K11a258
From Knot Atlas
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 (Knotscape image) | See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.
Visit K11a258's page at Knotilus! Visit K11a258's page at the original Knot Atlas! |
[edit] Knot presentations
| Planar diagram presentation | X6271 X8493 X14,5,15,6 X2837 X16,10,17,9 X20,11,21,12 X18,13,19,14 X4,15,5,16 X22,18,1,17 X12,19,13,20 X10,21,11,22 |
| Gauss code | 1, -4, 2, -8, 3, -1, 4, -2, 5, -11, 6, -10, 7, -3, 8, -5, 9, -7, 10, -6, 11, -9 |
| Dowker-Thistlethwaite code | 6 8 14 2 16 20 18 4 22 12 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 | 2t3−9t2 + 17t−19 + 17t−1−9t−2 + 2t−3 |
| Conway polynomial | 2z6 + 3z4−z2 + 1 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 75, 2 } |
| Jones polynomial | −q6 + 3q5−6q4 + 9q3−10q2 + 12q−11 + 9q−1−7q−2 + 4q−3−2q−4 + q−5 |
| HOMFLY-PT polynomial (db, data sources) | z6a−2 + z6−2a2z4 + 3z4a−2−z4a−4 + 3z4 + a4z2−6a2z2 + 3z2a−2−2z2a−4 + 3z2 + 2a4−4a2 + 2a−2−a−4 + 2 |
| Kauffman polynomial (db, data sources) | a2z10 + z10 + 2a3z9 + 6az9 + 4z9a−1 + a4z8 + 7z8a−2 + 6z8−11a3z7−26az7−6z7a−1 + 9z7a−3−6a4z6−18a2z6−13z6a−2 + 9z6a−4−34z6 + 20a3z5 + 35az5−7z5a−1−16z5a−3 + 6z5a−5 + 12a4z4 + 39a2z4−z4a−2−14z4a−4 + 3z4a−6 + 43z4−14a3z3−19az3 + 4z3a−1 + 4z3a−3−4z3a−5 + z3a−7−9a4z2−25a2z2 + 4z2a−2 + 7z2a−4−19z2 + 4a3z + 6az + 2za−1 + za−3 + za−5 + 2a4 + 4a2−2a−2−a−4 + 2 |
| The A2 invariant | Data:K11a258/QuantumInvariant/A2/1,0 |
| The G2 invariant | Data:K11a258/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["K11a258"];
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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]=
| 2t3−9t2 + 17t−19 + 17t−1−9t−2 + 2t−3 |
In[5]:=
| Conway[K][z]
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Out[5]=
| 2z6 + 3z4−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]=
| { 75, 2 } |
In[8]:=
| Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
| −q6 + 3q5−6q4 + 9q3−10q2 + 12q−11 + 9q−1−7q−2 + 4q−3−2q−4 + q−5 |
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−2 + z6−2a2z4 + 3z4a−2−z4a−4 + 3z4 + a4z2−6a2z2 + 3z2a−2−2z2a−4 + 3z2 + 2a4−4a2 + 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]=
| a2z10 + z10 + 2a3z9 + 6az9 + 4z9a−1 + a4z8 + 7z8a−2 + 6z8−11a3z7−26az7−6z7a−1 + 9z7a−3−6a4z6−18a2z6−13z6a−2 + 9z6a−4−34z6 + 20a3z5 + 35az5−7z5a−1−16z5a−3 + 6z5a−5 + 12a4z4 + 39a2z4−z4a−2−14z4a−4 + 3z4a−6 + 43z4−14a3z3−19az3 + 4z3a−1 + 4z3a−3−4z3a−5 + z3a−7−9a4z2−25a2z2 + 4z2a−2 + 7z2a−4−19z2 + 4a3z + 6az + 2za−1 + za−3 + za−5 + 2a4 + 4a2−2a−2−a−4 + 2 |
[edit] "Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {K11n150,}
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["K11a258"];
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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]=
| { 2t3−9t2 + 17t−19 + 17t−1−9t−2 + 2t−3, −q6 + 3q5−6q4 + 9q3−10q2 + 12q−11 + 9q−1−7q−2 + 4q−3−2q−4 + q−5 } |
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]=
| {K11n150,} |
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 K11a258. 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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