K11n78
From Knot Atlas
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 (Knotscape image) | See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.
Visit K11n78's page at Knotilus! Visit K11n78's page at the original Knot Atlas! |
[edit] Knot presentations
| Planar diagram presentation | X4251 X8493 X5,15,6,14 X2837 X9,20,10,21 X11,17,12,16 X13,19,14,18 X15,7,16,6 X17,13,18,12 X19,22,20,1 X21,10,22,11 |
| 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 | t4−3t3 + 6t2−8t + 9−8t−1 + 6t−2−3t−3 + t−4 |
| Conway polynomial | z8 + 5z6 + 8z4 + 5z2 + 1 |
| 2nd Alexander ideal (db, data sources) |  |
| Determinant and Signature | { 45, 4 } |
| Jones polynomial | q8−4q7 + 5q6−7q5 + 8q4−6q3 + 7q2−4q + 2−q−1 |
| HOMFLY-PT polynomial (db, data sources) | z8a−4−z6a−2 + 7z6a−4−z6a−6−5z4a−2 + 19z4a−4−6z4a−6−8z2a−2 + 25z2a−4−13z2a−6 + z2a−8−4a−2 + 13a−4−10a−6 + 2a−8 |
| Kauffman polynomial (db, data sources) | z9a−3 + z9a−5 + 2z8a−2 + 6z8a−4 + 4z8a−6 + z7a−1 + z7a−3 + 5z7a−5 + 5z7a−7−9z6a−2−24z6a−4−13z6a−6 + 2z6a−8−5z5a−1−18z5a−3−31z5a−5−18z5a−7 + 13z4a−2 + 31z4a−4 + 16z4a−6−2z4a−8 + 8z3a−1 + 28z3a−3 + 42z3a−5 + 26z3a−7 + 4z3a−9−9z2a−2−26z2a−4−17z2a−6 + z2a−8 + z2a−10−4za−1−13za−3−21za−5−14za−7−2za−9 + 4a−2 + 13a−4 + 10a−6 + 2a−8 |
| The A2 invariant | −q2−2q−2 + q−6 + 2q−8 + 6q−10 + 2q−12 + 4q−14−2q−16−3q−18−3q−20−3q−22 + q−24 + q−28 |
| The G2 invariant | Data:K11n78/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["K11n78"];
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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 + 6t2−8t + 9−8t−1 + 6t−2−3t−3 + t−4 |
In[5]:=
| Conway[K][z]
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Out[5]=
| z8 + 5z6 + 8z4 + 5z2 + 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]=
| { 45, 4 } |
In[8]:=
| Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
| q8−4q7 + 5q6−7q5 + 8q4−6q3 + 7q2−4q + 2−q−1 |
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−4−z6a−2 + 7z6a−4−z6a−6−5z4a−2 + 19z4a−4−6z4a−6−8z2a−2 + 25z2a−4−13z2a−6 + z2a−8−4a−2 + 13a−4−10a−6 + 2a−8 |
In[10]:=
| Kauffman[K][a, z]
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KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
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Out[10]=
| z9a−3 + z9a−5 + 2z8a−2 + 6z8a−4 + 4z8a−6 + z7a−1 + z7a−3 + 5z7a−5 + 5z7a−7−9z6a−2−24z6a−4−13z6a−6 + 2z6a−8−5z5a−1−18z5a−3−31z5a−5−18z5a−7 + 13z4a−2 + 31z4a−4 + 16z4a−6−2z4a−8 + 8z3a−1 + 28z3a−3 + 42z3a−5 + 26z3a−7 + 4z3a−9−9z2a−2−26z2a−4−17z2a−6 + z2a−8 + z2a−10−4za−1−13za−3−21za−5−14za−7−2za−9 + 4a−2 + 13a−4 + 10a−6 + 2a−8 |
[edit] "Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {10_62, K11n76,}
Same Jones Polynomial (up to mirroring, 
):
{K11n76,}
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["K11n78"];
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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 + 6t2−8t + 9−8t−1 + 6t−2−3t−3 + t−4, q8−4q7 + 5q6−7q5 + 8q4−6q3 + 7q2−4q + 2−q−1 } |
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]=
| {10_62, K11n76,} |
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]=
| {K11n76,} |
[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 = 4 is the signature of K11n78. 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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