K11a257
From Knot Atlas
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 (Knotscape image) | See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.
Visit K11a257's page at Knotilus! Visit K11a257's page at the original Knot Atlas! |
[edit] Knot presentations
| Planar diagram presentation | X6271 X8493 X14,5,15,6 X2837 X16,10,17,9 X18,11,19,12 X20,13,21,14 X4,15,5,16 X22,18,1,17 X12,19,13,20 X10,21,11,22 |
| Gauss code | 1, -4, 2, -8, 3, -1, 4, -2, 5, -11, 6, -10, 7, -3, 8, -5, 9, -6, 10, -7, 11, -9 |
| Dowker-Thistlethwaite code | 6 8 14 2 16 18 20 4 22 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 | t4−5t3 + 12t2−19t + 23−19t−1 + 12t−2−5t−3 + t−4 |
| Conway polynomial | z8 + 3z6 + 2z4 + 1 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 97, 0 } |
| Jones polynomial | −q5 + 3q4−6q3 + 10q2−13q + 16−15q−1 + 13q−2−10q−3 + 6q−4−3q−5 + q−6 |
| HOMFLY-PT polynomial (db, data sources) | z8−2a2z6−z6a−2 + 6z6 + a4z4−9a2z4−4z4a−2 + 14z4 + 3a4z2−13a2z2−5z2a−2 + 15z2 + 2a4−6a2−2a−2 + 7 |
| Kauffman polynomial (db, data sources) | a2z10 + z10 + 3a3z9 + 7az9 + 4z9a−1 + 4a4z8 + 7a2z8 + 6z8a−2 + 9z8 + 3a5z7−3a3z7−17az7−6z7a−1 + 5z7a−3 + a6z6−10a4z6−28a2z6−16z6a−2 + 3z6a−4−36z6−9a5z5−7a3z5 + 14az5−11z5a−3 + z5a−5−3a6z4 + 6a4z4 + 38a2z4 + 20z4a−2−6z4a−4 + 55z4 + 7a5z3 + 8a3z3 + 2az3 + 10z3a−1 + 7z3a−3−2z3a−5 + 2a6z2−4a4z2−25a2z2−11z2a−2 + z2a−4−31z2−2a5z−4a3z−4az−4za−1−2za−3 + 2a4 + 6a2 + 2a−2 + 7 |
| The A2 invariant | q18 + q12−3q10 + q8−2q6−q4 + 3q2−1 + 5q−2−q−4 + q−6 + q−8−2q−10 + q−12−q−14 |
| The G2 invariant | Data:K11a257/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["K11a257"];
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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]=
| t4−5t3 + 12t2−19t + 23−19t−1 + 12t−2−5t−3 + t−4 |
In[5]:=
| Conway[K][z]
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Out[5]=
| z8 + 3z6 + 2z4 + 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]=
| { 97, 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 + 10q2−13q + 16−15q−1 + 13q−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]=
| z8−2a2z6−z6a−2 + 6z6 + a4z4−9a2z4−4z4a−2 + 14z4 + 3a4z2−13a2z2−5z2a−2 + 15z2 + 2a4−6a2−2a−2 + 7 |
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 + 7a2z8 + 6z8a−2 + 9z8 + 3a5z7−3a3z7−17az7−6z7a−1 + 5z7a−3 + a6z6−10a4z6−28a2z6−16z6a−2 + 3z6a−4−36z6−9a5z5−7a3z5 + 14az5−11z5a−3 + z5a−5−3a6z4 + 6a4z4 + 38a2z4 + 20z4a−2−6z4a−4 + 55z4 + 7a5z3 + 8a3z3 + 2az3 + 10z3a−1 + 7z3a−3−2z3a−5 + 2a6z2−4a4z2−25a2z2−11z2a−2 + z2a−4−31z2−2a5z−4a3z−4az−4za−1−2za−3 + 2a4 + 6a2 + 2a−2 + 7 |
[edit] "Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {10_118,}
Same Jones Polynomial (up to mirroring, 
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{K11a110,}
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["K11a257"];
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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]=
| { t4−5t3 + 12t2−19t + 23−19t−1 + 12t−2−5t−3 + t−4, −q5 + 3q4−6q3 + 10q2−13q + 16−15q−1 + 13q−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_118,} |
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]=
| {K11a110,} |
[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 K11a257. 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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