K11a195
From Knot Atlas
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 (Knotscape image) | See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.
Visit K11a195's page at Knotilus! Visit K11a195's page at the original Knot Atlas! |
[edit] Knot presentations
| Planar diagram presentation | X4251 X12,3,13,4 X14,6,15,5 X20,7,21,8 X18,9,19,10 X16,11,17,12 X2,13,3,14 X22,16,1,15 X10,17,11,18 X8,19,9,20 X6,21,7,22 |
| Gauss code | 1, -7, 2, -1, 3, -11, 4, -10, 5, -9, 6, -2, 7, -3, 8, -6, 9, -5, 10, -4, 11, -8 |
| Dowker-Thistlethwaite code | 4 12 14 20 18 16 2 22 10 8 6 |
| 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 | 3t2−13t + 21−13t−1 + 3t−2 |
| Conway polynomial | 3z4−z2 + 1 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 53, 0 } |
| Jones polynomial | −q3 + 3q2−4q + 6−7q−1 + 8q−2−7q−3 + 6q−4−5q−5 + 3q−6−2q−7 + q−8 |
| HOMFLY-PT polynomial (db, data sources) | a8−2z2a6−2a6 + z4a4 + z2a4 + a4 + z4a2 + z4 + z2 + 1−z2a−2 |
| Kauffman polynomial (db, data sources) | a6z10 + a4z10 + 2a7z9 + 5a5z9 + 3a3z9 + a8z8−2a6z8 + a4z8 + 4a2z8−12a7z7−26a5z7−10a3z7 + 4az7−6a8z6−10a6z6−19a4z6−11a2z6 + 4z6 + 23a7z5 + 42a5z5 + 9a3z5−6az5 + 4z5a−1 + 11a8z4 + 26a6z4 + 29a4z4 + 8a2z4 + 3z4a−2−3z4−17a7z3−27a5z3−6a3z3−3z3a−1 + z3a−3−7a8z2−16a6z2−12a4z2−3a2z2−2z2a−2−2z2 + 4a7z + 7a5z + 3a3z + a8 + 2a6 + a4 + 1 |
| The A2 invariant | Data:K11a195/QuantumInvariant/A2/1,0 |
| The G2 invariant | Data:K11a195/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["K11a195"];
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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]=
| 3t2−13t + 21−13t−1 + 3t−2 |
In[5]:=
| Conway[K][z]
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Out[5]=
| 3z4−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]=
| { 53, 0 } |
In[8]:=
| Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
| −q3 + 3q2−4q + 6−7q−1 + 8q−2−7q−3 + 6q−4−5q−5 + 3q−6−2q−7 + q−8 |
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]=
| a8−2z2a6−2a6 + z4a4 + z2a4 + a4 + z4a2 + z4 + z2 + 1−z2a−2 |
In[10]:=
| Kauffman[K][a, z]
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KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
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Out[10]=
| a6z10 + a4z10 + 2a7z9 + 5a5z9 + 3a3z9 + a8z8−2a6z8 + a4z8 + 4a2z8−12a7z7−26a5z7−10a3z7 + 4az7−6a8z6−10a6z6−19a4z6−11a2z6 + 4z6 + 23a7z5 + 42a5z5 + 9a3z5−6az5 + 4z5a−1 + 11a8z4 + 26a6z4 + 29a4z4 + 8a2z4 + 3z4a−2−3z4−17a7z3−27a5z3−6a3z3−3z3a−1 + z3a−3−7a8z2−16a6z2−12a4z2−3a2z2−2z2a−2−2z2 + 4a7z + 7a5z + 3a3z + a8 + 2a6 + a4 + 1 |
[edit] "Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {K11n114,}
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["K11a195"];
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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]=
| { 3t2−13t + 21−13t−1 + 3t−2, −q3 + 3q2−4q + 6−7q−1 + 8q−2−7q−3 + 6q−4−5q−5 + 3q−6−2q−7 + q−8 } |
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]=
| {K11n114,} |
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 = 0 is the signature of K11a195. 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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