K11a123
From Knot Atlas
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 (Knotscape image) | See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.
Visit K11a123's page at Knotilus! Visit K11a123'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 X8,12,9,11 X18,14,19,13 X22,16,1,15 X6,18,7,17 X12,20,13,19 X16,22,17,21 |
| Gauss code | 1, -5, 2, -1, 3, -9, 4, -6, 5, -2, 6, -10, 7, -3, 8, -11, 9, -7, 10, -4, 11, -8 |
| Dowker-Thistlethwaite code | 4 10 14 20 2 8 18 22 6 12 16 |
| 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 | 9t2−29t + 41−29t−1 + 9t−2 |
| Conway polynomial | 9z4 + 7z2 + 1 |
| 2nd Alexander ideal (db, data sources) | {3,t + 1} |
| Determinant and Signature | { 117, 4 } |
| Jones polynomial | −q13 + 4q12−8q11 + 12q10−17q9 + 18q8−18q7 + 17q6−11q5 + 7q4−3q3 + q2 |
| HOMFLY-PT polynomial (db, data sources) | z4a−4 + 3z4a−6 + 4z4a−8 + z4a−10 + z2a−4 + 4z2a−6 + 6z2a−8−3z2a−10−z2a−12 + 2a−6 + 2a−8−4a−10 + a−12 |
| Kauffman polynomial (db, data sources) | 2z10a−10 + 2z10a−12 + 6z9a−9 + 11z9a−11 + 5z9a−13 + 9z8a−8 + 11z8a−10 + 6z8a−12 + 4z8a−14 + 8z7a−7−2z7a−9−25z7a−11−14z7a−13 + z7a−15 + 6z6a−6−14z6a−8−40z6a−10−34z6a−12−14z6a−14 + 3z5a−5−9z5a−7−20z5a−9 + z5a−11 + 6z5a−13−3z5a−15 + z4a−4−7z4a−6 + 8z4a−8 + 33z4a−10 + 31z4a−12 + 14z4a−14−2z3a−5 + 4z3a−7 + 23z3a−9 + 20z3a−11 + 6z3a−13 + 3z3a−15−z2a−4 + 6z2a−6−2z2a−8−12z2a−10−6z2a−12−3z2a−14−9za−9−12za−11−4za−13−za−15−2a−6 + 2a−8 + 4a−10 + a−12 |
| The A2 invariant | Data:K11a123/QuantumInvariant/A2/1,0 |
| The G2 invariant | Data:K11a123/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["K11a123"];
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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]=
| 9t2−29t + 41−29t−1 + 9t−2 |
In[5]:=
| Conway[K][z]
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Out[5]=
| 9z4 + 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]=
| {3,t + 1} |
In[7]:=
| {KnotDet[K], KnotSignature[K]}
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Out[7]=
| { 117, 4 } |
In[8]:=
| Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
| −q13 + 4q12−8q11 + 12q10−17q9 + 18q8−18q7 + 17q6−11q5 + 7q4−3q3 + q2 |
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]=
| z4a−4 + 3z4a−6 + 4z4a−8 + z4a−10 + z2a−4 + 4z2a−6 + 6z2a−8−3z2a−10−z2a−12 + 2a−6 + 2a−8−4a−10 + a−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 + 6z9a−9 + 11z9a−11 + 5z9a−13 + 9z8a−8 + 11z8a−10 + 6z8a−12 + 4z8a−14 + 8z7a−7−2z7a−9−25z7a−11−14z7a−13 + z7a−15 + 6z6a−6−14z6a−8−40z6a−10−34z6a−12−14z6a−14 + 3z5a−5−9z5a−7−20z5a−9 + z5a−11 + 6z5a−13−3z5a−15 + z4a−4−7z4a−6 + 8z4a−8 + 33z4a−10 + 31z4a−12 + 14z4a−14−2z3a−5 + 4z3a−7 + 23z3a−9 + 20z3a−11 + 6z3a−13 + 3z3a−15−z2a−4 + 6z2a−6−2z2a−8−12z2a−10−6z2a−12−3z2a−14−9za−9−12za−11−4za−13−za−15−2a−6 + 2a−8 + 4a−10 + a−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["K11a123"];
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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]=
| { 9t2−29t + 41−29t−1 + 9t−2, −q13 + 4q12−8q11 + 12q10−17q9 + 18q8−18q7 + 17q6−11q5 + 7q4−3q3 + q2 } |
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 = 4 is the signature of K11a123. 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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