K11a87
From Knot Atlas
|
|
|
|
 (Knotscape image) | See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.
Visit K11a87's page at Knotilus! Visit K11a87's page at the original Knot Atlas! |
[edit] Knot presentations
| Planar diagram presentation | X4251 X10,3,11,4 X12,6,13,5 X16,8,17,7 X18,10,19,9 X2,11,3,12 X20,13,21,14 X8,16,9,15 X6,18,7,17 X22,19,1,20 X14,21,15,22 |
| Gauss code | 1, -6, 2, -1, 3, -9, 4, -8, 5, -2, 6, -3, 7, -11, 8, -4, 9, -5, 10, -7, 11, -10 |
| Dowker-Thistlethwaite code | 4 10 12 16 18 2 20 8 6 22 14 |
| A Braid Representative |
| ||||||
| A Morse Link Presentation | 
|
[edit] Three dimensional invariants
|
[edit] Four dimensional invariants
|
[edit] Polynomial invariants
| Alexander polynomial | −2t3 + 11t2−28t + 39−28t−1 + 11t−2−2t−3 |
| Conway polynomial | −2z6−z4−2z2 + 1 |
| 2nd Alexander ideal (db, data sources) |  |
| Determinant and Signature | { 121, 0 } |
| Jones polynomial | q6−3q5 + 7q4−12q3 + 16q2−19q + 20−17q−1 + 13q−2−8q−3 + 4q−4−q−5 |
| HOMFLY-PT polynomial (db, data sources) | −z6a−2−z6 + 2a2z4−3z4a−2 + z4a−4−z4−a4z2 + 2a2z2−6z2a−2 + 2z2a−4 + z2−4a−2 + 2a−4 + 3 |
| Kauffman polynomial (db, data sources) | z10a−2 + z10 + 4az9 + 8z9a−1 + 4z9a−3 + 7a2z8 + 12z8a−2 + 5z8a−4 + 14z8 + 7a3z7 + 6az7−8z7a−1−4z7a−3 + 3z7a−5 + 4a4z6−7a2z6−37z6a−2−13z6a−4 + z6a−6−34z6 + a5z5−11a3z5−22az5−8z5a−1−6z5a−3−8z5a−5−6a4z4−a2z4 + 41z4a−2 + 12z4a−4−3z4a−6 + 31z4−a5z3 + 5a3z3 + 16az3 + 12z3a−1 + 8z3a−3 + 6z3a−5 + 2a4z2 + a2z2−21z2a−2−7z2a−4 + 2z2a−6−13z2−a3z−3az−3za−1−3za−3−2za−5 + 4a−2 + 2a−4 + 3 |
| The A2 invariant | Data:K11a87/QuantumInvariant/A2/1,0 |
| The G2 invariant | Data:K11a87/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[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
|
Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.
|
In[3]:=
| K = Knot["K11a87"];
|
In[4]:=
| Alexander[K][t]
|
KnotTheory::loading: Loading precomputed data in PD4Knots`.
|
Out[4]=
| −2t3 + 11t2−28t + 39−28t−1 + 11t−2−2t−3 |
In[5]:=
| Conway[K][z]
|
Out[5]=
| −2z6−z4−2z2 + 1 |
In[6]:=
| Alexander[K, 2][t]
|
KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.
|
Out[6]=
|
|
In[7]:=
| {KnotDet[K], KnotSignature[K]}
|
Out[7]=
| { 121, 0 } |
In[8]:=
| Jones[K][q]
|
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
|
Out[8]=
| q6−3q5 + 7q4−12q3 + 16q2−19q + 20−17q−1 + 13q−2−8q−3 + 4q−4−q−5 |
In[9]:=
| HOMFLYPT[K][a, z]
|
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
|
Out[9]=
| −z6a−2−z6 + 2a2z4−3z4a−2 + z4a−4−z4−a4z2 + 2a2z2−6z2a−2 + 2z2a−4 + z2−4a−2 + 2a−4 + 3 |
In[10]:=
| Kauffman[K][a, z]
|
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
|
Out[10]=
| z10a−2 + z10 + 4az9 + 8z9a−1 + 4z9a−3 + 7a2z8 + 12z8a−2 + 5z8a−4 + 14z8 + 7a3z7 + 6az7−8z7a−1−4z7a−3 + 3z7a−5 + 4a4z6−7a2z6−37z6a−2−13z6a−4 + z6a−6−34z6 + a5z5−11a3z5−22az5−8z5a−1−6z5a−3−8z5a−5−6a4z4−a2z4 + 41z4a−2 + 12z4a−4−3z4a−6 + 31z4−a5z3 + 5a3z3 + 16az3 + 12z3a−1 + 8z3a−3 + 6z3a−5 + 2a4z2 + a2z2−21z2a−2−7z2a−4 + 2z2a−6−13z2−a3z−3az−3za−1−3za−3−2za−5 + 4a−2 + 2a−4 + 3 |
[edit] "Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {}
Same Jones Polynomial (up to mirroring, 
):
{K11a28, K11a96,}
Computer Talk
The above data is available with the Mathematica package
KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.
(The path below may be different on your system, and possibly also the KnotTheory` date)
In[1]:=
| AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
|
Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
|
In[3]:=
| K = Knot["K11a87"];
|
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 + 11t2−28t + 39−28t−1 + 11t−2−2t−3, q6−3q5 + 7q4−12q3 + 16q2−19q + 20−17q−1 + 13q−2−8q−3 + 4q−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]=
| {} |
In[6]:=
| DeleteCases[
Select[
AllKnots[],
(J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) &
],
K
]
|
KnotTheory::loading: Loading precomputed data in Jones4Knots11`.
|
Out[6]=
| {K11a28, K11a96,} |
[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 K11a87. 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. |
|
