K11a5
From Knot Atlas
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 (Knotscape image) | See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.
Visit K11a5's page at Knotilus! Visit K11a5's page at the original Knot Atlas! |
[edit] Knot presentations
| Planar diagram presentation | X4251 X8394 X10,6,11,5 X14,7,15,8 X2,9,3,10 X20,12,21,11 X16,14,17,13 X6,15,7,16 X22,17,1,18 X12,20,13,19 X18,21,19,22 |
| Gauss code | 1, -5, 2, -1, 3, -8, 4, -2, 5, -3, 6, -10, 7, -4, 8, -7, 9, -11, 10, -6, 11, -9 |
| Dowker-Thistlethwaite code | 4 8 10 14 2 20 16 6 22 12 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 | −t3 + 9t2−30t + 45−30t−1 + 9t−2−t−3 |
| Conway polynomial | −z6 + 3z4−3z2 + 1 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 125, 0 } |
| Jones polynomial | −q5 + 4q4−8q3 + 14q2−18q + 20−20q−1 + 17q−2−12q−3 + 7q−4−3q−5 + q−6 |
| HOMFLY-PT polynomial (db, data sources) | a6−3z2a4−2a4 + 3z4a2 + 4z2a2 + 3a2−z6−2z4−5z2−3 + 2z4a−2 + 2z2a−2 + 2a−2−z2a−4 |
| Kauffman polynomial (db, data sources) | a2z10 + z10 + 3a3z9 + 7az9 + 4z9a−1 + 4a4z8 + 10a2z8 + 7z8a−2 + 13z8 + 3a5z7 + 3a3z7 + az7 + 8z7a−1 + 7z7a−3 + a6z6−6a4z6−19a2z6−4z6a−2 + 4z6a−4−20z6−8a5z5−18a3z5−23az5−24z5a−1−10z5a−3 + z5a−5−3a6z4−2a4z4 + 3a2z4−8z4a−2−6z4a−4 + 7a5z3 + 15a3z3 + 18az3 + 16z3a−1 + 5z3a−3−z3a−5 + 3a6z2 + 6a4z2 + 7a2z2 + 8z2a−2 + 3z2a−4 + 9z2−2a5z−3a3z−4az−4za−1−za−3−a6−2a4−3a2−2a−2−3 |
| The A2 invariant | Data:K11a5/QuantumInvariant/A2/1,0 |
| The G2 invariant | Data:K11a5/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["K11a5"];
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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]=
| −t3 + 9t2−30t + 45−30t−1 + 9t−2−t−3 |
In[5]:=
| Conway[K][z]
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Out[5]=
| −z6 + 3z4−3z2 + 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]=
| { 125, 0 } |
In[8]:=
| Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
| −q5 + 4q4−8q3 + 14q2−18q + 20−20q−1 + 17q−2−12q−3 + 7q−4−3q−5 + q−6 |
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]=
| a6−3z2a4−2a4 + 3z4a2 + 4z2a2 + 3a2−z6−2z4−5z2−3 + 2z4a−2 + 2z2a−2 + 2a−2−z2a−4 |
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 + 3a3z9 + 7az9 + 4z9a−1 + 4a4z8 + 10a2z8 + 7z8a−2 + 13z8 + 3a5z7 + 3a3z7 + az7 + 8z7a−1 + 7z7a−3 + a6z6−6a4z6−19a2z6−4z6a−2 + 4z6a−4−20z6−8a5z5−18a3z5−23az5−24z5a−1−10z5a−3 + z5a−5−3a6z4−2a4z4 + 3a2z4−8z4a−2−6z4a−4 + 7a5z3 + 15a3z3 + 18az3 + 16z3a−1 + 5z3a−3−z3a−5 + 3a6z2 + 6a4z2 + 7a2z2 + 8z2a−2 + 3z2a−4 + 9z2−2a5z−3a3z−4az−4za−1−za−3−a6−2a4−3a2−2a−2−3 |
[edit] "Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {}
Same Jones Polynomial (up to mirroring, 
):
{K11a112,}
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["K11a5"];
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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]=
| { −t3 + 9t2−30t + 45−30t−1 + 9t−2−t−3, −q5 + 4q4−8q3 + 14q2−18q + 20−20q−1 + 17q−2−12q−3 + 7q−4−3q−5 + q−6 } |
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]=
| {K11a112,} |
[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 = 0 is the signature of K11a5. 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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