K11n53
From Knot Atlas
|
|
|
|
 (Knotscape image) | See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.
Visit K11n53's page at Knotilus! Visit K11n53's page at the original Knot Atlas! |
[edit] Knot presentations
| Planar diagram presentation | X4251 X8493 X5,14,6,15 X2837 X9,16,10,17 X11,19,12,18 X13,6,14,7 X15,22,16,1 X17,20,18,21 X19,11,20,10 X21,12,22,13 |
| Gauss code | 1, -4, 2, -1, -3, 7, 4, -2, -5, 10, -6, 11, -7, 3, -8, 5, -9, 6, -10, 9, -11, 8 |
| Dowker-Thistlethwaite code | 4 8 -14 2 -16 -18 -6 -22 -20 -10 -12 |
| A Braid Representative |
| ||||||
| A Morse Link Presentation | 
|
[edit] Three dimensional invariants
|
[edit] Four dimensional invariants
|
[edit] Polynomial invariants
| Alexander polynomial | −t3 + 4t2−8t + 11−8t−1 + 4t−2−t−3 |
| Conway polynomial | −z6−2z4−z2 + 1 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 37, 0 } |
| Jones polynomial | −q3 + 3q2−4q + 6−6q−1 + 6q−2−5q−3 + 3q−4−2q−5 + q−6 |
| HOMFLY-PT polynomial (db, data sources) | −a2z6 + a4z4−5a2z4 + 2z4 + 3a4z2−9a2z2−z2a−2 + 6z2 + 2a4−5a2−a−2 + 5 |
| Kauffman polynomial (db, data sources) | a3z9 + az9 + 2a4z8 + 4a2z8 + 2z8 + 2a5z7−a3z7−2az7 + z7a−1 + a6z6−7a4z6−17a2z6−9z6−8a5z5−7a3z5−az5−2z5a−1−4a6z4 + 6a4z4 + 27a2z4 + 3z4a−2 + 20z4 + 8a5z3 + 12a3z3 + 8az3 + 5z3a−1 + z3a−3 + 3a6z2−4a4z2−19a2z2−4z2a−2−16z2−3a5z−6a3z−5az−3za−1−za−3 + 2a4 + 5a2 + a−2 + 5 |
| The A2 invariant | Data:K11n53/QuantumInvariant/A2/1,0 |
| The G2 invariant | Data:K11n53/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[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
|
Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.
|
In[3]:=
| K = Knot["K11n53"];
|
In[4]:=
| Alexander[K][t]
|
KnotTheory::loading: Loading precomputed data in PD4Knots`.
|
Out[4]=
| −t3 + 4t2−8t + 11−8t−1 + 4t−2−t−3 |
In[5]:=
| Conway[K][z]
|
Out[5]=
| −z6−2z4−z2 + 1 |
In[6]:=
| Alexander[K, 2][t]
|
KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.
|
Out[6]=
| {1} |
In[7]:=
| {KnotDet[K], KnotSignature[K]}
|
Out[7]=
| { 37, 0 } |
In[8]:=
| Jones[K][q]
|
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
|
Out[8]=
| −q3 + 3q2−4q + 6−6q−1 + 6q−2−5q−3 + 3q−4−2q−5 + q−6 |
In[9]:=
| HOMFLYPT[K][a, z]
|
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
|
Out[9]=
| −a2z6 + a4z4−5a2z4 + 2z4 + 3a4z2−9a2z2−z2a−2 + 6z2 + 2a4−5a2−a−2 + 5 |
In[10]:=
| Kauffman[K][a, z]
|
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
|
Out[10]=
| a3z9 + az9 + 2a4z8 + 4a2z8 + 2z8 + 2a5z7−a3z7−2az7 + z7a−1 + a6z6−7a4z6−17a2z6−9z6−8a5z5−7a3z5−az5−2z5a−1−4a6z4 + 6a4z4 + 27a2z4 + 3z4a−2 + 20z4 + 8a5z3 + 12a3z3 + 8az3 + 5z3a−1 + z3a−3 + 3a6z2−4a4z2−19a2z2−4z2a−2−16z2−3a5z−6a3z−5az−3za−1−za−3 + 2a4 + 5a2 + a−2 + 5 |
[edit] "Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {8_17,}
Same Jones Polynomial (up to mirroring, 
):
{9_14,}
Computer Talk
The above data is available with the Mathematica package
KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.
(The path below may be different on your system, and possibly also the KnotTheory` date)
In[1]:=
| AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
|
Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
|
In[3]:=
| K = Knot["K11n53"];
|
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 + 4t2−8t + 11−8t−1 + 4t−2−t−3, −q3 + 3q2−4q + 6−6q−1 + 6q−2−5q−3 + 3q−4−2q−5 + q−6 } |
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]=
| {8_17,} |
In[6]:=
| DeleteCases[
Select[
AllKnots[],
(J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) &
],
K
]
|
KnotTheory::loading: Loading precomputed data in Jones4Knots11`.
|
Out[6]=
| {9_14,} |
[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 K11n53. 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. |
|
