K11a323
From Knot Atlas
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 (Knotscape image) | See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.
Visit K11a323's page at Knotilus! Visit K11a323's page at the original Knot Atlas! |
[edit] Knot presentations
| Planar diagram presentation | X6271 X12,4,13,3 X16,5,17,6 X22,8,1,7 X20,10,21,9 X4,12,5,11 X18,13,19,14 X10,15,11,16 X2,17,3,18 X14,19,15,20 X8,22,9,21 |
| Gauss code | 1, -9, 2, -6, 3, -1, 4, -11, 5, -8, 6, -2, 7, -10, 8, -3, 9, -7, 10, -5, 11, -4 |
| Dowker-Thistlethwaite code | 6 12 16 22 20 4 18 10 2 14 8 |
| 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 | 2t3−9t2 + 19t−23 + 19t−1−9t−2 + 2t−3 |
| Conway polynomial | 2z6 + 3z4 + z2 + 1 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 83, -2 } |
| Jones polynomial | q5−3q4 + 5q3−8q2 + 11q−12 + 13q−1−11q−2 + 9q−3−6q−4 + 3q−5−q−6 |
| HOMFLY-PT polynomial (db, data sources) | a2z6 + z6−a4z4 + 3a2z4−2z4a−2 + 3z4−2a4z2 + 3a2z2−5z2a−2 + z2a−4 + 4z2−a4 + a2−3a−2 + a−4 + 3 |
| Kauffman polynomial (db, data sources) | 2z10a−2 + 2z10 + 6az9 + 9z9a−1 + 3z9a−3 + 9a2z8−5z8a−2 + z8a−4 + 3z8 + 10a3z7−14az7−40z7a−1−16z7a−3 + 9a4z6−21a2z6−9z6a−2−5z6a−4−34z6 + 6a5z5−19a3z5 + 52z5a−1 + 27z5a−3 + 3a6z4−13a4z4 + 8a2z4 + 29z4a−2 + 8z4a−4 + 45z4 + a7z3−4a5z3 + 6a3z3 + 6az3−20z3a−1−15z3a−3 + 5a4z2−18z2a−2−5z2a−4−18z2 + a5z−az + za−1 + za−3−a4−a2 + 3a−2 + a−4 + 3 |
| The A2 invariant | Data:K11a323/QuantumInvariant/A2/1,0 |
| The G2 invariant | Data:K11a323/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["K11a323"];
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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]=
| 2t3−9t2 + 19t−23 + 19t−1−9t−2 + 2t−3 |
In[5]:=
| Conway[K][z]
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Out[5]=
| 2z6 + 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]=
| { 83, -2 } |
In[8]:=
| Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
| q5−3q4 + 5q3−8q2 + 11q−12 + 13q−1−11q−2 + 9q−3−6q−4 + 3q−5−q−6 |
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]=
| a2z6 + z6−a4z4 + 3a2z4−2z4a−2 + 3z4−2a4z2 + 3a2z2−5z2a−2 + z2a−4 + 4z2−a4 + a2−3a−2 + a−4 + 3 |
In[10]:=
| Kauffman[K][a, z]
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KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
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Out[10]=
| 2z10a−2 + 2z10 + 6az9 + 9z9a−1 + 3z9a−3 + 9a2z8−5z8a−2 + z8a−4 + 3z8 + 10a3z7−14az7−40z7a−1−16z7a−3 + 9a4z6−21a2z6−9z6a−2−5z6a−4−34z6 + 6a5z5−19a3z5 + 52z5a−1 + 27z5a−3 + 3a6z4−13a4z4 + 8a2z4 + 29z4a−2 + 8z4a−4 + 45z4 + a7z3−4a5z3 + 6a3z3 + 6az3−20z3a−1−15z3a−3 + 5a4z2−18z2a−2−5z2a−4−18z2 + a5z−az + za−1 + za−3−a4−a2 + 3a−2 + a−4 + 3 |
[edit] "Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {10_83, K11a307,}
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["K11a323"];
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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]=
| { 2t3−9t2 + 19t−23 + 19t−1−9t−2 + 2t−3, q5−3q4 + 5q3−8q2 + 11q−12 + 13q−1−11q−2 + 9q−3−6q−4 + 3q−5−q−6 } |
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_83, K11a307,} |
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 = -2 is the signature of K11a323. 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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