1 + 2q−1−q−2 + q−3−q−4−q−5 + q−6−q−7 + q−8 |
| HOMFLY-PT polynomial (db, data sources) | a8−z2a6−a6−z4a2−3z2a2−a2 + z2 + 2 |
| Kauffman polynomial (db, data sources) | a7z9 + a5z9 + a8z8 + 2a6z8 + a4z8−7a7z7−7a5z7−7a8z6−14a6z6−7a4z6 + 15a7z5 + 14a5z5 + az5 + 15a8z4 + 27a6z4 + 13a4z4 + 2a2z4 + z4−13a7z3−11a5z3−2az3−11a8z2−15a6z2−6a4z2−5a2z2−3z2 + 5a7z + 6a5z−az + a8 + a6 + a2 + 2 |
| The A2 invariant | Data:K11n102/QuantumInvariant/A2/1,0 |
| The G2 invariant | Data:K11n102/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["K11n102"];
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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]=
| −t2 + t + 1 + t−1−t−2 |
In[5]:=
| Conway[K][z]
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Out[5]=
| −z4−3z2 + 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]=
| { 3, -2 } |
In[8]:=
| Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
| q−1 + 2q−1−q−2 + q−3−q−4−q−5 + q−6−q−7 + q−8 |
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]=
| a8−z2a6−a6−z4a2−3z2a2−a2 + z2 + 2 |
In[10]:=
| Kauffman[K][a, z]
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KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
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Out[10]=
| a7z9 + a5z9 + a8z8 + 2a6z8 + a4z8−7a7z7−7a5z7−7a8z6−14a6z6−7a4z6 + 15a7z5 + 14a5z5 + az5 + 15a8z4 + 27a6z4 + 13a4z4 + 2a2z4 + z4−13a7z3−11a5z3−2az3−11a8z2−15a6z2−6a4z2−5a2z2−3z2 + 5a7z + 6a5z−az + a8 + a6 + a2 + 2 |
[edit] "Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {K11n38,}
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["K11n102"];
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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]=
| { −t2 + t + 1 + t−1−t−2, q−1 + 2q−1−q−2 + q−3−q−4−q−5 + q−6−q−7 + q−8 } |
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]=
| {K11n38,} |
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 K11n102. 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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