K11a322
From Knot Atlas
Jump to navigationJump to search
|
|
|
 (Knotscape image) |
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. |
Knot presentations
| Planar diagram presentation | X6271 X12,4,13,3 X16,5,17,6 X22,8,1,7 X20,9,21,10 X4,12,5,11 X18,13,19,14 X8,15,9,16 X2,17,3,18 X10,19,11,20 X14,21,15,22 |
| Gauss code | 1, -9, 2, -6, 3, -1, 4, -8, 5, -10, 6, -2, 7, -11, 8, -3, 9, -7, 10, -5, 11, -4 |
| Dowker-Thistlethwaite code | 6 12 16 22 20 4 18 8 2 10 14 |
| A Braid Representative |
| ||||||
| A Morse Link Presentation | 
|
Three dimensional invariants
|
Four dimensional invariants
|
Polynomial invariants
| Alexander polynomial | [math]\displaystyle{ 2 t^3-13 t^2+36 t-49+36 t^{-1} -13 t^{-2} +2 t^{-3} }[/math] |
| Conway polynomial | [math]\displaystyle{ 2 z^6-z^4+2 z^2+1 }[/math] |
| 2nd Alexander ideal (db, data sources) | [math]\displaystyle{ \left\{2,t^2+t+1\right\} }[/math] |
| Determinant and Signature | { 151, -2 } |
| Jones polynomial | [math]\displaystyle{ q^3-4 q^2+9 q-15+21 q^{-1} -24 q^{-2} +25 q^{-3} -21 q^{-4} +16 q^{-5} -10 q^{-6} +4 q^{-7} - q^{-8} }[/math] |
| HOMFLY-PT polynomial (db, data sources) | [math]\displaystyle{ -z^4 a^6-z^2 a^6-2 a^6+z^6 a^4+2 z^4 a^4+5 z^2 a^4+4 a^4+z^6 a^2-2 z^2 a^2-2 a^2-2 z^4-z^2+1+z^2 a^{-2} }[/math] |
| Kauffman polynomial (db, data sources) | [math]\displaystyle{ 3 a^4 z^{10}+3 a^2 z^{10}+9 a^5 z^9+16 a^3 z^9+7 a z^9+12 a^6 z^8+14 a^4 z^8+9 a^2 z^8+7 z^8+9 a^7 z^7-9 a^5 z^7-33 a^3 z^7-11 a z^7+4 z^7 a^{-1} +4 a^8 z^6-22 a^6 z^6-42 a^4 z^6-32 a^2 z^6+z^6 a^{-2} -15 z^6+a^9 z^5-14 a^7 z^5-3 a^5 z^5+24 a^3 z^5+3 a z^5-9 z^5 a^{-1} -4 a^8 z^4+18 a^6 z^4+40 a^4 z^4+29 a^2 z^4-2 z^4 a^{-2} +9 z^4-a^9 z^3+8 a^7 z^3+8 a^5 z^3-8 a^3 z^3-2 a z^3+5 z^3 a^{-1} -8 a^6 z^2-17 a^4 z^2-13 a^2 z^2+z^2 a^{-2} -3 z^2-3 a^7 z-3 a^5 z+a^3 z+a z+2 a^6+4 a^4+2 a^2+1 }[/math] |
| The A2 invariant | Data:K11a322/QuantumInvariant/A2/1,0 |
| The G2 invariant | Data:K11a322/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["K11a322"];
|
In[4]:=
|
Alexander[K][t]
|
KnotTheory::loading: Loading precomputed data in PD4Knots`.
|
Out[4]=
|
[math]\displaystyle{ 2 t^3-13 t^2+36 t-49+36 t^{-1} -13 t^{-2} +2 t^{-3} }[/math] |
In[5]:=
|
Conway[K][z]
|
Out[5]=
|
[math]\displaystyle{ 2 z^6-z^4+2 z^2+1 }[/math] |
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]=
|
[math]\displaystyle{ \left\{2,t^2+t+1\right\} }[/math] |
In[7]:=
|
{KnotDet[K], KnotSignature[K]}
|
Out[7]=
|
{ 151, -2 } |
In[8]:=
|
Jones[K][q]
|
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
|
Out[8]=
|
[math]\displaystyle{ q^3-4 q^2+9 q-15+21 q^{-1} -24 q^{-2} +25 q^{-3} -21 q^{-4} +16 q^{-5} -10 q^{-6} +4 q^{-7} - q^{-8} }[/math] |
In[9]:=
|
HOMFLYPT[K][a, z]
|
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
|
Out[9]=
|
[math]\displaystyle{ -z^4 a^6-z^2 a^6-2 a^6+z^6 a^4+2 z^4 a^4+5 z^2 a^4+4 a^4+z^6 a^2-2 z^2 a^2-2 a^2-2 z^4-z^2+1+z^2 a^{-2} }[/math] |
In[10]:=
|
Kauffman[K][a, z]
|
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
|
Out[10]=
|
[math]\displaystyle{ 3 a^4 z^{10}+3 a^2 z^{10}+9 a^5 z^9+16 a^3 z^9+7 a z^9+12 a^6 z^8+14 a^4 z^8+9 a^2 z^8+7 z^8+9 a^7 z^7-9 a^5 z^7-33 a^3 z^7-11 a z^7+4 z^7 a^{-1} +4 a^8 z^6-22 a^6 z^6-42 a^4 z^6-32 a^2 z^6+z^6 a^{-2} -15 z^6+a^9 z^5-14 a^7 z^5-3 a^5 z^5+24 a^3 z^5+3 a z^5-9 z^5 a^{-1} -4 a^8 z^4+18 a^6 z^4+40 a^4 z^4+29 a^2 z^4-2 z^4 a^{-2} +9 z^4-a^9 z^3+8 a^7 z^3+8 a^5 z^3-8 a^3 z^3-2 a z^3+5 z^3 a^{-1} -8 a^6 z^2-17 a^4 z^2-13 a^2 z^2+z^2 a^{-2} -3 z^2-3 a^7 z-3 a^5 z+a^3 z+a z+2 a^6+4 a^4+2 a^2+1 }[/math] |
"Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {}
Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {K11a147,}
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["K11a322"];
|
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]=
|
{ [math]\displaystyle{ 2 t^3-13 t^2+36 t-49+36 t^{-1} -13 t^{-2} +2 t^{-3} }[/math], [math]\displaystyle{ q^3-4 q^2+9 q-15+21 q^{-1} -24 q^{-2} +25 q^{-3} -21 q^{-4} +16 q^{-5} -10 q^{-6} +4 q^{-7} - q^{-8} }[/math] } |
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]=
|
{K11a147,} |
Vassiliev invariants
| V2 and V3: | (2, -4) |
| V2,1 through V6,9: |
|
V2,1 through V6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V2 and V3.
Khovanov Homology
| The coefficients of the monomials [math]\displaystyle{ t^rq^j }[/math] are shown, along with their alternating sums [math]\displaystyle{ \chi }[/math] (fixed [math]\displaystyle{ j }[/math], alternation over [math]\displaystyle{ r }[/math]). The squares with yellow highlighting are those on the "critical diagonals", where [math]\displaystyle{ j-2r=s+1 }[/math] or [math]\displaystyle{ j-2r=s-1 }[/math], where [math]\displaystyle{ s= }[/math]-2 is the signature of K11a322. Nonzero entries off the critical diagonals (if any exist) are highlighted in red. |
|
| Integral Khovanov Homology
(db, data source) |
|
Computer Talk
Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.
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. |
|
