K11a124
From Knot Atlas
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 (Knotscape image) | See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.
Visit K11a124's page at Knotilus! Visit K11a124's page at the original Knot Atlas! |
[edit] Knot presentations
| Planar diagram presentation | X4251 X10,4,11,3 X14,6,15,5 X20,8,21,7 X2,10,3,9 X16,12,17,11 X8,14,9,13 X22,16,1,15 X12,18,13,17 X6,20,7,19 X18,22,19,21 |
| Gauss code | 1, -5, 2, -1, 3, -10, 4, -7, 5, -2, 6, -9, 7, -3, 8, -6, 9, -11, 10, -4, 11, -8 |
| Dowker-Thistlethwaite code | 4 10 14 20 2 16 8 22 12 6 18 |
| 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 | 5t3−18t2 + 34t−41 + 34t−1−18t−2 + 5t−3 |
| Conway polynomial | 5z6 + 12z4 + 7z2 + 1 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 155, 6 } |
| Jones polynomial | −q14 + 5q13−11q12 + 17q11−23q10 + 25q9−25q8 + 21q7−14q6 + 9q5−3q4 + q3 |
| HOMFLY-PT polynomial (db, data sources) | z6a−6 + 3z6a−8 + z6a−10 + 3z4a−6 + 11z4a−8−z4a−10−z4a−12 + 3z2a−6 + 13z2a−8−9z2a−10 + a−6 + 5a−8−7a−10 + 2a−12 |
| Kauffman polynomial (db, data sources) | 2z10a−10 + 2z10a−12 + 5z9a−9 + 13z9a−11 + 8z9a−13 + 6z8a−8 + 14z8a−10 + 21z8a−12 + 13z8a−14 + 3z7a−7−10z7a−11 + 4z7a−13 + 11z7a−15 + z6a−6−14z6a−8−38z6a−10−45z6a−12−17z6a−14 + 5z6a−16−6z5a−7−20z5a−9−25z5a−11−27z5a−13−15z5a−15 + z5a−17−3z4a−6 + 16z4a−8 + 34z4a−10 + 23z4a−12 + 4z4a−14−4z4a−16 + 3z3a−7 + 25z3a−9 + 34z3a−11 + 17z3a−13 + 5z3a−15 + 3z2a−6−13z2a−8−19z2a−10−3z2a−12−10za−9−13za−11−3za−13−a−6 + 5a−8 + 7a−10 + 2a−12 |
| The A2 invariant | Data:K11a124/QuantumInvariant/A2/1,0 |
| The G2 invariant | Data:K11a124/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["K11a124"];
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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]=
| 5t3−18t2 + 34t−41 + 34t−1−18t−2 + 5t−3 |
In[5]:=
| Conway[K][z]
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Out[5]=
| 5z6 + 12z4 + 7z2 + 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]=
| { 155, 6 } |
In[8]:=
| Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
| −q14 + 5q13−11q12 + 17q11−23q10 + 25q9−25q8 + 21q7−14q6 + 9q5−3q4 + 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 + 3z6a−8 + z6a−10 + 3z4a−6 + 11z4a−8−z4a−10−z4a−12 + 3z2a−6 + 13z2a−8−9z2a−10 + a−6 + 5a−8−7a−10 + 2a−12 |
In[10]:=
| Kauffman[K][a, z]
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KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
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Out[10]=
| 2z10a−10 + 2z10a−12 + 5z9a−9 + 13z9a−11 + 8z9a−13 + 6z8a−8 + 14z8a−10 + 21z8a−12 + 13z8a−14 + 3z7a−7−10z7a−11 + 4z7a−13 + 11z7a−15 + z6a−6−14z6a−8−38z6a−10−45z6a−12−17z6a−14 + 5z6a−16−6z5a−7−20z5a−9−25z5a−11−27z5a−13−15z5a−15 + z5a−17−3z4a−6 + 16z4a−8 + 34z4a−10 + 23z4a−12 + 4z4a−14−4z4a−16 + 3z3a−7 + 25z3a−9 + 34z3a−11 + 17z3a−13 + 5z3a−15 + 3z2a−6−13z2a−8−19z2a−10−3z2a−12−10za−9−13za−11−3za−13−a−6 + 5a−8 + 7a−10 + 2a−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["K11a124"];
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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]=
| { 5t3−18t2 + 34t−41 + 34t−1−18t−2 + 5t−3, −q14 + 5q13−11q12 + 17q11−23q10 + 25q9−25q8 + 21q7−14q6 + 9q5−3q4 + 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 K11a124. 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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