K11a328
From Knot Atlas
|
|
|
|
 (Knotscape image) | See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.
Visit K11a328's page at Knotilus! Visit K11a328's page at the original Knot Atlas! |
[edit] Knot presentations
| Planar diagram presentation | X6271 X12,4,13,3 X18,6,19,5 X16,7,17,8 X22,10,1,9 X4,12,5,11 X20,14,21,13 X8,15,9,16 X2,18,3,17 X10,20,11,19 X14,22,15,21 |
| Gauss code | 1, -9, 2, -6, 3, -1, 4, -8, 5, -10, 6, -2, 7, -11, 8, -4, 9, -3, 10, -7, 11, -5 |
| Dowker-Thistlethwaite code | 6 12 18 16 22 4 20 8 2 10 14 |
| A Braid Representative |
| ||||||
| A Morse Link Presentation | 
|
[edit] Three dimensional invariants
|
[edit] Four dimensional invariants
|
[edit] Polynomial invariants
| Alexander polynomial | −3t3 + 16t2−34t + 43−34t−1 + 16t−2−3t−3 |
| Conway polynomial | −3z6−2z4 + 3z2 + 1 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 149, 4 } |
| Jones polynomial | −q11 + 4q10−9q9 + 15q8−21q7 + 24q6−24q5 + 21q4−15q3 + 10q2−4q + 1 |
| HOMFLY-PT polynomial (db, data sources) | −z6a−4−2z6a−6 + z4a−2−6z4a−6 + 3z4a−8 + z2a−2 + 5z2a−4−8z2a−6 + 6z2a−8−z2a−10 + 4a−4−5a−6 + 3a−8−a−10 |
| Kauffman polynomial (db, data sources) | 3z10a−6 + 3z10a−8 + 8z9a−5 + 16z9a−7 + 8z9a−9 + 8z8a−4 + 11z8a−6 + 13z8a−8 + 10z8a−10 + 4z7a−3−15z7a−5−33z7a−7−6z7a−9 + 8z7a−11 + z6a−2−19z6a−4−43z6a−6−42z6a−8−15z6a−10 + 4z6a−12−8z5a−3 + 6z5a−5 + 19z5a−7−8z5a−9−12z5a−11 + z5a−13−2z4a−2 + 15z4a−4 + 46z4a−6 + 43z4a−8 + 9z4a−10−5z4a−12 + 3z3a−3 + 2z3a−7 + 12z3a−9 + 6z3a−11−z3a−13 + z2a−2−10z2a−4−23z2a−6−17z2a−8−4z2a−10 + z2a−12−2za−5−3za−7−3za−9−2za−11 + 4a−4 + 5a−6 + 3a−8 + a−10 |
| The A2 invariant | Data:K11a328/QuantumInvariant/A2/1,0 |
| The G2 invariant | Data:K11a328/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["K11a328"];
|
In[4]:=
| Alexander[K][t]
|
KnotTheory::loading: Loading precomputed data in PD4Knots`.
|
Out[4]=
| −3t3 + 16t2−34t + 43−34t−1 + 16t−2−3t−3 |
In[5]:=
| Conway[K][z]
|
Out[5]=
| −3z6−2z4 + 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]=
| { 149, 4 } |
In[8]:=
| Jones[K][q]
|
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
|
Out[8]=
| −q11 + 4q10−9q9 + 15q8−21q7 + 24q6−24q5 + 21q4−15q3 + 10q2−4q + 1 |
In[9]:=
| HOMFLYPT[K][a, z]
|
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
|
Out[9]=
| −z6a−4−2z6a−6 + z4a−2−6z4a−6 + 3z4a−8 + z2a−2 + 5z2a−4−8z2a−6 + 6z2a−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]=
| 3z10a−6 + 3z10a−8 + 8z9a−5 + 16z9a−7 + 8z9a−9 + 8z8a−4 + 11z8a−6 + 13z8a−8 + 10z8a−10 + 4z7a−3−15z7a−5−33z7a−7−6z7a−9 + 8z7a−11 + z6a−2−19z6a−4−43z6a−6−42z6a−8−15z6a−10 + 4z6a−12−8z5a−3 + 6z5a−5 + 19z5a−7−8z5a−9−12z5a−11 + z5a−13−2z4a−2 + 15z4a−4 + 46z4a−6 + 43z4a−8 + 9z4a−10−5z4a−12 + 3z3a−3 + 2z3a−7 + 12z3a−9 + 6z3a−11−z3a−13 + z2a−2−10z2a−4−23z2a−6−17z2a−8−4z2a−10 + z2a−12−2za−5−3za−7−3za−9−2za−11 + 4a−4 + 5a−6 + 3a−8 + a−10 |
[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["K11a328"];
|
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]=
| { −3t3 + 16t2−34t + 43−34t−1 + 16t−2−3t−3, −q11 + 4q10−9q9 + 15q8−21q7 + 24q6−24q5 + 21q4−15q3 + 10q2−4q + 1 } |
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 = 4 is the signature of K11a328. 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. |
|
