K11a348
From Knot Atlas
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 (Knotscape image) | See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.
Visit K11a348's page at Knotilus! Visit K11a348's page at the original Knot Atlas! |
[edit] Knot presentations
| Planar diagram presentation | X6271 X18,4,19,3 X16,5,17,6 X12,8,13,7 X4,10,5,9 X2,11,3,12 X20,14,21,13 X22,16,1,15 X10,18,11,17 X8,19,9,20 X14,22,15,21 |
| Gauss code | 1, -6, 2, -5, 3, -1, 4, -10, 5, -9, 6, -4, 7, -11, 8, -3, 9, -2, 10, -7, 11, -8 |
| Dowker-Thistlethwaite code | 6 18 16 12 4 2 20 22 10 8 14 |
| 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 | t4−7t3 + 19t2−29t + 33−29t−1 + 19t−2−7t−3 + t−4 |
| Conway polynomial | z8 + z6−3z4 + 1 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 145, 4 } |
| Jones polynomial | q10−4q9 + 9q8−15q7 + 20q6−23q5 + 23q4−20q3 + 15q2−9q + 5−q−1 |
| HOMFLY-PT polynomial (db, data sources) | z8a−4−z6a−2 + 4z6a−4−2z6a−6−2z4a−2 + 4z4a−4−6z4a−6 + z4a−8 + 2z2a−2−z2a−4−3z2a−6 + 2z2a−8 + 3a−2−3a−4 + a−6 |
| Kauffman polynomial (db, data sources) | 4z10a−4 + 4z10a−6 + 8z9a−3 + 19z9a−5 + 11z9a−7 + 5z8a−2 + z8a−4 + 11z8a−6 + 15z8a−8 + z7a−1−27z7a−3−56z7a−5−14z7a−7 + 14z7a−9−16z6a−2−33z6a−4−50z6a−6−24z6a−8 + 9z6a−10−2z5a−1 + 26z5a−3 + 46z5a−5−5z5a−7−19z5a−9 + 4z5a−11 + 12z4a−2 + 34z4a−4 + 41z4a−6 + 11z4a−8−7z4a−10 + z4a−12−7z3a−3−12z3a−5 + 5z3a−7 + 9z3a−9−z3a−11 + 2z2a−2−2z2a−4−9z2a−6−3z2a−8 + 2z2a−10 + za−3 + za−5−za−7−za−9−3a−2−3a−4−a−6 |
| The A2 invariant | −q2 + 3−q−2 + 3q−4 + 2q−6−3q−8 + 4q−10−6q−12 + 2q−14−q−16−q−18 + 4q−20−3q−22 + 2q−24−q−26−q−28 + q−30 |
| The G2 invariant | Data:K11a348/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["K11a348"];
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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]=
| t4−7t3 + 19t2−29t + 33−29t−1 + 19t−2−7t−3 + t−4 |
In[5]:=
| Conway[K][z]
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Out[5]=
| z8 + z6−3z4 + 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]=
| { 145, 4 } |
In[8]:=
| Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
| q10−4q9 + 9q8−15q7 + 20q6−23q5 + 23q4−20q3 + 15q2−9q + 5−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 + 4z6a−4−2z6a−6−2z4a−2 + 4z4a−4−6z4a−6 + z4a−8 + 2z2a−2−z2a−4−3z2a−6 + 2z2a−8 + 3a−2−3a−4 + a−6 |
In[10]:=
| Kauffman[K][a, z]
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KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
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Out[10]=
| 4z10a−4 + 4z10a−6 + 8z9a−3 + 19z9a−5 + 11z9a−7 + 5z8a−2 + z8a−4 + 11z8a−6 + 15z8a−8 + z7a−1−27z7a−3−56z7a−5−14z7a−7 + 14z7a−9−16z6a−2−33z6a−4−50z6a−6−24z6a−8 + 9z6a−10−2z5a−1 + 26z5a−3 + 46z5a−5−5z5a−7−19z5a−9 + 4z5a−11 + 12z4a−2 + 34z4a−4 + 41z4a−6 + 11z4a−8−7z4a−10 + z4a−12−7z3a−3−12z3a−5 + 5z3a−7 + 9z3a−9−z3a−11 + 2z2a−2−2z2a−4−9z2a−6−3z2a−8 + 2z2a−10 + za−3 + za−5−za−7−za−9−3a−2−3a−4−a−6 |
[edit] "Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {}
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["K11a348"];
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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]=
| { t4−7t3 + 19t2−29t + 33−29t−1 + 19t−2−7t−3 + t−4, q10−4q9 + 9q8−15q7 + 20q6−23q5 + 23q4−20q3 + 15q2−9q + 5−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]=
| {} |
[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 K11a348. 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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