K11n59
From Knot Atlas
|
|
|
|
 (Knotscape image) | See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.
Visit K11n59's page at Knotilus! Visit K11n59's page at the original Knot Atlas! |
[edit] Knot presentations
| Planar diagram presentation | X4251 X8493 X5,15,6,14 X2837 X16,9,17,10 X18,12,19,11 X20,14,21,13 X15,7,16,6 X22,17,1,18 X12,20,13,19 X10,22,11,21 |
| Gauss code | 1, -4, 2, -1, -3, 8, 4, -2, 5, -11, 6, -10, 7, 3, -8, -5, 9, -6, 10, -7, 11, -9 |
| Dowker-Thistlethwaite code | 4 8 -14 2 16 18 20 -6 22 12 10 |
| A Braid Representative |
| ||||
| A Morse Link Presentation | 
|
[edit] Three dimensional invariants
|
[edit] Four dimensional invariants
|
[edit] Polynomial invariants
| Alexander polynomial | −t3 + 6t2−12t + 15−12t−1 + 6t−2−t−3 |
| Conway polynomial | −z6 + 3z2 + 1 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 53, 4 } |
| Jones polynomial | −q11 + 3q10−5q9 + 7q8−9q7 + 9q6−8q5 + 6q4−3q3 + 2q2 |
| HOMFLY-PT polynomial (db, data sources) | −z6a−6 + 2z4a−4−4z4a−6 + 2z4a−8 + 6z2a−4−7z2a−6 + 5z2a−8−z2a−10 + 4a−4−5a−6 + 3a−8−a−10 |
| Kauffman polynomial (db, data sources) | z9a−7 + z9a−9 + 2z8a−6 + 5z8a−8 + 3z8a−10 + z7a−5 + 3z7a−9 + 4z7a−11−7z6a−6−16z6a−8−6z6a−10 + 3z6a−12−z5a−5−4z5a−7−14z5a−9−10z5a−11 + z5a−13 + 3z4a−4 + 16z4a−6 + 24z4a−8 + 4z4a−10−7z4a−12 + z3a−5 + 11z3a−7 + 19z3a−9 + 7z3a−11−2z3a−13−8z2a−4−16z2a−6−12z2a−8−2z2a−10 + 2z2a−12−3za−5−6za−7−6za−9−3za−11 + 4a−4 + 5a−6 + 3a−8 + a−10 |
| The A2 invariant | 2q−6 + 3q−10 + q−12−q−14 + q−16−3q−18−q−22 + 2q−26−q−28 + q−30−q−34 |
| The G2 invariant | Data:K11n59/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[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
|
Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.
|
In[3]:=
| K = Knot["K11n59"];
|
In[4]:=
| Alexander[K][t]
|
KnotTheory::loading: Loading precomputed data in PD4Knots`.
|
Out[4]=
| −t3 + 6t2−12t + 15−12t−1 + 6t−2−t−3 |
In[5]:=
| Conway[K][z]
|
Out[5]=
| −z6 + 3z2 + 1 |
In[6]:=
| Alexander[K, 2][t]
|
KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.
|
Out[6]=
| {1} |
In[7]:=
| {KnotDet[K], KnotSignature[K]}
|
Out[7]=
| { 53, 4 } |
In[8]:=
| Jones[K][q]
|
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
|
Out[8]=
| −q11 + 3q10−5q9 + 7q8−9q7 + 9q6−8q5 + 6q4−3q3 + 2q2 |
In[9]:=
| HOMFLYPT[K][a, z]
|
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
|
Out[9]=
| −z6a−6 + 2z4a−4−4z4a−6 + 2z4a−8 + 6z2a−4−7z2a−6 + 5z2a−8−z2a−10 + 4a−4−5a−6 + 3a−8−a−10 |
In[10]:=
| Kauffman[K][a, z]
|
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
|
Out[10]=
| z9a−7 + z9a−9 + 2z8a−6 + 5z8a−8 + 3z8a−10 + z7a−5 + 3z7a−9 + 4z7a−11−7z6a−6−16z6a−8−6z6a−10 + 3z6a−12−z5a−5−4z5a−7−14z5a−9−10z5a−11 + z5a−13 + 3z4a−4 + 16z4a−6 + 24z4a−8 + 4z4a−10−7z4a−12 + z3a−5 + 11z3a−7 + 19z3a−9 + 7z3a−11−2z3a−13−8z2a−4−16z2a−6−12z2a−8−2z2a−10 + 2z2a−12−3za−5−6za−7−6za−9−3za−11 + 4a−4 + 5a−6 + 3a−8 + a−10 |
[edit] "Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {K11n8,}
Same Jones Polynomial (up to mirroring, 
):
{}
Computer Talk
The above data is available with the Mathematica package
KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.
(The path below may be different on your system, and possibly also the KnotTheory` date)
In[1]:=
| AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
|
Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
|
In[3]:=
| K = Knot["K11n59"];
|
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 + 6t2−12t + 15−12t−1 + 6t−2−t−3, −q11 + 3q10−5q9 + 7q8−9q7 + 9q6−8q5 + 6q4−3q3 + 2q2 } |
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]=
| {K11n8,} |
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 = 4 is the signature of K11n59. 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. |
|
