K11n7
From Knot Atlas
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 (Knotscape image) | See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.
Visit K11n7's page at Knotilus! Visit K11n7's page at the original Knot Atlas! |
[edit] Knot presentations
| Planar diagram presentation | X4251 X8394 X10,6,11,5 X7,14,8,15 X2,9,3,10 X18,12,19,11 X13,6,14,7 X20,16,21,15 X12,18,13,17 X22,20,1,19 X16,22,17,21 |
| Gauss code | 1, -5, 2, -1, 3, 7, -4, -2, 5, -3, 6, -9, -7, 4, 8, -11, 9, -6, 10, -8, 11, -10 |
| Dowker-Thistlethwaite code | 4 8 10 -14 2 18 -6 20 12 22 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 | t3−6t2 + 16t−21 + 16t−1−6t−2 + t−3 |
| Conway polynomial | z6 + z2 + 1 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 67, 2 } |
| Jones polynomial | −2q6 + 5q5−8q4 + 11q3−11q2 + 11q−9 + 6q−1−3q−2 + q−3 |
| HOMFLY-PT polynomial (db, data sources) | z6a−2 + 3z4a−2−z4a−4−2z4 + a2z2 + 4z2a−2−4z2 + a2 + 2a−2 + a−4−a−6−2 |
| Kauffman polynomial (db, data sources) | z9a−1 + z9a−3 + 6z8a−2 + 3z8a−4 + 3z8 + 3az7 + 7z7a−1 + 7z7a−3 + 3z7a−5 + a2z6−7z6a−2−z6a−4 + z6a−6−4z6−9az5−24z5a−1−17z5a−3−2z5a−5−3a2z4−9z4a−2−z4a−4 + 4z4a−6−7z4 + 8az3 + 17z3a−1 + 11z3a−3 + 5z3a−5 + 3z3a−7 + 3a2z2 + 9z2a−2−3z2a−6 + 9z2−2az−4za−1−3za−3−3za−5−2za−7−a2−2a−2 + a−4 + a−6−2 |
| The A2 invariant | Data:K11n7/QuantumInvariant/A2/1,0 |
| The G2 invariant | Data:K11n7/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["K11n7"];
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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]=
| t3−6t2 + 16t−21 + 16t−1−6t−2 + t−3 |
In[5]:=
| Conway[K][z]
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Out[5]=
| z6 + 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]=
| { 67, 2 } |
In[8]:=
| Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
| −2q6 + 5q5−8q4 + 11q3−11q2 + 11q−9 + 6q−1−3q−2 + q−3 |
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]=
| z6a−2 + 3z4a−2−z4a−4−2z4 + a2z2 + 4z2a−2−4z2 + a2 + 2a−2 + a−4−a−6−2 |
In[10]:=
| Kauffman[K][a, z]
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KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
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Out[10]=
| z9a−1 + z9a−3 + 6z8a−2 + 3z8a−4 + 3z8 + 3az7 + 7z7a−1 + 7z7a−3 + 3z7a−5 + a2z6−7z6a−2−z6a−4 + z6a−6−4z6−9az5−24z5a−1−17z5a−3−2z5a−5−3a2z4−9z4a−2−z4a−4 + 4z4a−6−7z4 + 8az3 + 17z3a−1 + 11z3a−3 + 5z3a−5 + 3z3a−7 + 3a2z2 + 9z2a−2−3z2a−6 + 9z2−2az−4za−1−3za−3−3za−5−2za−7−a2−2a−2 + a−4 + a−6−2 |
[edit] "Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {K11n131, K11n160,}
Same Jones Polynomial (up to mirroring, 
):
{K11n36, K11n44,}
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["K11n7"];
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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]=
| { t3−6t2 + 16t−21 + 16t−1−6t−2 + t−3, −2q6 + 5q5−8q4 + 11q3−11q2 + 11q−9 + 6q−1−3q−2 + q−3 } |
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]=
| {K11n131, K11n160,} |
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]=
| {K11n36, K11n44,} |
[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 K11n7. 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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