K11a28
From Knot Atlas
|
|
|
|
 (Knotscape image) | See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.
Visit K11a28's page at Knotilus! Visit K11a28's page at the original Knot Atlas! |
[edit] Knot presentations
| Planar diagram presentation | X4251 X8394 X12,6,13,5 X16,7,17,8 X2,9,3,10 X18,11,19,12 X20,14,21,13 X22,16,1,15 X10,17,11,18 X6,19,7,20 X14,22,15,21 |
| Gauss code | 1, -5, 2, -1, 3, -10, 4, -2, 5, -9, 6, -3, 7, -11, 8, -4, 9, -6, 10, -7, 11, -8 |
| Dowker-Thistlethwaite code | 4 8 12 16 2 18 20 22 10 6 14 |
| A Braid Representative |
| ||||
| A Morse Link Presentation | 
|
[edit] Three dimensional invariants
|
[edit] Four dimensional invariants
|
[edit] Polynomial invariants
| Alexander polynomial | t4−6t3 + 15t2−24t + 29−24t−1 + 15t−2−6t−3 + t−4 |
| Conway polynomial | z8 + 2z6−z4−2z2 + 1 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 121, 0 } |
| Jones polynomial | −q5 + 4q4−8q3 + 13q2−17q + 20−19q−1 + 16q−2−12q−3 + 7q−4−3q−5 + q−6 |
| HOMFLY-PT polynomial (db, data sources) | z8−2a2z6−z6a−2 + 5z6 + a4z4−8a2z4−3z4a−2 + 9z4 + 3a4z2−10a2z2−2z2a−2 + 7z2 + 2a4−4a2 + 3 |
| Kauffman polynomial (db, data sources) | 2a2z10 + 2z10 + 5a3z9 + 11az9 + 6z9a−1 + 5a4z8 + 6a2z8 + 8z8a−2 + 9z8 + 3a5z7−11a3z7−28az7−7z7a−1 + 7z7a−3 + a6z6−13a4z6−28a2z6−13z6a−2 + 4z6a−4−31z6−8a5z5 + 9a3z5 + 31az5 + 2z5a−1−11z5a−3 + z5a−5−3a6z4 + 11a4z4 + 38a2z4 + 7z4a−2−6z4a−4 + 37z4 + 5a5z3−6a3z3−15az3 + 3z3a−3−z3a−5 + 2a6z2−7a4z2−22a2z2−2z2a−2 + z2a−4−16z2−a5z + a3z + 3az + za−1 + 2a4 + 4a2 + 3 |
| The A2 invariant | q18 + 2q12−3q10 + 2q8−2q6−2q4 + 3q2−3 + 5q−2−2q−4 + q−6 + 2q−8−2q−10 + 2q−12−q−14 |
| The G2 invariant | q94−2q92 + 5q90−9q88 + 11q86−12q84 + 5q82 + 11q80−33q78 + 61q76−81q74 + 79q72−47q70−21q68 + 120q66−215q64 + 274q62−251q60 + 121q58 + 88q56−314q54 + 476q52−483q50 + 327q48−38q46−283q44 + 499q42−523q40 + 337q38−25q36−273q34 + 422q32−359q30 + 115q28 + 202q26−448q24 + 502q22−341q20−3q18 + 378q16−653q14 + 716q12−528q10 + 163q8 + 264q6−606q4 + 735q2−612 + 288q−2 + 107q−4−413q−6 + 520q−8−381q−10 + 100q−12 + 206q−14−387q−16 + 362q−18−157q−20−144q−22 + 397q−24−489q−26 + 401q−28−161q−30−121q−32 + 341q−34−440q−36 + 397q−38−250q−40 + 59q−42 + 109q−44−215q−46 + 243q−48−203q−50 + 132q−52−44q−54−30q−56 + 72q−58−88q−60 + 73q−62−46q−64 + 21q−66 + 2q−68−13q−70 + 15q−72−13q−74 + 7q−76−3q−78 + q−80 |
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[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
|
Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.
|
In[3]:=
| K = Knot["K11a28"];
|
In[4]:=
| Alexander[K][t]
|
KnotTheory::loading: Loading precomputed data in PD4Knots`.
|
Out[4]=
| t4−6t3 + 15t2−24t + 29−24t−1 + 15t−2−6t−3 + t−4 |
In[5]:=
| Conway[K][z]
|
Out[5]=
| z8 + 2z6−z4−2z2 + 1 |
In[6]:=
| Alexander[K, 2][t]
|
KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.
|
Out[6]=
| {1} |
In[7]:=
| {KnotDet[K], KnotSignature[K]}
|
Out[7]=
| { 121, 0 } |
In[8]:=
| Jones[K][q]
|
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
|
Out[8]=
| −q5 + 4q4−8q3 + 13q2−17q + 20−19q−1 + 16q−2−12q−3 + 7q−4−3q−5 + q−6 |
In[9]:=
| HOMFLYPT[K][a, z]
|
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
|
Out[9]=
| z8−2a2z6−z6a−2 + 5z6 + a4z4−8a2z4−3z4a−2 + 9z4 + 3a4z2−10a2z2−2z2a−2 + 7z2 + 2a4−4a2 + 3 |
In[10]:=
| Kauffman[K][a, z]
|
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
|
Out[10]=
| 2a2z10 + 2z10 + 5a3z9 + 11az9 + 6z9a−1 + 5a4z8 + 6a2z8 + 8z8a−2 + 9z8 + 3a5z7−11a3z7−28az7−7z7a−1 + 7z7a−3 + a6z6−13a4z6−28a2z6−13z6a−2 + 4z6a−4−31z6−8a5z5 + 9a3z5 + 31az5 + 2z5a−1−11z5a−3 + z5a−5−3a6z4 + 11a4z4 + 38a2z4 + 7z4a−2−6z4a−4 + 37z4 + 5a5z3−6a3z3−15az3 + 3z3a−3−z3a−5 + 2a6z2−7a4z2−22a2z2−2z2a−2 + z2a−4−16z2−a5z + a3z + 3az + za−1 + 2a4 + 4a2 + 3 |
[edit] "Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {10_123,}
Same Jones Polynomial (up to mirroring, 
):
{K11a87, K11a96,}
Computer Talk
The above data is available with the Mathematica package
KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.
(The path below may be different on your system, and possibly also the KnotTheory` date)
In[1]:=
| AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
|
Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
|
In[3]:=
| K = Knot["K11a28"];
|
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]=
| { t4−6t3 + 15t2−24t + 29−24t−1 + 15t−2−6t−3 + t−4, −q5 + 4q4−8q3 + 13q2−17q + 20−19q−1 + 16q−2−12q−3 + 7q−4−3q−5 + q−6 } |
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_123,} |
In[6]:=
| DeleteCases[
Select[
AllKnots[],
(J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) &
],
K
]
|
KnotTheory::loading: Loading precomputed data in Jones4Knots11`.
|
Out[6]=
| {K11a87, 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 K11a28. 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. |
|
