K11n85
From Knot Atlas
|
|
|
|
 (Knotscape image) | See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.
Visit K11n85's page at Knotilus! Visit K11n85's page at the original Knot Atlas! |
[edit] Knot presentations
| Planar diagram presentation | X4251 X10,3,11,4 X12,6,13,5 X7,15,8,14 X16,10,17,9 X2,11,3,12 X13,19,14,18 X20,16,21,15 X22,17,1,18 X19,6,20,7 X8,21,9,22 |
| Gauss code | 1, -6, 2, -1, 3, 10, -4, -11, 5, -2, 6, -3, -7, 4, 8, -5, 9, 7, -10, -8, 11, -9 |
| Dowker-Thistlethwaite code | 4 10 12 -14 16 2 -18 20 22 -6 8 |
| A Braid Representative |
| ||||
| A Morse Link Presentation | 
|
[edit] Three dimensional invariants
|
[edit] Four dimensional invariants
|
[edit] Polynomial invariants
| Alexander polynomial | −t3 + 5t2−10t + 13−10t−1 + 5t−2−t−3 |
| Conway polynomial | −z6−z4 + z2 + 1 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 45, 0 } |
| Jones polynomial | q6−3q5 + 4q4−6q3 + 8q2−7q + 7−5q−1 + 3q−2−q−3 |
| HOMFLY-PT polynomial (db, data sources) | −z6a−2−4z4a−2 + z4a−4 + 2z4−a2z2−5z2a−2 + 2z2a−4 + 5z2−a2−a−2 + 3 |
| Kauffman polynomial (db, data sources) | z9a−1 + z9a−3 + 5z8a−2 + 3z8a−4 + 2z8 + az7 + 2z7a−3 + 3z7a−5−18z6a−2−10z6a−4 + z6a−6−7z6−az5−6z5a−1−16z5a−3−11z5a−5 + 3a2z4 + 23z4a−2 + 8z4a−4−3z4a−6 + 15z4 + a3z3 + 4az3 + 11z3a−1 + 17z3a−3 + 9z3a−5−3a2z2−12z2a−2−3z2a−4 + z2a−6−11z2−a3z−3az−5za−1−4za−3−za−5 + a2 + a−2 + 3 |
| The A2 invariant | −q10 + q6−q4 + 2q2 + q−2 + 2q−4 + 2q−8−2q−10−q−12−q−16 + q−18 |
| The G2 invariant | Data:K11n85/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[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
|
Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.
|
In[3]:=
| K = Knot["K11n85"];
|
In[4]:=
| Alexander[K][t]
|
KnotTheory::loading: Loading precomputed data in PD4Knots`.
|
Out[4]=
| −t3 + 5t2−10t + 13−10t−1 + 5t−2−t−3 |
In[5]:=
| Conway[K][z]
|
Out[5]=
| −z6−z4 + z2 + 1 |
In[6]:=
| Alexander[K, 2][t]
|
KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.
|
Out[6]=
| {1} |
In[7]:=
| {KnotDet[K], KnotSignature[K]}
|
Out[7]=
| { 45, 0 } |
In[8]:=
| Jones[K][q]
|
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
|
Out[8]=
| q6−3q5 + 4q4−6q3 + 8q2−7q + 7−5q−1 + 3q−2−q−3 |
In[9]:=
| HOMFLYPT[K][a, z]
|
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
|
Out[9]=
| −z6a−2−4z4a−2 + z4a−4 + 2z4−a2z2−5z2a−2 + 2z2a−4 + 5z2−a2−a−2 + 3 |
In[10]:=
| Kauffman[K][a, z]
|
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
|
Out[10]=
| z9a−1 + z9a−3 + 5z8a−2 + 3z8a−4 + 2z8 + az7 + 2z7a−3 + 3z7a−5−18z6a−2−10z6a−4 + z6a−6−7z6−az5−6z5a−1−16z5a−3−11z5a−5 + 3a2z4 + 23z4a−2 + 8z4a−4−3z4a−6 + 15z4 + a3z3 + 4az3 + 11z3a−1 + 17z3a−3 + 9z3a−5−3a2z2−12z2a−2−3z2a−4 + z2a−6−11z2−a3z−3az−5za−1−4za−3−za−5 + a2 + a−2 + 3 |
[edit] "Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {8_18, 9_24, K11n164,}
Same Jones Polynomial (up to mirroring, 
):
{}
Computer Talk
The above data is available with the Mathematica package
KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.
(The path below may be different on your system, and possibly also the KnotTheory` date)
In[1]:=
| AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
|
Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
|
In[3]:=
| K = Knot["K11n85"];
|
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 + 5t2−10t + 13−10t−1 + 5t−2−t−3, q6−3q5 + 4q4−6q3 + 8q2−7q + 7−5q−1 + 3q−2−q−3 } |
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_18, 9_24, K11n164,} |
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 K11n85. 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. |
|
