K11n136
From Knot Atlas
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 (Knotscape image) | See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.
Visit K11n136's page at Knotilus! Visit K11n136's page at the original Knot Atlas! |
[edit] Knot presentations
| Planar diagram presentation | X4251 X12,4,13,3 X5,17,6,16 X14,8,15,7 X18,10,19,9 X2,12,3,11 X8,14,9,13 X15,1,16,22 X20,18,21,17 X10,20,11,19 X21,7,22,6 |
| Gauss code | 1, -6, 2, -1, -3, 11, 4, -7, 5, -10, 6, -2, 7, -4, -8, 3, 9, -5, 10, -9, -11, 8 |
| Dowker-Thistlethwaite code | 4 12 -16 14 18 2 8 -22 20 10 -6 |
| 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 | 3t3−8t2 + 13t−15 + 13t−1−8t−2 + 3t−3 |
| Conway polynomial | 3z6 + 10z4 + 8z2 + 1 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 63, 6 } |
| Jones polynomial | −2q12 + 5q11−8q10 + 10q9−11q8 + 10q7−8q6 + 6q5−2q4 + q3 |
| HOMFLY-PT polynomial (db, data sources) | z6a−6 + 2z6a−8 + 4z4a−6 + 8z4a−8−2z4a−10 + 5z2a−6 + 8z2a−8−5z2a−10 + 2a−6 + a−8−2a−10 |
| Kauffman polynomial (db, data sources) | z9a−9 + z9a−11 + 3z8a−8 + 6z8a−10 + 3z8a−12 + 2z7a−7 + 5z7a−9 + 6z7a−11 + 3z7a−13 + z6a−6−8z6a−8−12z6a−10−2z6a−12 + z6a−14−5z5a−7−19z5a−9−16z5a−11−2z5a−13−4z4a−6 + 7z4a−8 + 7z4a−10 + 4z4a−14 + 2z3a−7 + 14z3a−9 + 12z3a−11 + 3z3a−13 + 3z3a−15 + 5z2a−6−5z2a−8−6z2a−10 + z2a−12−3z2a−14 + za−7−3za−9−3za−11−za−13−2za−15−2a−6 + a−8 + 2a−10 |
| The A2 invariant | Data:K11n136/QuantumInvariant/A2/1,0 |
| The G2 invariant | Data:K11n136/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["K11n136"];
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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]=
| 3t3−8t2 + 13t−15 + 13t−1−8t−2 + 3t−3 |
In[5]:=
| Conway[K][z]
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Out[5]=
| 3z6 + 10z4 + 8z2 + 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]=
| { 63, 6 } |
In[8]:=
| Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
| −2q12 + 5q11−8q10 + 10q9−11q8 + 10q7−8q6 + 6q5−2q4 + q3 |
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]=
| z6a−6 + 2z6a−8 + 4z4a−6 + 8z4a−8−2z4a−10 + 5z2a−6 + 8z2a−8−5z2a−10 + 2a−6 + a−8−2a−10 |
In[10]:=
| Kauffman[K][a, z]
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KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
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Out[10]=
| z9a−9 + z9a−11 + 3z8a−8 + 6z8a−10 + 3z8a−12 + 2z7a−7 + 5z7a−9 + 6z7a−11 + 3z7a−13 + z6a−6−8z6a−8−12z6a−10−2z6a−12 + z6a−14−5z5a−7−19z5a−9−16z5a−11−2z5a−13−4z4a−6 + 7z4a−8 + 7z4a−10 + 4z4a−14 + 2z3a−7 + 14z3a−9 + 12z3a−11 + 3z3a−13 + 3z3a−15 + 5z2a−6−5z2a−8−6z2a−10 + z2a−12−3z2a−14 + za−7−3za−9−3za−11−za−13−2za−15−2a−6 + a−8 + 2a−10 |
[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["K11n136"];
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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]=
| { 3t3−8t2 + 13t−15 + 13t−1−8t−2 + 3t−3, −2q12 + 5q11−8q10 + 10q9−11q8 + 10q7−8q6 + 6q5−2q4 + q3 } |
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 = 6 is the signature of K11n136. 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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