K11a288
From Knot Atlas
|
|
|
|
 (Knotscape image) | See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.
Visit K11a288's page at Knotilus! Visit K11a288's page at the original Knot Atlas! |
[edit] Knot presentations
| Planar diagram presentation | X6271 X10,3,11,4 X16,6,17,5 X18,7,19,8 X20,10,21,9 X4,11,5,12 X8,14,9,13 X2,16,3,15 X22,17,1,18 X14,19,15,20 X12,22,13,21 |
| Gauss code | 1, -8, 2, -6, 3, -1, 4, -7, 5, -2, 6, -11, 7, -10, 8, -3, 9, -4, 10, -5, 11, -9 |
| Dowker-Thistlethwaite code | 6 10 16 18 20 4 8 2 22 14 12 |
| A Braid Representative |
| ||||
| A Morse Link Presentation | 
|
[edit] Three dimensional invariants
|
[edit] Four dimensional invariants
|
[edit] Polynomial invariants
| Alexander polynomial | t4−7t3 + 23t2−44t + 55−44t−1 + 23t−2−7t−3 + t−4 |
| Conway polynomial | z8 + z6 + z4 + z2 + 1 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 205, 0 } |
| Jones polynomial | q6−5q5 + 12q4−21q3 + 29q2−33q + 34−29q−1 + 22q−2−13q−3 + 5q−4−q−5 |
| HOMFLY-PT polynomial (db, data sources) | z8−a2z6−2z6a−2 + 4z6−2a2z4−5z4a−2 + z4a−4 + 7z4−2a2z2−4z2a−2 + z2a−4 + 6z2−a2−a−2 + 3 |
| Kauffman polynomial (db, data sources) | 5z10a−2 + 5z10 + 16az9 + 28z9a−1 + 12z9a−3 + 20a2z8 + 21z8a−2 + 11z8a−4 + 30z8 + 13a3z7−12az7−45z7a−1−15z7a−3 + 5z7a−5 + 5a4z6−31a2z6−65z6a−2−21z6a−4 + z6a−6−79z6 + a5z5−14a3z5−13az5 + 6z5a−1−4z5a−3−8z5a−5−2a4z4 + 19a2z4 + 47z4a−2 + 14z4a−4−z4a−6 + 53z4 + 6a3z3 + 10az3 + 8z3a−1 + 8z3a−3 + 4z3a−5−6a2z2−12z2a−2−3z2a−4−15z2−a3z−2az−2za−1−za−3 + a2 + a−2 + 3 |
| The A2 invariant | −q14 + 3q12−5q10 + 3q8 + q6−5q4 + 8q2−5 + 6q−2−q−4−2q−6 + 5q−8−6q−10 + 3q−12−2q−16 + q−18 |
| The G2 invariant | Data:K11a288/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[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
|
Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.
|
In[3]:=
| K = Knot["K11a288"];
|
In[4]:=
| Alexander[K][t]
|
KnotTheory::loading: Loading precomputed data in PD4Knots`.
|
Out[4]=
| t4−7t3 + 23t2−44t + 55−44t−1 + 23t−2−7t−3 + t−4 |
In[5]:=
| Conway[K][z]
|
Out[5]=
| z8 + z6 + z4 + z2 + 1 |
In[6]:=
| Alexander[K, 2][t]
|
KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.
|
Out[6]=
| {1} |
In[7]:=
| {KnotDet[K], KnotSignature[K]}
|
Out[7]=
| { 205, 0 } |
In[8]:=
| Jones[K][q]
|
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
|
Out[8]=
| q6−5q5 + 12q4−21q3 + 29q2−33q + 34−29q−1 + 22q−2−13q−3 + 5q−4−q−5 |
In[9]:=
| HOMFLYPT[K][a, z]
|
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
|
Out[9]=
| z8−a2z6−2z6a−2 + 4z6−2a2z4−5z4a−2 + z4a−4 + 7z4−2a2z2−4z2a−2 + z2a−4 + 6z2−a2−a−2 + 3 |
In[10]:=
| Kauffman[K][a, z]
|
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
|
Out[10]=
| 5z10a−2 + 5z10 + 16az9 + 28z9a−1 + 12z9a−3 + 20a2z8 + 21z8a−2 + 11z8a−4 + 30z8 + 13a3z7−12az7−45z7a−1−15z7a−3 + 5z7a−5 + 5a4z6−31a2z6−65z6a−2−21z6a−4 + z6a−6−79z6 + a5z5−14a3z5−13az5 + 6z5a−1−4z5a−3−8z5a−5−2a4z4 + 19a2z4 + 47z4a−2 + 14z4a−4−z4a−6 + 53z4 + 6a3z3 + 10az3 + 8z3a−1 + 8z3a−3 + 4z3a−5−6a2z2−12z2a−2−3z2a−4−15z2−a3z−2az−2za−1−za−3 + a2 + a−2 + 3 |
[edit] "Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {}
Same Jones Polynomial (up to mirroring, 
):
{}
Computer Talk
The above data is available with the Mathematica package
KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.
(The path below may be different on your system, and possibly also the KnotTheory` date)
In[1]:=
| AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
|
Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
|
In[3]:=
| K = Knot["K11a288"];
|
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]=
| { t4−7t3 + 23t2−44t + 55−44t−1 + 23t−2−7t−3 + t−4, q6−5q5 + 12q4−21q3 + 29q2−33q + 34−29q−1 + 22q−2−13q−3 + 5q−4−q−5 } |
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]=
| {} |
[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 K11a288. 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. |
|
