K11n31
From Knot Atlas
|
|
|
|
 (Knotscape image) | See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.
Visit K11n31's page at Knotilus! Visit K11n31's page at the original Knot Atlas! |
[edit] Knot presentations
| Planar diagram presentation | X4251 X8493 X12,6,13,5 X2837 X9,17,10,16 X6,12,7,11 X13,20,14,21 X15,11,16,10 X17,1,18,22 X19,14,20,15 X21,19,22,18 |
| Gauss code | 1, -4, 2, -1, 3, -6, 4, -2, -5, 8, 6, -3, -7, 10, -8, 5, -9, 11, -10, 7, -11, 9 |
| Dowker-Thistlethwaite code | 4 8 12 2 -16 6 -20 -10 -22 -14 -18 |
| A Braid Representative |
| ||||||
| A Morse Link Presentation | 
|
[edit] Three dimensional invariants
|
[edit] Four dimensional invariants
|
[edit] Polynomial invariants
| Alexander polynomial | −t3 + 2t2 + 2t−5 + 2t−1 + 2t−2−t−3 |
| Conway polynomial | −z6−4z4 + z2 + 1 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 3, 2 } |
| Jones polynomial | −q10 + 2q9−2q8 + 2q7−q6−q3 + 2q2−q + 1 |
| HOMFLY-PT polynomial (db, data sources) | −z6a−4 + z4a−2−5z4a−4 + 4z2a−2−6z2a−4 + z2a−6 + 2z2a−8 + 3a−2−3a−4 + 2a−8−a−10 |
| Kauffman polynomial (db, data sources) | z9a−7 + z9a−9 + z8a−6 + 3z8a−8 + 2z8a−10 + z7a−3−6z7a−7−4z7a−9 + z7a−11 + z6a−2 + 2z6a−4−7z6a−6−19z6a−8−11z6a−10−4z5a−3 + 8z5a−7−z5a−9−5z5a−11−5z4a−2−10z4a−4 + 11z4a−6 + 32z4a−8 + 16z4a−10 + 2z3a−3−4z3a−5−4z3a−7 + 8z3a−9 + 6z3a−11 + 7z2a−2 + 10z2a−4−6z2a−6−17z2a−8−8z2a−10 + za−3 + 3za−5 + 2za−7−2za−9−2za−11−3a−2−3a−4 + 2a−8 + a−10 |
| The A2 invariant | Data:K11n31/QuantumInvariant/A2/1,0 |
| The G2 invariant | Data:K11n31/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[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
|
Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.
|
In[3]:=
| K = Knot["K11n31"];
|
In[4]:=
| Alexander[K][t]
|
KnotTheory::loading: Loading precomputed data in PD4Knots`.
|
Out[4]=
| −t3 + 2t2 + 2t−5 + 2t−1 + 2t−2−t−3 |
In[5]:=
| Conway[K][z]
|
Out[5]=
| −z6−4z4 + z2 + 1 |
In[6]:=
| Alexander[K, 2][t]
|
KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.
|
Out[6]=
| {1} |
In[7]:=
| {KnotDet[K], KnotSignature[K]}
|
Out[7]=
| { 3, 2 } |
In[8]:=
| Jones[K][q]
|
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
|
Out[8]=
| −q10 + 2q9−2q8 + 2q7−q6−q3 + 2q2−q + 1 |
In[9]:=
| HOMFLYPT[K][a, z]
|
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
|
Out[9]=
| −z6a−4 + z4a−2−5z4a−4 + 4z2a−2−6z2a−4 + z2a−6 + 2z2a−8 + 3a−2−3a−4 + 2a−8−a−10 |
In[10]:=
| Kauffman[K][a, z]
|
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
|
Out[10]=
| z9a−7 + z9a−9 + z8a−6 + 3z8a−8 + 2z8a−10 + z7a−3−6z7a−7−4z7a−9 + z7a−11 + z6a−2 + 2z6a−4−7z6a−6−19z6a−8−11z6a−10−4z5a−3 + 8z5a−7−z5a−9−5z5a−11−5z4a−2−10z4a−4 + 11z4a−6 + 32z4a−8 + 16z4a−10 + 2z3a−3−4z3a−5−4z3a−7 + 8z3a−9 + 6z3a−11 + 7z2a−2 + 10z2a−4−6z2a−6−17z2a−8−8z2a−10 + za−3 + 3za−5 + 2za−7−2za−9−2za−11−3a−2−3a−4 + 2a−8 + a−10 |
[edit] "Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {}
Same Jones Polynomial (up to mirroring, 
):
{}
Computer Talk
The above data is available with the Mathematica package
KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.
(The path below may be different on your system, and possibly also the KnotTheory` date)
In[1]:=
| AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
|
Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
|
In[3]:=
| K = Knot["K11n31"];
|
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]=
| { −t3 + 2t2 + 2t−5 + 2t−1 + 2t−2−t−3, −q10 + 2q9−2q8 + 2q7−q6−q3 + 2q2−q + 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 = 2 is the signature of K11n31. 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. |
|
