K11a229
From Knot Atlas
|
|
|
|
 (Knotscape image) | See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.
Visit K11a229's page at Knotilus! Visit K11a229's page at the original Knot Atlas! |
[edit] Knot presentations
| Planar diagram presentation | X4251 X12,4,13,3 X20,6,21,5 X18,8,19,7 X14,10,15,9 X2,12,3,11 X10,14,11,13 X22,15,1,16 X8,18,9,17 X6,20,7,19 X16,21,17,22 |
| Gauss code | 1, -6, 2, -1, 3, -10, 4, -9, 5, -7, 6, -2, 7, -5, 8, -11, 9, -4, 10, -3, 11, -8 |
| Dowker-Thistlethwaite code | 4 12 20 18 14 2 10 22 8 6 16 |
| A Braid Representative |
| ||||||||
| A Morse Link Presentation | 
|
[edit] Three dimensional invariants
|
[edit] Four dimensional invariants
|
[edit] Polynomial invariants
| Alexander polynomial | −4t2 + 18t−27 + 18t−1−4t−2 |
| Conway polynomial | −4z4 + 2z2 + 1 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 71, 2 } |
| Jones polynomial | −q10 + 2q9−4q8 + 7q7−9q6 + 11q5−11q4 + 10q3−8q2 + 5q−2 + q−1 |
| HOMFLY-PT polynomial (db, data sources) | −z4a−2−2z4a−4−z4a−6−2z2a−4 + z2a−6 + 2z2a−8 + z2−a−4 + a−6 + a−8−a−10 + 1 |
| Kauffman polynomial (db, data sources) | z10a−6 + z10a−8 + 2z9a−5 + 4z9a−7 + 2z9a−9 + 3z8a−4−z8a−8 + 2z8a−10 + 4z7a−3−12z7a−7−7z7a−9 + z7a−11 + 3z6a−2−z6a−4 + 2z6a−6−3z6a−8−9z6a−10 + 2z5a−1−6z5a−3−5z5a−5 + 14z5a−7 + 6z5a−9−5z5a−11−2z4a−2−5z4a−4−12z4a−6 + 2z4a−8 + 12z4a−10 + z4−2z3a−1 + 7z3a−3 + 5z3a−5−12z3a−7−z3a−9 + 7z3a−11 + 7z2a−4 + 8z2a−6−2z2a−8−5z2a−10−2z2−2za−3−2za−5 + 2za−7−2za−11−a−4−a−6 + a−8 + a−10 + 1 |
| The A2 invariant | Data:K11a229/QuantumInvariant/A2/1,0 |
| The G2 invariant | Data:K11a229/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[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
|
Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.
|
In[3]:=
| K = Knot["K11a229"];
|
In[4]:=
| Alexander[K][t]
|
KnotTheory::loading: Loading precomputed data in PD4Knots`.
|
Out[4]=
| −4t2 + 18t−27 + 18t−1−4t−2 |
In[5]:=
| Conway[K][z]
|
Out[5]=
| −4z4 + 2z2 + 1 |
In[6]:=
| Alexander[K, 2][t]
|
KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.
|
Out[6]=
| {1} |
In[7]:=
| {KnotDet[K], KnotSignature[K]}
|
Out[7]=
| { 71, 2 } |
In[8]:=
| Jones[K][q]
|
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
|
Out[8]=
| −q10 + 2q9−4q8 + 7q7−9q6 + 11q5−11q4 + 10q3−8q2 + 5q−2 + q−1 |
In[9]:=
| HOMFLYPT[K][a, z]
|
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
|
Out[9]=
| −z4a−2−2z4a−4−z4a−6−2z2a−4 + z2a−6 + 2z2a−8 + z2−a−4 + a−6 + a−8−a−10 + 1 |
In[10]:=
| Kauffman[K][a, z]
|
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
|
Out[10]=
| z10a−6 + z10a−8 + 2z9a−5 + 4z9a−7 + 2z9a−9 + 3z8a−4−z8a−8 + 2z8a−10 + 4z7a−3−12z7a−7−7z7a−9 + z7a−11 + 3z6a−2−z6a−4 + 2z6a−6−3z6a−8−9z6a−10 + 2z5a−1−6z5a−3−5z5a−5 + 14z5a−7 + 6z5a−9−5z5a−11−2z4a−2−5z4a−4−12z4a−6 + 2z4a−8 + 12z4a−10 + z4−2z3a−1 + 7z3a−3 + 5z3a−5−12z3a−7−z3a−9 + 7z3a−11 + 7z2a−4 + 8z2a−6−2z2a−8−5z2a−10−2z2−2za−3−2za−5 + 2za−7−2za−11−a−4−a−6 + a−8 + a−10 + 1 |
[edit] "Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {K11a226,}
Same Jones Polynomial (up to mirroring, 
):
{}
Computer Talk
The above data is available with the Mathematica package
KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.
(The path below may be different on your system, and possibly also the KnotTheory` date)
In[1]:=
| AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
|
Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
|
In[3]:=
| K = Knot["K11a229"];
|
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]=
| { −4t2 + 18t−27 + 18t−1−4t−2, −q10 + 2q9−4q8 + 7q7−9q6 + 11q5−11q4 + 10q3−8q2 + 5q−2 + q−1 } |
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]=
| {K11a226,} |
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 = 2 is the signature of K11a229. 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. |
|
