K11a127
From Knot Atlas
|
|
|
|
 (Knotscape image) | See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.
Visit K11a127's page at Knotilus! Visit K11a127's page at the original Knot Atlas! |
[edit] Knot presentations
| Planar diagram presentation | X4251 X10,4,11,3 X14,5,15,6 X20,8,21,7 X2,10,3,9 X22,11,1,12 X18,14,19,13 X8,15,9,16 X12,18,13,17 X6,20,7,19 X16,22,17,21 |
| Gauss code | 1, -5, 2, -1, 3, -10, 4, -8, 5, -2, 6, -9, 7, -3, 8, -11, 9, -7, 10, -4, 11, -6 |
| Dowker-Thistlethwaite code | 4 10 14 20 2 22 18 8 12 6 16 |
| A Braid Representative |
| ||||
| A Morse Link Presentation | 
|
[edit] Three dimensional invariants
|
[edit] Four dimensional invariants
|
[edit] Polynomial invariants
| Alexander polynomial | t4−6t3 + 17t2−28t + 33−28t−1 + 17t−2−6t−3 + t−4 |
| Conway polynomial | z8 + 2z6 + z4 + 2z2 + 1 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 137, 4 } |
| Jones polynomial | q10−4q9 + 9q8−15q7 + 19q6−22q5 + 22q4−18q3 + 14q2−8q + 4−q−1 |
| HOMFLY-PT polynomial (db, data sources) | z8a−4−z6a−2 + 5z6a−4−2z6a−6−3z4a−2 + 10z4a−4−7z4a−6 + z4a−8−2z2a−2 + 10z2a−4−8z2a−6 + 2z2a−8 + 4a−4−4a−6 + a−8 |
| Kauffman polynomial (db, data sources) | 2z10a−4 + 2z10a−6 + 5z9a−3 + 13z9a−5 + 8z9a−7 + 4z8a−2 + 10z8a−4 + 20z8a−6 + 14z8a−8 + z7a−1−12z7a−3−25z7a−5 + 2z7a−7 + 14z7a−9−14z6a−2−48z6a−4−64z6a−6−21z6a−8 + 9z6a−10−3z5a−1 + z5a−3−8z5a−5−36z5a−7−20z5a−9 + 4z5a−11 + 16z4a−2 + 53z4a−4 + 54z4a−6 + 9z4a−8−7z4a−10 + z4a−12 + 3z3a−1 + 11z3a−3 + 27z3a−5 + 32z3a−7 + 12z3a−9−z3a−11−6z2a−2−21z2a−4−20z2a−6−3z2a−8 + 2z2a−10−za−1−4za−3−10za−5−10za−7−3za−9 + 4a−4 + 4a−6 + a−8 |
| The A2 invariant | −q2 + 2−2q−2 + 2q−4 + 2q−6−2q−8 + 6q−10−3q−12 + 3q−14−q−16−3q−18 + 2q−20−4q−22 + 2q−24−q−28 + q−30 |
| The G2 invariant | q12−3q10 + 9q8−19q6 + 28q4−33q2 + 17 + 26q−2−91q−4 + 164q−6−204q−8 + 168q−10−41q−12−167q−14 + 393q−16−528q−18 + 497q−20−263q−22−119q−24 + 510q−26−760q−28 + 752q−30−457q−32−4q−34 + 460q−36−713q−38 + 662q−40−333q−42−116q−44 + 492q−46−624q−48 + 450q−50−37q−52−434q−54 + 775q−56−811q−58 + 524q−60 + q−62−577q−64 + 976q−66−1060q−68 + 779q−70−227q−72−388q−74 + 840q−76−967q−78 + 728q−80−250q−82−270q−84 + 594q−86−624q−88 + 356q−90 + 62q−92−428q−94 + 588q−96−465q−98 + 124q−100 + 274q−102−578q−104 + 662q−106−511q−108 + 211q−110 + 135q−112−399q−114 + 517q−116−473q−118 + 312q−120−101q−122−88q−124 + 207q−126−253q−128 + 225q−130−153q−132 + 72q−134 + 6q−136−54q−138 + 72q−140−73q−142 + 54q−144−31q−146 + 12q−148 + 4q−150−10q−152 + 11q−154−10q−156 + 6q−158−3q−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["K11a127"];
|
In[4]:=
| Alexander[K][t]
|
KnotTheory::loading: Loading precomputed data in PD4Knots`.
|
Out[4]=
| t4−6t3 + 17t2−28t + 33−28t−1 + 17t−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]=
| { 137, 4 } |
In[8]:=
| Jones[K][q]
|
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
|
Out[8]=
| q10−4q9 + 9q8−15q7 + 19q6−22q5 + 22q4−18q3 + 14q2−8q + 4−q−1 |
In[9]:=
| HOMFLYPT[K][a, z]
|
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
|
Out[9]=
| z8a−4−z6a−2 + 5z6a−4−2z6a−6−3z4a−2 + 10z4a−4−7z4a−6 + z4a−8−2z2a−2 + 10z2a−4−8z2a−6 + 2z2a−8 + 4a−4−4a−6 + a−8 |
In[10]:=
| Kauffman[K][a, z]
|
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
|
Out[10]=
| 2z10a−4 + 2z10a−6 + 5z9a−3 + 13z9a−5 + 8z9a−7 + 4z8a−2 + 10z8a−4 + 20z8a−6 + 14z8a−8 + z7a−1−12z7a−3−25z7a−5 + 2z7a−7 + 14z7a−9−14z6a−2−48z6a−4−64z6a−6−21z6a−8 + 9z6a−10−3z5a−1 + z5a−3−8z5a−5−36z5a−7−20z5a−9 + 4z5a−11 + 16z4a−2 + 53z4a−4 + 54z4a−6 + 9z4a−8−7z4a−10 + z4a−12 + 3z3a−1 + 11z3a−3 + 27z3a−5 + 32z3a−7 + 12z3a−9−z3a−11−6z2a−2−21z2a−4−20z2a−6−3z2a−8 + 2z2a−10−za−1−4za−3−10za−5−10za−7−3za−9 + 4a−4 + 4a−6 + a−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["K11a127"];
|
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 + 17t2−28t + 33−28t−1 + 17t−2−6t−3 + t−4, q10−4q9 + 9q8−15q7 + 19q6−22q5 + 22q4−18q3 + 14q2−8q + 4−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 K11a127. 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. |
|
