K11n41
From Knot Atlas
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 (Knotscape image) | See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.
Visit K11n41's page at Knotilus! Visit K11n41's page at the original Knot Atlas! |
[edit] Knot presentations
| Planar diagram presentation | X4251 X8493 X5,13,6,12 X2837 X9,18,10,19 X11,17,12,16 X13,7,14,6 X15,21,16,20 X17,22,18,1 X19,15,20,14 X21,10,22,11 |
| Gauss code | 1, -4, 2, -1, -3, 7, 4, -2, -5, 11, -6, 3, -7, 10, -8, 6, -9, 5, -10, 8, -11, 9 |
| Dowker-Thistlethwaite code | 4 8 -12 2 -18 -16 -6 -20 -22 -14 -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 | t4−4t3 + 8t2−9t + 9−9t−1 + 8t−2−4t−3 + t−4 |
| Conway polynomial | z8 + 4z6 + 4z4 + 3z2 + 1 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 53, 4 } |
| Jones polynomial | q8−4q7 + 6q6−8q5 + 9q4−8q3 + 8q2−5q + 3−q−1 |
| HOMFLY-PT polynomial (db, data sources) | z8a−4−z6a−2 + 6z6a−4−z6a−6−4z4a−2 + 13z4a−4−5z4a−6−4z2a−2 + 14z2a−4−8z2a−6 + z2a−8−a−2 + 6a−4−5a−6 + a−8 |
| Kauffman polynomial (db, data sources) | 2z9a−3 + 2z9a−5 + 3z8a−2 + 9z8a−4 + 6z8a−6 + z7a−1−3z7a−3 + 2z7a−5 + 6z7a−7−13z6a−2−36z6a−4−21z6a−6 + 2z6a−8−4z5a−1−12z5a−3−28z5a−5−20z5a−7 + 17z4a−2 + 42z4a−4 + 24z4a−6−z4a−8 + 5z3a−1 + 20z3a−3 + 35z3a−5 + 24z3a−7 + 4z3a−9−8z2a−2−22z2a−4−15z2a−6 + z2a−10−2za−1−7za−3−13za−5−10za−7−2za−9 + a−2 + 6a−4 + 5a−6 + a−8 |
| The A2 invariant | −q2 + 1−q−2 + q−4 + q−6 + q−8 + 4q−10−q−12 + 3q−14−2q−16−q−18−q−20−2q−22 + q−24−q−26 + q−28 |
| The G2 invariant | Data:K11n41/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["K11n41"];
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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]=
| t4−4t3 + 8t2−9t + 9−9t−1 + 8t−2−4t−3 + t−4 |
In[5]:=
| Conway[K][z]
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Out[5]=
| z8 + 4z6 + 4z4 + 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]=
| { 53, 4 } |
In[8]:=
| Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
| q8−4q7 + 6q6−8q5 + 9q4−8q3 + 8q2−5q + 3−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 + 6z6a−4−z6a−6−4z4a−2 + 13z4a−4−5z4a−6−4z2a−2 + 14z2a−4−8z2a−6 + z2a−8−a−2 + 6a−4−5a−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]=
| 2z9a−3 + 2z9a−5 + 3z8a−2 + 9z8a−4 + 6z8a−6 + z7a−1−3z7a−3 + 2z7a−5 + 6z7a−7−13z6a−2−36z6a−4−21z6a−6 + 2z6a−8−4z5a−1−12z5a−3−28z5a−5−20z5a−7 + 17z4a−2 + 42z4a−4 + 24z4a−6−z4a−8 + 5z3a−1 + 20z3a−3 + 35z3a−5 + 24z3a−7 + 4z3a−9−8z2a−2−22z2a−4−15z2a−6 + z2a−10−2za−1−7za−3−13za−5−10za−7−2za−9 + a−2 + 6a−4 + 5a−6 + a−8 |
[edit] "Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {K11n47,}
Same Jones Polynomial (up to mirroring, 
):
{K11n47,}
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["K11n41"];
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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]=
| { t4−4t3 + 8t2−9t + 9−9t−1 + 8t−2−4t−3 + t−4, q8−4q7 + 6q6−8q5 + 9q4−8q3 + 8q2−5q + 3−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]=
| {K11n47,} |
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]=
| {K11n47,} |
[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 K11n41. 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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