K11n81
From Knot Atlas
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 (Knotscape image) | See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.
Visit K11n81's page at Knotilus! Visit K11n81's page at the original Knot Atlas! |
[edit] Knot presentations
| Planar diagram presentation | X4251 X8493 X5,16,6,17 X2837 X9,21,10,20 X11,1,12,22 X13,19,14,18 X15,6,16,7 X17,13,18,12 X19,15,20,14 X21,11,22,10 |
| Gauss code | 1, -4, 2, -1, -3, 8, 4, -2, -5, 11, -6, 9, -7, 10, -8, 3, -9, 7, -10, 5, -11, 6 |
| Dowker-Thistlethwaite code | 4 8 -16 2 -20 -22 -18 -6 -12 -14 -10 |
| 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 + 3t3−4t2 + 4t−3 + 4t−1−4t−2 + 3t−3−t−4 |
| Conway polynomial | −z8−5z6−6z4−z2 + 1 |
| 2nd Alexander ideal (db, data sources) |  |
| Determinant and Signature | { 27, 6 } |
| Jones polynomial | 2q9−3q8 + 3q7−5q6 + 4q5−4q4 + 4q3−q2 + q |
| HOMFLY-PT polynomial (db, data sources) | −z8a−6 + z6a−4−7z6a−6 + z6a−8 + 6z4a−4−18z4a−6 + 6z4a−8 + 12z2a−4−23z2a−6 + 11z2a−8−z2a−10 + 8a−4−13a−6 + 7a−8−a−10 |
| Kauffman polynomial (db, data sources) | z9a−5 + z9a−7 + z8a−4 + 5z8a−6 + 4z8a−8−4z7a−5 + 2z7a−7 + 6z7a−9−7z6a−4−28z6a−6−18z6a−8 + 3z6a−10−30z5a−7−30z5a−9 + 18z4a−4 + 50z4a−6 + 20z4a−8−12z4a−10 + 13z3a−5 + 53z3a−7 + 41z3a−9 + z3a−11−20z2a−4−38z2a−6−10z2a−8 + 8z2a−10−11za−5−26za−7−18za−9−3za−11 + 8a−4 + 13a−6 + 7a−8 + a−10 |
| The A2 invariant | q−4 + q−6 + 3q−8 + 3q−10 + 2q−12−4q−16−3q−18−5q−20 + q−24 + 2q−26 + 2q−28 + q−32−q−34 |
| The G2 invariant | Data:K11n81/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["K11n81"];
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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 + 3t3−4t2 + 4t−3 + 4t−1−4t−2 + 3t−3−t−4 |
In[5]:=
| Conway[K][z]
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Out[5]=
| −z8−5z6−6z4−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]=
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In[7]:=
| {KnotDet[K], KnotSignature[K]}
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Out[7]=
| { 27, 6 } |
In[8]:=
| Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
| 2q9−3q8 + 3q7−5q6 + 4q5−4q4 + 4q3−q2 + q |
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]=
| −z8a−6 + z6a−4−7z6a−6 + z6a−8 + 6z4a−4−18z4a−6 + 6z4a−8 + 12z2a−4−23z2a−6 + 11z2a−8−z2a−10 + 8a−4−13a−6 + 7a−8−a−10 |
In[10]:=
| Kauffman[K][a, z]
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KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
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Out[10]=
| z9a−5 + z9a−7 + z8a−4 + 5z8a−6 + 4z8a−8−4z7a−5 + 2z7a−7 + 6z7a−9−7z6a−4−28z6a−6−18z6a−8 + 3z6a−10−30z5a−7−30z5a−9 + 18z4a−4 + 50z4a−6 + 20z4a−8−12z4a−10 + 13z3a−5 + 53z3a−7 + 41z3a−9 + z3a−11−20z2a−4−38z2a−6−10z2a−8 + 8z2a−10−11za−5−26za−7−18za−9−3za−11 + 8a−4 + 13a−6 + 7a−8 + a−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["K11n81"];
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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 + 3t3−4t2 + 4t−3 + 4t−1−4t−2 + 3t−3−t−4, 2q9−3q8 + 3q7−5q6 + 4q5−4q4 + 4q3−q2 + q } |
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 K11n81. 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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