K11a78
From Knot Atlas
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 (Knotscape image) | See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.
Visit K11a78's page at Knotilus! Visit K11a78's page at the original Knot Atlas! |
[edit] Knot presentations
| Planar diagram presentation | X4251 X10,3,11,4 X12,6,13,5 X14,7,15,8 X20,10,21,9 X2,11,3,12 X16,14,17,13 X6,15,7,16 X22,18,1,17 X8,20,9,19 X18,22,19,21 |
| Gauss code | 1, -6, 2, -1, 3, -8, 4, -10, 5, -2, 6, -3, 7, -4, 8, -7, 9, -11, 10, -5, 11, -9 |
| Dowker-Thistlethwaite code | 4 10 12 14 20 2 16 6 22 8 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 | 2t3−12t2 + 29t−37 + 29t−1−12t−2 + 2t−3 |
| Conway polynomial | 2z6−z2 + 1 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 123, 2 } |
| Jones polynomial | −q8 + 4q7−8q6 + 13q5−18q4 + 20q3−19q2 + 17q−12 + 7q−1−3q−2 + q−3 |
| HOMFLY-PT polynomial (db, data sources) | z6a−2 + z6a−4 + z4a−2 + 2z4a−4−z4a−6−2z4 + a2z2 + 2z2a−4−z2a−6−3z2 + a2 + a−2−1 |
| Kauffman polynomial (db, data sources) | z10a−2 + z10a−4 + 3z9a−1 + 7z9a−3 + 4z9a−5 + 10z8a−2 + 13z8a−4 + 7z8a−6 + 4z8 + 3az7 + 3z7a−1 + 7z7a−5 + 7z7a−7 + a2z6−21z6a−2−24z6a−4−6z6a−6 + 4z6a−8−6z6−8az5−19z5a−1−23z5a−3−24z5a−5−11z5a−7 + z5a−9−3a2z4 + 9z4a−2 + 11z4a−4−3z4a−6−6z4a−8−2z4 + 7az3 + 17z3a−1 + 22z3a−3 + 19z3a−5 + 6z3a−7−z3a−9 + 3a2z2−z2a−4 + 3z2a−6 + 2z2a−8 + 5z2−2az−5za−1−6za−3−4za−5−za−7−a2−a−2−1 |
| The A2 invariant | Data:K11a78/QuantumInvariant/A2/1,0 |
| The G2 invariant | Data:K11a78/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["K11a78"];
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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]=
| 2t3−12t2 + 29t−37 + 29t−1−12t−2 + 2t−3 |
In[5]:=
| Conway[K][z]
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Out[5]=
| 2z6−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]=
| { 123, 2 } |
In[8]:=
| Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
| −q8 + 4q7−8q6 + 13q5−18q4 + 20q3−19q2 + 17q−12 + 7q−1−3q−2 + q−3 |
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 + z6a−4 + z4a−2 + 2z4a−4−z4a−6−2z4 + a2z2 + 2z2a−4−z2a−6−3z2 + a2 + a−2−1 |
In[10]:=
| Kauffman[K][a, z]
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KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
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Out[10]=
| z10a−2 + z10a−4 + 3z9a−1 + 7z9a−3 + 4z9a−5 + 10z8a−2 + 13z8a−4 + 7z8a−6 + 4z8 + 3az7 + 3z7a−1 + 7z7a−5 + 7z7a−7 + a2z6−21z6a−2−24z6a−4−6z6a−6 + 4z6a−8−6z6−8az5−19z5a−1−23z5a−3−24z5a−5−11z5a−7 + z5a−9−3a2z4 + 9z4a−2 + 11z4a−4−3z4a−6−6z4a−8−2z4 + 7az3 + 17z3a−1 + 22z3a−3 + 19z3a−5 + 6z3a−7−z3a−9 + 3a2z2−z2a−4 + 3z2a−6 + 2z2a−8 + 5z2−2az−5za−1−6za−3−4za−5−za−7−a2−a−2−1 |
[edit] "Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {K11a130,}
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["K11a78"];
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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]=
| { 2t3−12t2 + 29t−37 + 29t−1−12t−2 + 2t−3, −q8 + 4q7−8q6 + 13q5−18q4 + 20q3−19q2 + 17q−12 + 7q−1−3q−2 + q−3 } |
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]=
| {K11a130,} |
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 K11a78. 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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