K11a84
From Knot Atlas
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 (Knotscape image) | See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.
Visit K11a84's page at Knotilus! Visit K11a84's page at the original Knot Atlas! |
[edit] Knot presentations
| Planar diagram presentation | X4251 X10,4,11,3 X12,5,13,6 X16,8,17,7 X2,10,3,9 X22,11,1,12 X20,13,21,14 X18,15,19,16 X6,18,7,17 X14,19,15,20 X8,21,9,22 |
| Gauss code | 1, -5, 2, -1, 3, -9, 4, -11, 5, -2, 6, -3, 7, -10, 8, -4, 9, -8, 10, -7, 11, -6 |
| Dowker-Thistlethwaite code | 4 10 12 16 2 22 20 18 6 14 8 |
| 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 + 10t2−23t + 31−23t−1 + 10t−2−2t−3 |
| Conway polynomial | −2z6−2z4−z2 + 1 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 101, 0 } |
| Jones polynomial | −q5 + 3q4−6q3 + 11q2−14q + 16−16q−1 + 14q−2−10q−3 + 6q−4−3q−5 + q−6 |
| HOMFLY-PT polynomial (db, data sources) | −a2z6−z6 + a4z4−3a2z4 + 2z4a−2−2z4 + 2a4z2−4a2z2 + 4z2a−2−z2a−4−2z2 + a4−a2 + 3a−2−a−4−1 |
| Kauffman polynomial (db, data sources) | a2z10 + z10 + 3a3z9 + 6az9 + 3z9a−1 + 4a4z8 + 6a2z8 + 4z8a−2 + 6z8 + 3a5z7−3a3z7−10az7 + 4z7a−3 + a6z6−10a4z6−18a2z6−z6a−2 + 3z6a−4−11z6−9a5z5−4a3z5 + 9az5−z5a−1−4z5a−3 + z5a−5−3a6z4 + 7a4z4 + 18a2z4−8z4a−2−6z4a−4 + 6z4 + 7a5z3 + 3a3z3−7az3−3z3a−1−2z3a−3−2z3a−5 + 2a6z2−3a4z2−9a2z2 + 9z2a−2 + 4z2a−4 + z2−a5z + 2az + 2za−1 + 2za−3 + za−5 + a4 + a2−3a−2−a−4−1 |
| The A2 invariant | Data:K11a84/QuantumInvariant/A2/1,0 |
| The G2 invariant | Data:K11a84/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["K11a84"];
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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 + 10t2−23t + 31−23t−1 + 10t−2−2t−3 |
In[5]:=
| Conway[K][z]
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Out[5]=
| −2z6−2z4−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]=
| { 101, 0 } |
In[8]:=
| Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
| −q5 + 3q4−6q3 + 11q2−14q + 16−16q−1 + 14q−2−10q−3 + 6q−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]=
| −a2z6−z6 + a4z4−3a2z4 + 2z4a−2−2z4 + 2a4z2−4a2z2 + 4z2a−2−z2a−4−2z2 + a4−a2 + 3a−2−a−4−1 |
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 + 6az9 + 3z9a−1 + 4a4z8 + 6a2z8 + 4z8a−2 + 6z8 + 3a5z7−3a3z7−10az7 + 4z7a−3 + a6z6−10a4z6−18a2z6−z6a−2 + 3z6a−4−11z6−9a5z5−4a3z5 + 9az5−z5a−1−4z5a−3 + z5a−5−3a6z4 + 7a4z4 + 18a2z4−8z4a−2−6z4a−4 + 6z4 + 7a5z3 + 3a3z3−7az3−3z3a−1−2z3a−3−2z3a−5 + 2a6z2−3a4z2−9a2z2 + 9z2a−2 + 4z2a−4 + z2−a5z + 2az + 2za−1 + 2za−3 + za−5 + a4 + a2−3a−2−a−4−1 |
[edit] "Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {10_119,}
Same Jones Polynomial (up to mirroring, 
):
{K11a88,}
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["K11a84"];
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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 + 10t2−23t + 31−23t−1 + 10t−2−2t−3, −q5 + 3q4−6q3 + 11q2−14q + 16−16q−1 + 14q−2−10q−3 + 6q−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]=
| {10_119,} |
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]=
| {K11a88,} |
[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 K11a84. 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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