K11a280
From Knot Atlas
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 (Knotscape image) | See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.
Visit K11a280's page at Knotilus! Visit K11a280's page at the original Knot Atlas! |
[edit] Knot presentations
| Planar diagram presentation | X6271 X10,3,11,4 X16,5,17,6 X12,8,13,7 X18,9,19,10 X2,11,3,12 X22,14,1,13 X20,16,21,15 X4,17,5,18 X8,19,9,20 X14,22,15,21 |
| Gauss code | 1, -6, 2, -9, 3, -1, 4, -10, 5, -2, 6, -4, 7, -11, 8, -3, 9, -5, 10, -8, 11, -7 |
| Dowker-Thistlethwaite code | 6 10 16 12 18 2 22 20 4 8 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 | 6t2−26t + 41−26t−1 + 6t−2 |
| Conway polynomial | 6z4−2z2 + 1 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 105, 0 } |
| Jones polynomial | −q5 + 3q4−6q3 + 11q2−14q + 17−17q−1 + 14q−2−11q−3 + 7q−4−3q−5 + q−6 |
| HOMFLY-PT polynomial (db, data sources) | a6−3z2a4−a4 + 2z4a2−a2 + 3z4 + 3z2 + 2 + z4a−2−z2a−2−z2a−4 |
| Kauffman polynomial (db, data sources) | 2a2z10 + 2z10 + 4a3z9 + 10az9 + 6z9a−1 + 4a4z8 + 2a2z8 + 7z8a−2 + 5z8 + 3a5z7−7a3z7−32az7−17z7a−1 + 5z7a−3 + a6z6−7a4z6−15a2z6−20z6a−2 + 3z6a−4−30z6−8a5z5 + 2a3z5 + 44az5 + 23z5a−1−10z5a−3 + z5a−5−3a6z4−3a4z4 + 19a2z4 + 25z4a−2−6z4a−4 + 50z4 + 5a5z3−5a3z3−26az3−10z3a−1 + 4z3a−3−2z3a−5 + 3a6z2 + 4a4z2−11a2z2−11z2a−2 + z2a−4−24z2−a5z + 3a3z + 7az + 3za−1−a6−a4 + a2 + 2 |
| The A2 invariant | Data:K11a280/QuantumInvariant/A2/1,0 |
| The G2 invariant | Data:K11a280/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["K11a280"];
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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]=
| 6t2−26t + 41−26t−1 + 6t−2 |
In[5]:=
| Conway[K][z]
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Out[5]=
| 6z4−2z2 + 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]=
| { 105, 0 } |
In[8]:=
| Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
| −q5 + 3q4−6q3 + 11q2−14q + 17−17q−1 + 14q−2−11q−3 + 7q−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]=
| a6−3z2a4−a4 + 2z4a2−a2 + 3z4 + 3z2 + 2 + z4a−2−z2a−2−z2a−4 |
In[10]:=
| Kauffman[K][a, z]
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KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
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Out[10]=
| 2a2z10 + 2z10 + 4a3z9 + 10az9 + 6z9a−1 + 4a4z8 + 2a2z8 + 7z8a−2 + 5z8 + 3a5z7−7a3z7−32az7−17z7a−1 + 5z7a−3 + a6z6−7a4z6−15a2z6−20z6a−2 + 3z6a−4−30z6−8a5z5 + 2a3z5 + 44az5 + 23z5a−1−10z5a−3 + z5a−5−3a6z4−3a4z4 + 19a2z4 + 25z4a−2−6z4a−4 + 50z4 + 5a5z3−5a3z3−26az3−10z3a−1 + 4z3a−3−2z3a−5 + 3a6z2 + 4a4z2−11a2z2−11z2a−2 + z2a−4−24z2−a5z + 3a3z + 7az + 3za−1−a6−a4 + a2 + 2 |
[edit] "Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {}
Same Jones Polynomial (up to mirroring, 
):
{K11a16,}
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["K11a280"];
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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]=
| { 6t2−26t + 41−26t−1 + 6t−2, −q5 + 3q4−6q3 + 11q2−14q + 17−17q−1 + 14q−2−11q−3 + 7q−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]=
| {} |
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]=
| {K11a16,} |
[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 K11a280. 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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