K11a21
From Knot Atlas
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 (Knotscape image) | See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.
Visit K11a21's page at Knotilus! Visit K11a21's page at the original Knot Atlas! |
[edit] Knot presentations
| Planar diagram presentation | X4251 X8493 X12,6,13,5 X2837 X18,9,19,10 X6,12,7,11 X22,14,1,13 X20,16,21,15 X10,17,11,18 X16,20,17,19 X14,22,15,21 |
| Gauss code | 1, -4, 2, -1, 3, -6, 4, -2, 5, -9, 6, -3, 7, -11, 8, -10, 9, -5, 10, -8, 11, -7 |
| Dowker-Thistlethwaite code | 4 8 12 2 18 6 22 20 10 16 14 |
| 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 | −5t2 + 19t−27 + 19t−1−5t−2 |
| Conway polynomial | −5z4−z2 + 1 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 75, 2 } |
| Jones polynomial | −q10 + 3q9−5q8 + 8q7−10q6 + 11q5−12q4 + 10q3−7q2 + 5q−2 + q−1 |
| HOMFLY-PT polynomial (db, data sources) | −z4a−2−2z4a−4−2z4a−6−2z2a−4−3z2a−6 + 3z2a−8 + z2 + a−2−a−4−2a−6 + 3a−8−a−10 + 1 |
| Kauffman polynomial (db, data sources) | z10a−6 + z10a−8 + 3z9a−5 + 6z9a−7 + 3z9a−9 + 3z8a−4 + 4z8a−6 + 4z8a−8 + 3z8a−10 + 3z7a−3−7z7a−5−21z7a−7−10z7a−9 + z7a−11 + 3z6a−2−2z6a−4−21z6a−6−29z6a−8−13z6a−10 + 2z5a−1−z5a−3 + 12z5a−5 + 24z5a−7 + 5z5a−9−4z5a−11−3z4a−2−2z4a−4 + 28z4a−6 + 42z4a−8 + 16z4a−10 + z4−2z3a−1−2z3a−3−15z3a−5−16z3a−7 + 3z3a−9 + 4z3a−11 + 2z2a−2 + 3z2a−4−16z2a−6−22z2a−8−7z2a−10−2z2 + 2za−3 + 7za−5 + 6za−7−za−11−a−2−a−4 + 2a−6 + 3a−8 + a−10 + 1 |
| The A2 invariant | Data:K11a21/QuantumInvariant/A2/1,0 |
| The G2 invariant | Data:K11a21/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["K11a21"];
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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]=
| −5t2 + 19t−27 + 19t−1−5t−2 |
In[5]:=
| Conway[K][z]
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Out[5]=
| −5z4−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]=
| { 75, 2 } |
In[8]:=
| Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
| −q10 + 3q9−5q8 + 8q7−10q6 + 11q5−12q4 + 10q3−7q2 + 5q−2 + 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]=
| −z4a−2−2z4a−4−2z4a−6−2z2a−4−3z2a−6 + 3z2a−8 + z2 + a−2−a−4−2a−6 + 3a−8−a−10 + 1 |
In[10]:=
| Kauffman[K][a, z]
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KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
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Out[10]=
| z10a−6 + z10a−8 + 3z9a−5 + 6z9a−7 + 3z9a−9 + 3z8a−4 + 4z8a−6 + 4z8a−8 + 3z8a−10 + 3z7a−3−7z7a−5−21z7a−7−10z7a−9 + z7a−11 + 3z6a−2−2z6a−4−21z6a−6−29z6a−8−13z6a−10 + 2z5a−1−z5a−3 + 12z5a−5 + 24z5a−7 + 5z5a−9−4z5a−11−3z4a−2−2z4a−4 + 28z4a−6 + 42z4a−8 + 16z4a−10 + z4−2z3a−1−2z3a−3−15z3a−5−16z3a−7 + 3z3a−9 + 4z3a−11 + 2z2a−2 + 3z2a−4−16z2a−6−22z2a−8−7z2a−10−2z2 + 2za−3 + 7za−5 + 6za−7−za−11−a−2−a−4 + 2a−6 + 3a−8 + a−10 + 1 |
[edit] "Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {}
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["K11a21"];
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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]=
| { −5t2 + 19t−27 + 19t−1−5t−2, −q10 + 3q9−5q8 + 8q7−10q6 + 11q5−12q4 + 10q3−7q2 + 5q−2 + 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]=
| {} |
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 K11a21. 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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