K11a22
From Knot Atlas
|
|
|
|
 (Knotscape image) | See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.
Visit K11a22's page at Knotilus! Visit K11a22's page at the original Knot Atlas! |
[edit] Knot presentations
| Planar diagram presentation | X4251 X8493 X12,5,13,6 X2837 X18,10,19,9 X14,11,15,12 X6,13,7,14 X20,16,21,15 X22,18,1,17 X10,20,11,19 X16,22,17,21 |
| Gauss code | 1, -4, 2, -1, 3, -7, 4, -2, 5, -10, 6, -3, 7, -6, 8, -11, 9, -5, 10, -8, 11, -9 |
| Dowker-Thistlethwaite code | 4 8 12 2 18 14 6 20 22 10 16 |
| A Braid Representative |
| ||||
| A Morse Link Presentation | 
|
[edit] Three dimensional invariants
|
[edit] Four dimensional invariants
|
[edit] Polynomial invariants
| Alexander polynomial | t4−5t3 + 13t2−20t + 23−20t−1 + 13t−2−5t−3 + t−4 |
| Conway polynomial | z8 + 3z6 + 3z4 + 3z2 + 1 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 101, 4 } |
| Jones polynomial | q10−3q9 + 6q8−11q7 + 14q6−16q5 + 16q4−13q3 + 11q2−6q + 3−q−1 |
| HOMFLY-PT polynomial (db, data sources) | z8a−4−z6a−2 + 6z6a−4−2z6a−6−4z4a−2 + 15z4a−4−9z4a−6 + z4a−8−5z2a−2 + 19z2a−4−14z2a−6 + 3z2a−8−2a−2 + 9a−4−8a−6 + 2a−8 |
| Kauffman polynomial (db, data sources) | z10a−4 + z10a−6 + 3z9a−3 + 7z9a−5 + 4z9a−7 + 3z8a−2 + 8z8a−4 + 12z8a−6 + 7z8a−8 + z7a−1−6z7a−3−11z7a−5 + 3z7a−7 + 7z7a−9−12z6a−2−38z6a−4−41z6a−6−10z6a−8 + 5z6a−10−4z5a−1−7z5a−3−17z5a−5−26z5a−7−9z5a−9 + 3z5a−11 + 16z4a−2 + 48z4a−4 + 43z4a−6 + 6z4a−8−4z4a−10 + z4a−12 + 5z3a−1 + 17z3a−3 + 32z3a−5 + 29z3a−7 + 6z3a−9−3z3a−11−9z2a−2−29z2a−4−25z2a−6−3z2a−8 + z2a−10−z2a−12−2za−1−7za−3−14za−5−11za−7−za−9 + za−11 + 2a−2 + 9a−4 + 8a−6 + 2a−8 |
| The A2 invariant | −q2 + 1−2q−2 + q−4 + 2q−6 + 6q−10−q−12 + 3q−14−q−16−3q−18−4q−22 + q−24 + q−30 |
| The G2 invariant | q12−2q10 + 6q8−11q6 + 14q4−16q2 + 5 + 17q−2−49q−4 + 82q−6−97q−8 + 72q−10−7q−12−91q−14 + 185q−16−233q−18 + 204q−20−95q−22−72q−24 + 227q−26−309q−28 + 285q−30−147q−32−33q−34 + 193q−36−259q−38 + 215q−40−72q−42−87q−44 + 211q−46−220q−48 + 128q−50 + 45q−52−215q−54 + 321q−56−304q−58 + 173q−60 + 35q−62−247q−64 + 384q−66−395q−68 + 272q−70−61q−72−166q−74 + 309q−76−336q−78 + 227q−80−51q−82−122q−84 + 211q−86−194q−88 + 76q−90 + 69q−92−183q−94 + 208q−96−141q−98 + 7q−100 + 127q−102−216q−104 + 231q−106−168q−108 + 63q−110 + 49q−112−134q−114 + 170q−116−158q−118 + 115q−120−46q−122−13q−124 + 59q−126−83q−128 + 81q−130−64q−132 + 40q−134−11q−136−11q−138 + 25q−140−30q−142 + 26q−144−18q−146 + 10q−148−2q−150−4q−152 + 5q−154−6q−156 + 4q−158−2q−160 + q−162 |
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["K11a22"];
|
In[4]:=
| Alexander[K][t]
|
KnotTheory::loading: Loading precomputed data in PD4Knots`.
|
Out[4]=
| t4−5t3 + 13t2−20t + 23−20t−1 + 13t−2−5t−3 + t−4 |
In[5]:=
| Conway[K][z]
|
Out[5]=
| z8 + 3z6 + 3z4 + 3z2 + 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]=
| { 101, 4 } |
In[8]:=
| Jones[K][q]
|
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
|
Out[8]=
| q10−3q9 + 6q8−11q7 + 14q6−16q5 + 16q4−13q3 + 11q2−6q + 3−q−1 |
In[9]:=
| HOMFLYPT[K][a, z]
|
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
|
Out[9]=
| z8a−4−z6a−2 + 6z6a−4−2z6a−6−4z4a−2 + 15z4a−4−9z4a−6 + z4a−8−5z2a−2 + 19z2a−4−14z2a−6 + 3z2a−8−2a−2 + 9a−4−8a−6 + 2a−8 |
In[10]:=
| Kauffman[K][a, z]
|
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
|
Out[10]=
| z10a−4 + z10a−6 + 3z9a−3 + 7z9a−5 + 4z9a−7 + 3z8a−2 + 8z8a−4 + 12z8a−6 + 7z8a−8 + z7a−1−6z7a−3−11z7a−5 + 3z7a−7 + 7z7a−9−12z6a−2−38z6a−4−41z6a−6−10z6a−8 + 5z6a−10−4z5a−1−7z5a−3−17z5a−5−26z5a−7−9z5a−9 + 3z5a−11 + 16z4a−2 + 48z4a−4 + 43z4a−6 + 6z4a−8−4z4a−10 + z4a−12 + 5z3a−1 + 17z3a−3 + 32z3a−5 + 29z3a−7 + 6z3a−9−3z3a−11−9z2a−2−29z2a−4−25z2a−6−3z2a−8 + z2a−10−z2a−12−2za−1−7za−3−14za−5−11za−7−za−9 + za−11 + 2a−2 + 9a−4 + 8a−6 + 2a−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["K11a22"];
|
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]=
| { t4−5t3 + 13t2−20t + 23−20t−1 + 13t−2−5t−3 + t−4, q10−3q9 + 6q8−11q7 + 14q6−16q5 + 16q4−13q3 + 11q2−6q + 3−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 = 4 is the signature of K11a22. 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. |
|
