K11a218
From Knot Atlas
|
|
|
|
 (Knotscape image) | See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.
Visit K11a218's page at Knotilus! Visit K11a218's page at the original Knot Atlas! |
[edit] Knot presentations
| Planar diagram presentation | X4251 X12,3,13,4 X18,5,19,6 X16,8,17,7 X20,9,21,10 X14,12,15,11 X2,13,3,14 X8,16,9,15 X22,17,1,18 X10,19,11,20 X6,21,7,22 |
| Gauss code | 1, -7, 2, -1, 3, -11, 4, -8, 5, -10, 6, -2, 7, -6, 8, -4, 9, -3, 10, -5, 11, -9 |
| Dowker-Thistlethwaite code | 4 12 18 16 20 14 2 8 22 10 6 |
| A Braid Representative |
| ||||||
| A Morse Link Presentation | 
|
[edit] Three dimensional invariants
|
[edit] Four dimensional invariants
|
[edit] Polynomial invariants
| Alexander polynomial | t3−10t2 + 32t−45 + 32t−1−10t−2 + t−3 |
| Conway polynomial | z6−4z4 + z2 + 1 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 131, -2 } |
| Jones polynomial | q3−4q2 + 9q−14 + 19q−1−21q−2 + 21q−3−18q−4 + 13q−5−7q−6 + 3q−7−q−8 |
| HOMFLY-PT polynomial (db, data sources) | −a8 + 3z2a6 + 2a6−3z4a4−3z2a4−a4 + z6a2 + z4a2 + z2a2−2z4−z2 + 1 + z2a−2 |
| Kauffman polynomial (db, data sources) | 2a4z10 + 2a2z10 + 5a5z9 + 11a3z9 + 6az9 + 6a6z8 + 9a4z8 + 10a2z8 + 7z8 + 5a7z7−a5z7−18a3z7−8az7 + 4z7a−1 + 3a8z6−5a6z6−20a4z6−29a2z6 + z6a−2−16z6 + a9z5−6a7z5−6a5z5 + 8a3z5−2az5−9z5a−1−5a8z4−2a6z4 + 11a4z4 + 20a2z4−2z4a−2 + 10z4−2a9z3 + 2a7z3 + 6a5z3−3a3z3 + 5z3a−1 + 3a8z2 + 6a6z2 + a4z2−6a2z2 + z2a−2−3z2 + a9z−a5z + a3z + az−a8−2a6−a4 + 1 |
| The A2 invariant | Data:K11a218/QuantumInvariant/A2/1,0 |
| The G2 invariant | Data:K11a218/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[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
|
Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.
|
In[3]:=
| K = Knot["K11a218"];
|
In[4]:=
| Alexander[K][t]
|
KnotTheory::loading: Loading precomputed data in PD4Knots`.
|
Out[4]=
| t3−10t2 + 32t−45 + 32t−1−10t−2 + t−3 |
In[5]:=
| Conway[K][z]
|
Out[5]=
| z6−4z4 + z2 + 1 |
In[6]:=
| Alexander[K, 2][t]
|
KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.
|
Out[6]=
| {1} |
In[7]:=
| {KnotDet[K], KnotSignature[K]}
|
Out[7]=
| { 131, -2 } |
In[8]:=
| Jones[K][q]
|
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
|
Out[8]=
| q3−4q2 + 9q−14 + 19q−1−21q−2 + 21q−3−18q−4 + 13q−5−7q−6 + 3q−7−q−8 |
In[9]:=
| HOMFLYPT[K][a, z]
|
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
|
Out[9]=
| −a8 + 3z2a6 + 2a6−3z4a4−3z2a4−a4 + z6a2 + z4a2 + z2a2−2z4−z2 + 1 + z2a−2 |
In[10]:=
| Kauffman[K][a, z]
|
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
|
Out[10]=
| 2a4z10 + 2a2z10 + 5a5z9 + 11a3z9 + 6az9 + 6a6z8 + 9a4z8 + 10a2z8 + 7z8 + 5a7z7−a5z7−18a3z7−8az7 + 4z7a−1 + 3a8z6−5a6z6−20a4z6−29a2z6 + z6a−2−16z6 + a9z5−6a7z5−6a5z5 + 8a3z5−2az5−9z5a−1−5a8z4−2a6z4 + 11a4z4 + 20a2z4−2z4a−2 + 10z4−2a9z3 + 2a7z3 + 6a5z3−3a3z3 + 5z3a−1 + 3a8z2 + 6a6z2 + a4z2−6a2z2 + z2a−2−3z2 + a9z−a5z + a3z + az−a8−2a6−a4 + 1 |
[edit] "Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {}
Same Jones Polynomial (up to mirroring, 
):
{K11a131,}
Computer Talk
The above data is available with the Mathematica package
KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.
(The path below may be different on your system, and possibly also the KnotTheory` date)
In[1]:=
| AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
|
Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
|
In[3]:=
| K = Knot["K11a218"];
|
In[4]:=
| {A = Alexander[K][t], J = Jones[K][q]}
|
KnotTheory::loading: Loading precomputed data in PD4Knots`.
|
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
|
Out[4]=
| { t3−10t2 + 32t−45 + 32t−1−10t−2 + t−3, q3−4q2 + 9q−14 + 19q−1−21q−2 + 21q−3−18q−4 + 13q−5−7q−6 + 3q−7−q−8 } |
In[5]:=
| DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]
|
KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.
|
KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.
|
Out[5]=
| {} |
In[6]:=
| DeleteCases[
Select[
AllKnots[],
(J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) &
],
K
]
|
KnotTheory::loading: Loading precomputed data in Jones4Knots11`.
|
Out[6]=
| {K11a131,} |
[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 = -2 is the signature of K11a218. Nonzero entries off the critical diagonals (if any exist) are highlighted in red. |
|
| Integral Khovanov Homology
(db, data source) |
|
[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. |
|
