K11a367
From Knot Atlas
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 (Knotscape image) | See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.
Visit K11a367's page at Knotilus! Visit K11a367's page at the original Knot Atlas! |
[edit] Knot presentations
| Planar diagram presentation | X12,2,13,1 X14,4,15,3 X16,6,17,5 X18,8,19,7 X20,10,21,9 X22,12,1,11 X2,14,3,13 X4,16,5,15 X6,18,7,17 X8,20,9,19 X10,22,11,21 |
| Gauss code | 1, -7, 2, -8, 3, -9, 4, -10, 5, -11, 6, -1, 7, -2, 8, -3, 9, -4, 10, -5, 11, -6 |
| Dowker-Thistlethwaite code | 12 14 16 18 20 22 2 4 6 8 10 |
| 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 | t5−t4 + t3−t2 + t−1 + t−1−t−2 + t−3−t−4 + t−5 |
| Conway polynomial | z10 + 9z8 + 28z6 + 35z4 + 15z2 + 1 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 11, 10 } |
| Jones polynomial | −q16 + q15−q14 + q13−q12 + q11−q10 + q9−q8 + q7 + q5 |
| HOMFLY-PT polynomial (db, data sources) | z10a−10 + 10z8a−10−z8a−12 + 36z6a−10−8z6a−12 + 56z4a−10−21z4a−12 + 35z2a−10−20z2a−12 + 6a−10−5a−12 |
| Kauffman polynomial (db, data sources) | z10a−10 + z10a−12 + z9a−11 + z9a−13−10z8a−10−9z8a−12 + z8a−14−8z7a−11−7z7a−13 + z7a−15 + 36z6a−10 + 29z6a−12−6z6a−14 + z6a−16 + 21z5a−11 + 15z5a−13−5z5a−15 + z5a−17−56z4a−10−41z4a−12 + 10z4a−14−4z4a−16 + z4a−18−20z3a−11−10z3a−13 + 6z3a−15−3z3a−17 + z3a−19 + 35z2a−10 + 25z2a−12−4z2a−14 + 3z2a−16−2z2a−18 + z2a−20 + 5za−11 + za−13−za−15 + za−17−za−19 + za−21−6a−10−5a−12 |
| The A2 invariant | q−18 + q−20 + 2q−22 + q−24 + q−26−q−42−q−44−q−46 |
| The G2 invariant | q−90 + q−92 + q−94 + q−98 + 2q−100 + 2q−102 + q−104 + q−106 + 2q−108 + 3q−110 + 2q−112 + q−116 + 2q−118 + q−120−q−124 + q−128−2q−132−q−134−q−138−q−140−q−142−q−144−q−150−q−188−q−190−q−196−q−198−q−200−q−206−q−208 + q−264 |
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["K11a367"];
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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]=
| t5−t4 + t3−t2 + t−1 + t−1−t−2 + t−3−t−4 + t−5 |
In[5]:=
| Conway[K][z]
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Out[5]=
| z10 + 9z8 + 28z6 + 35z4 + 15z2 + 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]=
| { 11, 10 } |
In[8]:=
| Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
| −q16 + q15−q14 + q13−q12 + q11−q10 + q9−q8 + q7 + q5 |
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]=
| z10a−10 + 10z8a−10−z8a−12 + 36z6a−10−8z6a−12 + 56z4a−10−21z4a−12 + 35z2a−10−20z2a−12 + 6a−10−5a−12 |
In[10]:=
| Kauffman[K][a, z]
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KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
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Out[10]=
| z10a−10 + z10a−12 + z9a−11 + z9a−13−10z8a−10−9z8a−12 + z8a−14−8z7a−11−7z7a−13 + z7a−15 + 36z6a−10 + 29z6a−12−6z6a−14 + z6a−16 + 21z5a−11 + 15z5a−13−5z5a−15 + z5a−17−56z4a−10−41z4a−12 + 10z4a−14−4z4a−16 + z4a−18−20z3a−11−10z3a−13 + 6z3a−15−3z3a−17 + z3a−19 + 35z2a−10 + 25z2a−12−4z2a−14 + 3z2a−16−2z2a−18 + z2a−20 + 5za−11 + za−13−za−15 + za−17−za−19 + za−21−6a−10−5a−12 |
[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["K11a367"];
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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]=
| { t5−t4 + t3−t2 + t−1 + t−1−t−2 + t−3−t−4 + t−5, −q16 + q15−q14 + q13−q12 + q11−q10 + q9−q8 + q7 + q5 } |
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 = 10 is the signature of K11a367. 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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