K11a347
From Knot Atlas
|
|
|
|
 (Knotscape image) | See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.
Visit K11a347's page at Knotilus! Visit K11a347's page at the original Knot Atlas! |
[edit] Knot presentations
| Planar diagram presentation | X6271 X16,3,17,4 X12,6,13,5 X20,8,21,7 X18,10,19,9 X4,12,5,11 X22,13,1,14 X10,16,11,15 X2,17,3,18 X8,20,9,19 X14,21,15,22 |
| Gauss code | 1, -9, 2, -6, 3, -1, 4, -10, 5, -8, 6, -3, 7, -11, 8, -2, 9, -5, 10, -4, 11, -7 |
| Dowker-Thistlethwaite code | 6 16 12 20 18 4 22 10 2 8 14 |
| A Braid Representative |
| ||||||
| A Morse Link Presentation | 
|
[edit] Three dimensional invariants
|
[edit] Four dimensional invariants
|
[edit] Polynomial invariants
| Alexander polynomial | 2t3−11t2 + 26t−33 + 26t−1−11t−2 + 2t−3 |
| Conway polynomial | 2z6 + z4 + 1 |
| 2nd Alexander ideal (db, data sources) |  |
| Determinant and Signature | { 111, 2 } |
| Jones polynomial | −q8 + 3q7−7q6 + 12q5−15q4 + 18q3−18q2 + 15q−11 + 7q−1−3q−2 + q−3 |
| HOMFLY-PT polynomial (db, data sources) | z6a−2 + z6a−4 + z4a−2 + 3z4a−4−z4a−6−2z4 + a2z2−2z2a−2 + 6z2a−4−2z2a−6−3z2 + a2−3a−2 + 5a−4−2a−6 |
| Kauffman polynomial (db, data sources) | 2z10a−2 + 2z10a−4 + 4z9a−1 + 10z9a−3 + 6z9a−5 + 2z8a−2 + 6z8a−4 + 8z8a−6 + 4z8 + 3az7−7z7a−1−29z7a−3−13z7a−5 + 6z7a−7 + a2z6−11z6a−2−28z6a−4−22z6a−6 + 3z6a−8−7z6−8az5 + 5z5a−1 + 40z5a−3 + 13z5a−5−13z5a−7 + z5a−9−3a2z4 + 13z4a−2 + 45z4a−4 + 28z4a−6−5z4a−8−2z4 + 5az3−9z3a−1−26z3a−3−z3a−5 + 9z3a−7−2z3a−9 + 3a2z2−12z2a−2−27z2a−4−14z2a−6 + 4z2 + 4za−1 + 6za−3−2za−7−a2 + 3a−2 + 5a−4 + 2a−6 |
| The A2 invariant | Data:K11a347/QuantumInvariant/A2/1,0 |
| The G2 invariant | Data:K11a347/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[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
|
Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.
|
In[3]:=
| K = Knot["K11a347"];
|
In[4]:=
| Alexander[K][t]
|
KnotTheory::loading: Loading precomputed data in PD4Knots`.
|
Out[4]=
| 2t3−11t2 + 26t−33 + 26t−1−11t−2 + 2t−3 |
In[5]:=
| Conway[K][z]
|
Out[5]=
| 2z6 + z4 + 1 |
In[6]:=
| Alexander[K, 2][t]
|
KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.
|
Out[6]=
|
|
In[7]:=
| {KnotDet[K], KnotSignature[K]}
|
Out[7]=
| { 111, 2 } |
In[8]:=
| Jones[K][q]
|
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
|
Out[8]=
| −q8 + 3q7−7q6 + 12q5−15q4 + 18q3−18q2 + 15q−11 + 7q−1−3q−2 + q−3 |
In[9]:=
| HOMFLYPT[K][a, z]
|
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
|
Out[9]=
| z6a−2 + z6a−4 + z4a−2 + 3z4a−4−z4a−6−2z4 + a2z2−2z2a−2 + 6z2a−4−2z2a−6−3z2 + a2−3a−2 + 5a−4−2a−6 |
In[10]:=
| Kauffman[K][a, z]
|
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
|
Out[10]=
| 2z10a−2 + 2z10a−4 + 4z9a−1 + 10z9a−3 + 6z9a−5 + 2z8a−2 + 6z8a−4 + 8z8a−6 + 4z8 + 3az7−7z7a−1−29z7a−3−13z7a−5 + 6z7a−7 + a2z6−11z6a−2−28z6a−4−22z6a−6 + 3z6a−8−7z6−8az5 + 5z5a−1 + 40z5a−3 + 13z5a−5−13z5a−7 + z5a−9−3a2z4 + 13z4a−2 + 45z4a−4 + 28z4a−6−5z4a−8−2z4 + 5az3−9z3a−1−26z3a−3−z3a−5 + 9z3a−7−2z3a−9 + 3a2z2−12z2a−2−27z2a−4−14z2a−6 + 4z2 + 4za−1 + 6za−3−2za−7−a2 + 3a−2 + 5a−4 + 2a−6 |
[edit] "Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {10_113, K11a107,}
Same Jones Polynomial (up to mirroring, 
):
{K11a159,}
Computer Talk
The above data is available with the Mathematica package
KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.
(The path below may be different on your system, and possibly also the KnotTheory` date)
In[1]:=
| AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
|
Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
|
In[3]:=
| K = Knot["K11a347"];
|
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 + 26t−33 + 26t−1−11t−2 + 2t−3, −q8 + 3q7−7q6 + 12q5−15q4 + 18q3−18q2 + 15q−11 + 7q−1−3q−2 + q−3 } |
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]=
| {10_113, K11a107,} |
In[6]:=
| DeleteCases[
Select[
AllKnots[],
(J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) &
],
K
]
|
KnotTheory::loading: Loading precomputed data in Jones4Knots11`.
|
Out[6]=
| {K11a159,} |
[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 K11a347. 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. |
|
