K11a51
From Knot Atlas
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 (Knotscape image) | See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.
Visit K11a51's page at Knotilus! Visit K11a51's page at the original Knot Atlas! |
[edit] Knot presentations
| Planar diagram presentation | X4251 X8394 X14,5,15,6 X10,8,11,7 X2,9,3,10 X20,11,21,12 X16,13,17,14 X6,15,7,16 X22,18,1,17 X12,19,13,20 X18,22,19,21 |
| Gauss code | 1, -5, 2, -1, 3, -8, 4, -2, 5, -4, 6, -10, 7, -3, 8, -7, 9, -11, 10, -6, 11, -9 |
| Dowker-Thistlethwaite code | 4 8 14 10 2 20 16 6 22 12 18 |
| 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 | t3−9t2 + 28t−39 + 28t−1−9t−2 + t−3 |
| Conway polynomial | z6−3z4 + z2 + 1 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 115, -2 } |
| Jones polynomial | q3−3q2 + 7q−12 + 16q−1−18q−2 + 19q−3−16q−4 + 12q−5−7q−6 + 3q−7−q−8 |
| HOMFLY-PT polynomial (db, data sources) | −a8 + 3z2a6 + 2a6−3z4a4−4z2a4−2a4 + z6a2 + 2z4a2 + 4z2a2 + 3a2−2z4−3z2−2 + z2a−2 + a−2 |
| Kauffman polynomial (db, data sources) | a4z10 + a2z10 + 3a5z9 + 6a3z9 + 3az9 + 5a6z8 + 9a4z8 + 8a2z8 + 4z8 + 5a7z7 + 7a5z7 + a3z7 + 2az7 + 3z7a−1 + 3a8z6−2a6z6−11a4z6−13a2z6 + z6a−2−6z6 + a9z5−7a7z5−20a5z5−20a3z5−16az5−8z5a−1−5a8z4−6a6z4−6a4z4−5a2z4−3z4a−2−3z4−2a9z3 + 4a7z3 + 16a5z3 + 16a3z3 + 13az3 + 7z3a−1 + 3a8z2 + 7a6z2 + 10a4z2 + 10a2z2 + 3z2a−2 + 7z2 + a9z−a7z−4a5z−4a3z−4az−2za−1−a8−2a6−2a4−3a2−a−2−2 |
| The A2 invariant | Data:K11a51/QuantumInvariant/A2/1,0 |
| The G2 invariant | Data:K11a51/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["K11a51"];
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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]=
| t3−9t2 + 28t−39 + 28t−1−9t−2 + t−3 |
In[5]:=
| Conway[K][z]
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Out[5]=
| z6−3z4 + 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]=
| { 115, -2 } |
In[8]:=
| Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
| q3−3q2 + 7q−12 + 16q−1−18q−2 + 19q−3−16q−4 + 12q−5−7q−6 + 3q−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 + 3z2a6 + 2a6−3z4a4−4z2a4−2a4 + z6a2 + 2z4a2 + 4z2a2 + 3a2−2z4−3z2−2 + z2a−2 + a−2 |
In[10]:=
| Kauffman[K][a, z]
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KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
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Out[10]=
| a4z10 + a2z10 + 3a5z9 + 6a3z9 + 3az9 + 5a6z8 + 9a4z8 + 8a2z8 + 4z8 + 5a7z7 + 7a5z7 + a3z7 + 2az7 + 3z7a−1 + 3a8z6−2a6z6−11a4z6−13a2z6 + z6a−2−6z6 + a9z5−7a7z5−20a5z5−20a3z5−16az5−8z5a−1−5a8z4−6a6z4−6a4z4−5a2z4−3z4a−2−3z4−2a9z3 + 4a7z3 + 16a5z3 + 16a3z3 + 13az3 + 7z3a−1 + 3a8z2 + 7a6z2 + 10a4z2 + 10a2z2 + 3z2a−2 + 7z2 + a9z−a7z−4a5z−4a3z−4az−2za−1−a8−2a6−2a4−3a2−a−2−2 |
[edit] "Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {}
Same Jones Polynomial (up to mirroring, 
):
{K11a3, K11a331,}
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["K11a51"];
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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]=
| { t3−9t2 + 28t−39 + 28t−1−9t−2 + t−3, q3−3q2 + 7q−12 + 16q−1−18q−2 + 19q−3−16q−4 + 12q−5−7q−6 + 3q−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]=
| {} |
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]=
| {K11a3, K11a331,} |
[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 K11a51. 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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