K11n11
From Knot Atlas
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 (Knotscape image) | See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.
Visit K11n11's page at Knotilus! Visit K11n11'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 X11,19,12,18 X13,6,14,7 X15,21,16,20 X17,1,18,22 X19,13,20,12 X21,17,22,16 |
| Gauss code | 1, -5, 2, -1, 3, 7, -4, -2, 5, -3, -6, 10, -7, 4, -8, 11, -9, 6, -10, 8, -11, 9 |
| Dowker-Thistlethwaite code | 4 8 10 -14 2 -18 -6 -20 -22 -12 -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 | t3−5t2 + 13t−17 + 13t−1−5t−2 + t−3 |
| Conway polynomial | z6 + z4 + 2z2 + 1 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 55, 2 } |
| Jones polynomial | −q8 + 3q7−6q6 + 8q5−9q4 + 10q3−8q2 + 6q−3 + q−1 |
| HOMFLY-PT polynomial (db, data sources) | z6a−4−2z4a−2 + 4z4a−4−z4a−6−4z2a−2 + 7z2a−4−2z2a−6 + z2−2a−2 + 4a−4−2a−6 + 1 |
| Kauffman polynomial (db, data sources) | z9a−3 + z9a−5 + z8a−2 + 4z8a−4 + 3z8a−6−2z7a−3 + 2z7a−5 + 4z7a−7−2z6a−2−9z6a−4−4z6a−6 + 3z6a−8 + 3z5a−1 + 8z5a−3−3z5a−5−7z5a−7 + z5a−9 + 8z4a−2 + 15z4a−4 + 2z4a−6−6z4a−8 + z4−4z3a−1−7z3a−3 + z3a−7−2z3a−9−8z2a−2−12z2a−4−4z2a−6 + 2z2a−8−2z2 + za−1 + 2za−3 + za−9 + 2a−2 + 4a−4 + 2a−6 + 1 |
| The A2 invariant | q4−1 + 2q−2−2q−4 + q−6 + q−8 + 3q−12−q−14 + 2q−16−q−18−2q−20 + q−22−q−24 |
| The G2 invariant | Data:K11n11/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["K11n11"];
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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]=
| t3−5t2 + 13t−17 + 13t−1−5t−2 + t−3 |
In[5]:=
| Conway[K][z]
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Out[5]=
| z6 + z4 + 2z2 + 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]=
| { 55, 2 } |
In[8]:=
| Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
| −q8 + 3q7−6q6 + 8q5−9q4 + 10q3−8q2 + 6q−3 + 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]=
| z6a−4−2z4a−2 + 4z4a−4−z4a−6−4z2a−2 + 7z2a−4−2z2a−6 + z2−2a−2 + 4a−4−2a−6 + 1 |
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 + z8a−2 + 4z8a−4 + 3z8a−6−2z7a−3 + 2z7a−5 + 4z7a−7−2z6a−2−9z6a−4−4z6a−6 + 3z6a−8 + 3z5a−1 + 8z5a−3−3z5a−5−7z5a−7 + z5a−9 + 8z4a−2 + 15z4a−4 + 2z4a−6−6z4a−8 + z4−4z3a−1−7z3a−3 + z3a−7−2z3a−9−8z2a−2−12z2a−4−4z2a−6 + 2z2a−8−2z2 + za−1 + 2za−3 + za−9 + 2a−2 + 4a−4 + 2a−6 + 1 |
[edit] "Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {9_31, K11n22, K11n112, K11n127,}
Same Jones Polynomial (up to mirroring, 
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{9_39, K11n112,}
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["K11n11"];
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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]=
| { t3−5t2 + 13t−17 + 13t−1−5t−2 + t−3, −q8 + 3q7−6q6 + 8q5−9q4 + 10q3−8q2 + 6q−3 + 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]=
| {9_31, K11n22, K11n112, K11n127,} |
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]=
| {9_39, K11n112,} |
[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 K11n11. 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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