K11a268
From Knot Atlas
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 (Knotscape image) | See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.
Visit K11a268's page at Knotilus! Visit K11a268's page at the original Knot Atlas! |
[edit] Knot presentations
| Planar diagram presentation | X6271 X10,3,11,4 X12,6,13,5 X18,8,19,7 X20,10,21,9 X16,11,17,12 X22,13,1,14 X4,16,5,15 X2,17,3,18 X8,20,9,19 X14,21,15,22 |
| Gauss code | 1, -9, 2, -8, 3, -1, 4, -10, 5, -2, 6, -3, 7, -11, 8, -6, 9, -4, 10, -5, 11, -7 |
| Dowker-Thistlethwaite code | 6 10 12 18 20 16 22 4 2 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 | −t4 + 6t3−17t2 + 29t−33 + 29t−1−17t−2 + 6t−3−t−4 |
| Conway polynomial | −z8−2z6−z4−z2 + 1 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 139, 2 } |
| Jones polynomial | q7−4q6 + 9q5−15q4 + 20q3−22q2 + 22q−19 + 14q−1−8q−2 + 4q−3−q−4 |
| HOMFLY-PT polynomial (db, data sources) | −z8a−2−5z6a−2 + z6a−4 + 2z6−a2z4−10z4a−2 + 3z4a−4 + 7z4−2a2z2−9z2a−2 + 3z2a−4 + 7z2−2a−2 + a−4 + 2 |
| Kauffman polynomial (db, data sources) | 3z10a−2 + 3z10 + 6az9 + 16z9a−1 + 10z9a−3 + 4a2z8 + 15z8a−2 + 15z8a−4 + 4z8 + a3z7−19az7−44z7a−1−10z7a−3 + 14z7a−5−14a2z6−57z6a−2−25z6a−4 + 9z6a−6−37z6−3a3z5 + 17az5 + 32z5a−1−11z5a−3−19z5a−5 + 4z5a−7 + 15a2z4 + 51z4a−2 + 13z4a−4−7z4a−6 + z4a−8 + 45z4 + 2a3z3−4az3−7z3a−1 + 9z3a−3 + 9z3a−5−z3a−7−5a2z2−18z2a−2−4z2a−4 + 2z2a−6−17z2−za−3−za−5 + 2a−2 + a−4 + 2 |
| The A2 invariant | −q12 + q10 + q8−q6 + 4q4−3q2 + 1 + q−2−3q−4 + 5q−6−4q−8 + 3q−10−q−12−2q−14 + 3q−16−2q−18 + q−20 |
| The G2 invariant | Data:K11a268/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["K11a268"];
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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]=
| −t4 + 6t3−17t2 + 29t−33 + 29t−1−17t−2 + 6t−3−t−4 |
In[5]:=
| Conway[K][z]
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Out[5]=
| −z8−2z6−z4−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]=
| { 139, 2 } |
In[8]:=
| Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
| q7−4q6 + 9q5−15q4 + 20q3−22q2 + 22q−19 + 14q−1−8q−2 + 4q−3−q−4 |
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]=
| −z8a−2−5z6a−2 + z6a−4 + 2z6−a2z4−10z4a−2 + 3z4a−4 + 7z4−2a2z2−9z2a−2 + 3z2a−4 + 7z2−2a−2 + a−4 + 2 |
In[10]:=
| Kauffman[K][a, z]
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KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
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Out[10]=
| 3z10a−2 + 3z10 + 6az9 + 16z9a−1 + 10z9a−3 + 4a2z8 + 15z8a−2 + 15z8a−4 + 4z8 + a3z7−19az7−44z7a−1−10z7a−3 + 14z7a−5−14a2z6−57z6a−2−25z6a−4 + 9z6a−6−37z6−3a3z5 + 17az5 + 32z5a−1−11z5a−3−19z5a−5 + 4z5a−7 + 15a2z4 + 51z4a−2 + 13z4a−4−7z4a−6 + z4a−8 + 45z4 + 2a3z3−4az3−7z3a−1 + 9z3a−3 + 9z3a−5−z3a−7−5a2z2−18z2a−2−4z2a−4 + 2z2a−6−17z2−za−3−za−5 + 2a−2 + a−4 + 2 |
[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["K11a268"];
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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]=
| { −t4 + 6t3−17t2 + 29t−33 + 29t−1−17t−2 + 6t−3−t−4, q7−4q6 + 9q5−15q4 + 20q3−22q2 + 22q−19 + 14q−1−8q−2 + 4q−3−q−4 } |
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 K11a268. 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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