K11a114
From Knot Atlas
|
|
|
|
 (Knotscape image) | See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.
Visit K11a114's page at Knotilus! Visit K11a114's page at the original Knot Atlas! |
[edit] Knot presentations
| Planar diagram presentation | X4251 X10,4,11,3 X14,6,15,5 X18,7,19,8 X2,10,3,9 X16,12,17,11 X6,14,7,13 X22,16,1,15 X20,17,21,18 X8,19,9,20 X12,22,13,21 |
| Gauss code | 1, -5, 2, -1, 3, -7, 4, -10, 5, -2, 6, -11, 7, -3, 8, -6, 9, -4, 10, -9, 11, -8 |
| Dowker-Thistlethwaite code | 4 10 14 18 2 16 6 22 20 8 12 |
| A Braid Representative |
| ||||||
| A Morse Link Presentation | 
|
[edit] Three dimensional invariants
|
[edit] Four dimensional invariants
|
[edit] Polynomial invariants
| Alexander polynomial | 3t3−15t2 + 35t−45 + 35t−1−15t−2 + 3t−3 |
| Conway polynomial | 3z6 + 3z4 + 2z2 + 1 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 151, 2 } |
| Jones polynomial | q9−4q8 + 9q7−16q6 + 21q5−24q4 + 25q3−21q2 + 16q−9 + 4q−1−q−2 |
| HOMFLY-PT polynomial (db, data sources) | z6a−2 + 2z6a−4 + z4a−2 + 6z4a−4−3z4a−6−z4 + 8z2a−4−6z2a−6 + z2a−8−z2 + 4a−4−4a−6 + a−8 |
| Kauffman polynomial (db, data sources) | 2z10a−4 + 2z10a−6 + 7z9a−3 + 13z9a−5 + 6z9a−7 + 10z8a−2 + 19z8a−4 + 16z8a−6 + 7z8a−8 + 8z7a−1 + 2z7a−3−12z7a−5−2z7a−7 + 4z7a−9−13z6a−2−46z6a−4−44z6a−6−14z6a−8 + z6a−10 + 4z6 + az5−11z5a−1−20z5a−3−18z5a−5−19z5a−7−9z5a−9 + 7z4a−2 + 37z4a−4 + 35z4a−6 + 8z4a−8−2z4a−10−5z4−az3 + 6z3a−1 + 16z3a−3 + 22z3a−5 + 20z3a−7 + 7z3a−9−2z2a−2−16z2a−4−15z2a−6−2z2a−8 + z2a−10 + 2z2−za−1−3za−3−7za−5−7za−7−2za−9 + 4a−4 + 4a−6 + a−8 |
| The A2 invariant | Data:K11a114/QuantumInvariant/A2/1,0 |
| The G2 invariant | Data:K11a114/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[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
|
Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.
|
In[3]:=
| K = Knot["K11a114"];
|
In[4]:=
| Alexander[K][t]
|
KnotTheory::loading: Loading precomputed data in PD4Knots`.
|
Out[4]=
| 3t3−15t2 + 35t−45 + 35t−1−15t−2 + 3t−3 |
In[5]:=
| Conway[K][z]
|
Out[5]=
| 3z6 + 3z4 + 2z2 + 1 |
In[6]:=
| Alexander[K, 2][t]
|
KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.
|
Out[6]=
| {1} |
In[7]:=
| {KnotDet[K], KnotSignature[K]}
|
Out[7]=
| { 151, 2 } |
In[8]:=
| Jones[K][q]
|
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
|
Out[8]=
| q9−4q8 + 9q7−16q6 + 21q5−24q4 + 25q3−21q2 + 16q−9 + 4q−1−q−2 |
In[9]:=
| HOMFLYPT[K][a, z]
|
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
|
Out[9]=
| z6a−2 + 2z6a−4 + z4a−2 + 6z4a−4−3z4a−6−z4 + 8z2a−4−6z2a−6 + z2a−8−z2 + 4a−4−4a−6 + a−8 |
In[10]:=
| Kauffman[K][a, z]
|
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
|
Out[10]=
| 2z10a−4 + 2z10a−6 + 7z9a−3 + 13z9a−5 + 6z9a−7 + 10z8a−2 + 19z8a−4 + 16z8a−6 + 7z8a−8 + 8z7a−1 + 2z7a−3−12z7a−5−2z7a−7 + 4z7a−9−13z6a−2−46z6a−4−44z6a−6−14z6a−8 + z6a−10 + 4z6 + az5−11z5a−1−20z5a−3−18z5a−5−19z5a−7−9z5a−9 + 7z4a−2 + 37z4a−4 + 35z4a−6 + 8z4a−8−2z4a−10−5z4−az3 + 6z3a−1 + 16z3a−3 + 22z3a−5 + 20z3a−7 + 7z3a−9−2z2a−2−16z2a−4−15z2a−6−2z2a−8 + z2a−10 + 2z2−za−1−3za−3−7za−5−7za−7−2za−9 + 4a−4 + 4a−6 + a−8 |
[edit] "Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {}
Same Jones Polynomial (up to mirroring, 
):
{}
Computer Talk
The above data is available with the Mathematica package
KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.
(The path below may be different on your system, and possibly also the KnotTheory` date)
In[1]:=
| AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
|
Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
|
In[3]:=
| K = Knot["K11a114"];
|
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]=
| { 3t3−15t2 + 35t−45 + 35t−1−15t−2 + 3t−3, q9−4q8 + 9q7−16q6 + 21q5−24q4 + 25q3−21q2 + 16q−9 + 4q−1−q−2 } |
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 K11a114. 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. |
|
