K11n82
From Knot Atlas
|
|
|
|
 (Knotscape image) | See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.
Visit K11n82's page at Knotilus! Visit K11n82's page at the original Knot Atlas! |
[edit] Knot presentations
| Planar diagram presentation | X4251 X8394 X5,17,6,16 X7,12,8,13 X2,9,3,10 X11,18,12,19 X13,21,14,20 X15,1,16,22 X17,10,18,11 X19,7,20,6 X21,15,22,14 |
| Gauss code | 1, -5, 2, -1, -3, 10, -4, -2, 5, 9, -6, 4, -7, 11, -8, 3, -9, 6, -10, 7, -11, 8 |
| Dowker-Thistlethwaite code | 4 8 -16 -12 2 -18 -20 -22 -10 -6 -14 |
| A Braid Representative |
| ||||
| A Morse Link Presentation | 
|
[edit] Three dimensional invariants
|
[edit] Four dimensional invariants
|
[edit] Polynomial invariants
| Alexander polynomial | t3−3t2 + 4t−3 + 4t−1−3t−2 + t−3 |
| Conway polynomial | z6 + 3z4 + z2 + 1 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 19, 2 } |
| Jones polynomial | −q4 + 2q3−2q2 + 3q−3 + 3q−1−2q−2 + 2q−3−q−4 |
| HOMFLY-PT polynomial (db, data sources) | z6−a2z4−z4a−2 + 5z4−3a2z2−3z2a−2 + 7z2−a2−a−2 + 3 |
| Kauffman polynomial (db, data sources) | az9 + z9a−1 + 2a2z8 + z8a−2 + 3z8 + a3z7−4az7−5z7a−1−11a2z6−5z6a−2−16z6−5a3z5 + 2az5 + 8z5a−1 + z5a−3 + 17a2z4 + 8z4a−2 + 25z4 + 6a3z3 + 2az3−7z3a−1−3z3a−3−8a2z2−6z2a−2 + z2a−4−15z2−a3z−az + za−1 + 2za−3 + za−5 + a2 + a−2 + 3 |
| The A2 invariant | −q12 + q6 + q4 + 1 + q−4 + q−6−q−12 |
| The G2 invariant | Data:K11n82/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[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
|
Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.
|
In[3]:=
| K = Knot["K11n82"];
|
In[4]:=
| Alexander[K][t]
|
KnotTheory::loading: Loading precomputed data in PD4Knots`.
|
Out[4]=
| t3−3t2 + 4t−3 + 4t−1−3t−2 + t−3 |
In[5]:=
| Conway[K][z]
|
Out[5]=
| z6 + 3z4 + z2 + 1 |
In[6]:=
| Alexander[K, 2][t]
|
KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.
|
Out[6]=
| {1} |
In[7]:=
| {KnotDet[K], KnotSignature[K]}
|
Out[7]=
| { 19, 2 } |
In[8]:=
| Jones[K][q]
|
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
|
Out[8]=
| −q4 + 2q3−2q2 + 3q−3 + 3q−1−2q−2 + 2q−3−q−4 |
In[9]:=
| HOMFLYPT[K][a, z]
|
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
|
Out[9]=
| z6−a2z4−z4a−2 + 5z4−3a2z2−3z2a−2 + 7z2−a2−a−2 + 3 |
In[10]:=
| Kauffman[K][a, z]
|
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
|
Out[10]=
| az9 + z9a−1 + 2a2z8 + z8a−2 + 3z8 + a3z7−4az7−5z7a−1−11a2z6−5z6a−2−16z6−5a3z5 + 2az5 + 8z5a−1 + z5a−3 + 17a2z4 + 8z4a−2 + 25z4 + 6a3z3 + 2az3−7z3a−1−3z3a−3−8a2z2−6z2a−2 + z2a−4−15z2−a3z−az + za−1 + 2za−3 + za−5 + a2 + a−2 + 3 |
[edit] "Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {}
Same Jones Polynomial (up to mirroring, 
):
{}
Computer Talk
The above data is available with the Mathematica package
KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.
(The path below may be different on your system, and possibly also the KnotTheory` date)
In[1]:=
| AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
|
Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
|
In[3]:=
| K = Knot["K11n82"];
|
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]=
| { t3−3t2 + 4t−3 + 4t−1−3t−2 + t−3, −q4 + 2q3−2q2 + 3q−3 + 3q−1−2q−2 + 2q−3−q−4 } |
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 = 2 is the signature of K11n82. 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. |
|
