K11a104
From Knot Atlas
|
|
|
|
 (Knotscape image) | See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.
Visit K11a104's page at Knotilus! Visit K11a104's page at the original Knot Atlas! |
[edit] Knot presentations
| Planar diagram presentation | X4251 X10,3,11,4 X14,5,15,6 X12,8,13,7 X20,10,21,9 X2,11,3,12 X16,13,17,14 X6,15,7,16 X22,18,1,17 X8,20,9,19 X18,22,19,21 |
| Gauss code | 1, -6, 2, -1, 3, -8, 4, -10, 5, -2, 6, -4, 7, -3, 8, -7, 9, -11, 10, -5, 11, -9 |
| Dowker-Thistlethwaite code | 4 10 14 12 20 2 16 6 22 8 18 |
| A Braid Representative |
| ||||||
| A Morse Link Presentation | 
|
[edit] Three dimensional invariants
|
[edit] Four dimensional invariants
|
[edit] Polynomial invariants
| Alexander polynomial | −2t3 + 12t2−29t + 39−29t−1 + 12t−2−2t−3 |
| Conway polynomial | −2z6 + z2 + 1 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 125, 0 } |
| Jones polynomial | q6−4q5 + 8q4−13q3 + 18q2−20q + 20−17q−1 + 13q−2−7q−3 + 3q−4−q−5 |
| HOMFLY-PT polynomial (db, data sources) | −z6a−2−z6 + 2a2z4−2z4a−2 + z4a−4−z4−a4z2 + 3a2z2−2z2a−2 + z2a−4−a4 + 2a2 |
| Kauffman polynomial (db, data sources) | z10a−2 + z10 + 4az9 + 8z9a−1 + 4z9a−3 + 6a2z8 + 14z8a−2 + 6z8a−4 + 14z8 + 5a3z7 + 4az7−3z7a−1 + 2z7a−3 + 4z7a−5 + 3a4z6−6a2z6−36z6a−2−13z6a−4 + z6a−6−31z6 + a5z5−6a3z5−16az5−21z5a−1−22z5a−3−10z5a−5−5a4z4 + a2z4 + 26z4a−2 + 7z4a−4−2z4a−6 + 23z4−2a5z3 + 2a3z3 + 14az3 + 21z3a−1 + 18z3a−3 + 7z3a−5 + 3a4z2 + 3a2z2−7z2a−2−z2a−4 + z2a−6−5z2 + a5z−4az−6za−1−4za−3−za−5−a4−2a2 |
| The A2 invariant | Data:K11a104/QuantumInvariant/A2/1,0 |
| The G2 invariant | Data:K11a104/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[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
|
Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.
|
In[3]:=
| K = Knot["K11a104"];
|
In[4]:=
| Alexander[K][t]
|
KnotTheory::loading: Loading precomputed data in PD4Knots`.
|
Out[4]=
| −2t3 + 12t2−29t + 39−29t−1 + 12t−2−2t−3 |
In[5]:=
| Conway[K][z]
|
Out[5]=
| −2z6 + z2 + 1 |
In[6]:=
| Alexander[K, 2][t]
|
KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.
|
Out[6]=
| {1} |
In[7]:=
| {KnotDet[K], KnotSignature[K]}
|
Out[7]=
| { 125, 0 } |
In[8]:=
| Jones[K][q]
|
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
|
Out[8]=
| q6−4q5 + 8q4−13q3 + 18q2−20q + 20−17q−1 + 13q−2−7q−3 + 3q−4−q−5 |
In[9]:=
| HOMFLYPT[K][a, z]
|
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
|
Out[9]=
| −z6a−2−z6 + 2a2z4−2z4a−2 + z4a−4−z4−a4z2 + 3a2z2−2z2a−2 + z2a−4−a4 + 2a2 |
In[10]:=
| Kauffman[K][a, z]
|
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
|
Out[10]=
| z10a−2 + z10 + 4az9 + 8z9a−1 + 4z9a−3 + 6a2z8 + 14z8a−2 + 6z8a−4 + 14z8 + 5a3z7 + 4az7−3z7a−1 + 2z7a−3 + 4z7a−5 + 3a4z6−6a2z6−36z6a−2−13z6a−4 + z6a−6−31z6 + a5z5−6a3z5−16az5−21z5a−1−22z5a−3−10z5a−5−5a4z4 + a2z4 + 26z4a−2 + 7z4a−4−2z4a−6 + 23z4−2a5z3 + 2a3z3 + 14az3 + 21z3a−1 + 18z3a−3 + 7z3a−5 + 3a4z2 + 3a2z2−7z2a−2−z2a−4 + z2a−6−5z2 + a5z−4az−6za−1−4za−3−za−5−a4−2a2 |
[edit] "Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {K11a67, K11a168,}
Same Jones Polynomial (up to mirroring, 
):
{K11a168,}
Computer Talk
The above data is available with the Mathematica package
KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.
(The path below may be different on your system, and possibly also the KnotTheory` date)
In[1]:=
| AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
|
Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
|
In[3]:=
| K = Knot["K11a104"];
|
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]=
| { −2t3 + 12t2−29t + 39−29t−1 + 12t−2−2t−3, q6−4q5 + 8q4−13q3 + 18q2−20q + 20−17q−1 + 13q−2−7q−3 + 3q−4−q−5 } |
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]=
| {K11a67, K11a168,} |
In[6]:=
| DeleteCases[
Select[
AllKnots[],
(J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) &
],
K
]
|
KnotTheory::loading: Loading precomputed data in Jones4Knots11`.
|
Out[6]=
| {K11a168,} |
[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 K11a104. 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. |
|
