K11n32
From Knot Atlas
|
|
|
|
 (Knotscape image) | See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.
Visit K11n32's page at Knotilus! Visit K11n32's page at the original Knot Atlas! |
[edit] Knot presentations
| Planar diagram presentation | X4251 X8493 X5,12,6,13 X2837 X16,9,17,10 X11,6,12,7 X20,14,21,13 X10,15,11,16 X22,17,1,18 X14,20,15,19 X18,21,19,22 |
| Gauss code | 1, -4, 2, -1, -3, 6, 4, -2, 5, -8, -6, 3, 7, -10, 8, -5, 9, -11, 10, -7, 11, -9 |
| Dowker-Thistlethwaite code | 4 8 -12 2 16 -6 20 10 22 14 18 |
| A Braid Representative |
| ||||||
| A Morse Link Presentation | 
|
[edit] Three dimensional invariants
|
[edit] Four dimensional invariants
|
[edit] Polynomial invariants
| Alexander polynomial | −t3 + 6t2−16t + 23−16t−1 + 6t−2−t−3 |
| Conway polynomial | −z6−z2 + 1 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 69, 0 } |
| Jones polynomial | −q5 + 3q4−6q3 + 10q2−11q + 12−11q−1 + 8q−2−5q−3 + 2q−4 |
| HOMFLY-PT polynomial (db, data sources) | −z6 + a2z4 + 2z4a−2−3z4 + 4z2a−2−z2a−4−4z2 + a4−a2 + 3a−2−a−4−1 |
| Kauffman polynomial (db, data sources) | az9 + z9a−1 + 2a2z8 + 3z8a−2 + 5z8 + a3z7 + 3az7 + 6z7a−1 + 4z7a−3−a2z6 + z6a−2 + 3z6a−4−3z6 + 3a3z5−9z5a−1−5z5a−3 + z5a−5 + 3a4z4 + 4a2z4−10z4a−2−6z4a−4−3z4−6a3z3−8az3−z3a−1−z3a−3−2z3a−5−4a4z2−6a2z2 + 8z2a−2 + 4z2a−4 + 2z2 + 3a3z + 5az + 3za−1 + 2za−3 + za−5 + a4 + a2−3a−2−a−4−1 |
| The A2 invariant | Data:K11n32/QuantumInvariant/A2/1,0 |
| The G2 invariant | Data:K11n32/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[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
|
Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.
|
In[3]:=
| K = Knot["K11n32"];
|
In[4]:=
| Alexander[K][t]
|
KnotTheory::loading: Loading precomputed data in PD4Knots`.
|
Out[4]=
| −t3 + 6t2−16t + 23−16t−1 + 6t−2−t−3 |
In[5]:=
| Conway[K][z]
|
Out[5]=
| −z6−z2 + 1 |
In[6]:=
| Alexander[K, 2][t]
|
KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.
|
Out[6]=
| {1} |
In[7]:=
| {KnotDet[K], KnotSignature[K]}
|
Out[7]=
| { 69, 0 } |
In[8]:=
| Jones[K][q]
|
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
|
Out[8]=
| −q5 + 3q4−6q3 + 10q2−11q + 12−11q−1 + 8q−2−5q−3 + 2q−4 |
In[9]:=
| HOMFLYPT[K][a, z]
|
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
|
Out[9]=
| −z6 + a2z4 + 2z4a−2−3z4 + 4z2a−2−z2a−4−4z2 + a4−a2 + 3a−2−a−4−1 |
In[10]:=
| Kauffman[K][a, z]
|
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
|
Out[10]=
| az9 + z9a−1 + 2a2z8 + 3z8a−2 + 5z8 + a3z7 + 3az7 + 6z7a−1 + 4z7a−3−a2z6 + z6a−2 + 3z6a−4−3z6 + 3a3z5−9z5a−1−5z5a−3 + z5a−5 + 3a4z4 + 4a2z4−10z4a−2−6z4a−4−3z4−6a3z3−8az3−z3a−1−z3a−3−2z3a−5−4a4z2−6a2z2 + 8z2a−2 + 4z2a−4 + 2z2 + 3a3z + 5az + 3za−1 + 2za−3 + za−5 + a4 + a2−3a−2−a−4−1 |
[edit] "Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {9_34, K11n119,}
Same Jones Polynomial (up to mirroring, 
):
{}
Computer Talk
The above data is available with the Mathematica package
KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.
(The path below may be different on your system, and possibly also the KnotTheory` date)
In[1]:=
| AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
|
Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
|
In[3]:=
| K = Knot["K11n32"];
|
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 + 6t2−16t + 23−16t−1 + 6t−2−t−3, −q5 + 3q4−6q3 + 10q2−11q + 12−11q−1 + 8q−2−5q−3 + 2q−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]=
| {9_34, K11n119,} |
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 K11n32. 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. |
|
