K11n77
From Knot Atlas
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 (Knotscape image) | See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.
Visit K11n77's page at Knotilus! Visit K11n77's page at the original Knot Atlas! |
[edit] Knot presentations
| Planar diagram presentation | X4251 X8493 X5,15,6,14 X2837 X20,10,21,9 X11,17,12,16 X13,19,14,18 X15,7,16,6 X17,13,18,12 X22,20,1,19 X10,22,11,21 |
| Gauss code | 1, -4, 2, -1, -3, 8, 4, -2, 5, -11, -6, 9, -7, 3, -8, 6, -9, 7, 10, -5, 11, -10 |
| Dowker-Thistlethwaite code | 4 8 -14 2 20 -16 -18 -6 -12 22 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−t3−2t2 + 8t−11 + 8t−1−2t−2−t−3 + t−4 |
| Conway polynomial | z8 + 7z6 + 12z4 + 7z2 + 1 |
| 2nd Alexander ideal (db, data sources) |  |
| Determinant and Signature | { 27, 6 } |
| Jones polynomial | −q14 + 3q13−4q12 + 5q11−6q10 + 4q9−4q8 + 2q7 + q6 + q4 |
| HOMFLY-PT polynomial (db, data sources) | z8a−8 + 8z6a−8−z6a−10 + 21z4a−8−9z4a−10 + 23z2a−8−20z2a−10 + 4z2a−12 + 9a−8−13a−10 + 6a−12−a−14 |
| Kauffman polynomial (db, data sources) | z8a−8 + z8a−10 + z8a−12 + z8a−14 + z7a−9 + 2z7a−11 + 4z7a−13 + 3z7a−15−8z6a−8−10z6a−10−4z6a−12 + z6a−14 + 3z6a−16−9z5a−9−15z5a−11−15z5a−13−8z5a−15 + z5a−17 + 21z4a−8 + 29z4a−10 + 8z4a−12−8z4a−14−8z4a−16 + 20z3a−9 + 36z3a−11 + 23z3a−13 + 5z3a−15−2z3a−17−23z2a−8−31z2a−10−6z2a−12 + 5z2a−14 + 3z2a−16−13za−9−22za−11−12za−13−3za−15 + 9a−8 + 13a−10 + 6a−12 + a−14 |
| The A2 invariant | q−14 + q−16 + 2q−18 + 4q−20 + 2q−22 + q−24−5q−28−3q−30−4q−32 + 2q−36 + q−38 + 3q−40−q−42−q−44 |
| The G2 invariant | Data:K11n77/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["K11n77"];
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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−t3−2t2 + 8t−11 + 8t−1−2t−2−t−3 + t−4 |
In[5]:=
| Conway[K][z]
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Out[5]=
| z8 + 7z6 + 12z4 + 7z2 + 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]=
| −q14 + 3q13−4q12 + 5q11−6q10 + 4q9−4q8 + 2q7 + q6 + q4 |
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−8 + 8z6a−8−z6a−10 + 21z4a−8−9z4a−10 + 23z2a−8−20z2a−10 + 4z2a−12 + 9a−8−13a−10 + 6a−12−a−14 |
In[10]:=
| Kauffman[K][a, z]
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KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
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Out[10]=
| z8a−8 + z8a−10 + z8a−12 + z8a−14 + z7a−9 + 2z7a−11 + 4z7a−13 + 3z7a−15−8z6a−8−10z6a−10−4z6a−12 + z6a−14 + 3z6a−16−9z5a−9−15z5a−11−15z5a−13−8z5a−15 + z5a−17 + 21z4a−8 + 29z4a−10 + 8z4a−12−8z4a−14−8z4a−16 + 20z3a−9 + 36z3a−11 + 23z3a−13 + 5z3a−15−2z3a−17−23z2a−8−31z2a−10−6z2a−12 + 5z2a−14 + 3z2a−16−13za−9−22za−11−12za−13−3za−15 + 9a−8 + 13a−10 + 6a−12 + a−14 |
[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["K11n77"];
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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−t3−2t2 + 8t−11 + 8t−1−2t−2−t−3 + t−4, −q14 + 3q13−4q12 + 5q11−6q10 + 4q9−4q8 + 2q7 + q6 + q4 } |
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 K11n77. 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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