K11n61
From Knot Atlas
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 (Knotscape image) | See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.
Visit K11n61's page at Knotilus! Visit K11n61's page at the original Knot Atlas! |
[edit] Knot presentations
| Planar diagram presentation | X4251 X8493 X5,15,6,14 X2837 X9,17,10,16 X11,18,12,19 X13,1,14,22 X15,7,16,6 X17,20,18,21 X19,12,20,13 X21,11,22,10 |
| Gauss code | 1, -4, 2, -1, -3, 8, 4, -2, -5, 11, -6, 10, -7, 3, -8, 5, -9, 6, -10, 9, -11, 7 |
| Dowker-Thistlethwaite code | 4 8 -14 2 -16 -18 -22 -6 -20 -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 | t4−3t3 + 4t2−t−1−t−1 + 4t−2−3t−3 + t−4 |
| Conway polynomial | z8 + 5z6 + 6z4 + 4z2 + 1 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 17, 4 } |
| Jones polynomial | −q7 + q6−2q5 + 3q4−2q3 + 3q2−2q + 2−q−1 |
| HOMFLY-PT polynomial (db, data sources) | z8a−4−z6a−2 + 7z6a−4−z6a−6−5z4a−2 + 17z4a−4−6z4a−6−6z2a−2 + 19z2a−4−10z2a−6 + z2a−8−2a−2 + 8a−4−6a−6 + a−8 |
| Kauffman polynomial (db, data sources) | z9a−3 + z9a−5 + 2z8a−2 + 4z8a−4 + 2z8a−6 + z7a−1−3z7a−3−3z7a−5 + z7a−7−11z6a−2−23z6a−4−12z6a−6−5z5a−1−5z5a−3−6z5a−5−6z5a−7 + 17z4a−2 + 39z4a−4 + 22z4a−6 + 6z3a−1 + 14z3a−3 + 18z3a−5 + 10z3a−7−10z2a−2−27z2a−4−18z2a−6−z2a−8−2za−1−6za−3−10za−5−6za−7 + 2a−2 + 8a−4 + 6a−6 + a−8 |
| The A2 invariant | −q2 + q−6 + 2q−8 + 3q−10 + q−12 + 2q−14−q−16−q−18−2q−20−q−22−q−26 + q−28 |
| The G2 invariant | Data:K11n61/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["K11n61"];
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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]=
| t4−3t3 + 4t2−t−1−t−1 + 4t−2−3t−3 + t−4 |
In[5]:=
| Conway[K][z]
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Out[5]=
| z8 + 5z6 + 6z4 + 4z2 + 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]=
| { 17, 4 } |
In[8]:=
| Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
| −q7 + q6−2q5 + 3q4−2q3 + 3q2−2q + 2−q−1 |
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]=
| z8a−4−z6a−2 + 7z6a−4−z6a−6−5z4a−2 + 17z4a−4−6z4a−6−6z2a−2 + 19z2a−4−10z2a−6 + z2a−8−2a−2 + 8a−4−6a−6 + a−8 |
In[10]:=
| Kauffman[K][a, z]
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KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
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Out[10]=
| z9a−3 + z9a−5 + 2z8a−2 + 4z8a−4 + 2z8a−6 + z7a−1−3z7a−3−3z7a−5 + z7a−7−11z6a−2−23z6a−4−12z6a−6−5z5a−1−5z5a−3−6z5a−5−6z5a−7 + 17z4a−2 + 39z4a−4 + 22z4a−6 + 6z3a−1 + 14z3a−3 + 18z3a−5 + 10z3a−7−10z2a−2−27z2a−4−18z2a−6−z2a−8−2za−1−6za−3−10za−5−6za−7 + 2a−2 + 8a−4 + 6a−6 + a−8 |
[edit] "Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {}
Same Jones Polynomial (up to mirroring, 
):
{10_130,}
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["K11n61"];
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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]=
| { t4−3t3 + 4t2−t−1−t−1 + 4t−2−3t−3 + t−4, −q7 + q6−2q5 + 3q4−2q3 + 3q2−2q + 2−q−1 } |
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]=
| {10_130,} |
[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 K11n61. 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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