K11a355
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 | X8291 X14,4,15,3 X16,6,17,5 X18,8,19,7 X20,10,21,9 X22,12,1,11 X2,14,3,13 X4,16,5,15 X6,18,7,17 X12,20,13,19 X10,22,11,21 |
| Gauss code | 1, -7, 2, -8, 3, -9, 4, -1, 5, -11, 6, -10, 7, -2, 8, -3, 9, -4, 10, -5, 11, -6 |
| Dowker-Thistlethwaite code | 8 14 16 18 20 22 2 4 6 12 10 |
| A Braid Representative |
| ||||
| A Morse Link Presentation | 
|
Three dimensional invariants
|
Four dimensional invariants
|
Polynomial invariants
| Alexander polynomial | [math]\displaystyle{ 2 t^4-4 t^3+6 t^2-7 t+7-7 t^{-1} +6 t^{-2} -4 t^{-3} +2 t^{-4} }[/math] |
| Conway polynomial | [math]\displaystyle{ 2 z^8+12 z^6+22 z^4+13 z^2+1 }[/math] |
| 2nd Alexander ideal (db, data sources) | [math]\displaystyle{ \{1\} }[/math] |
| Determinant and Signature | { 45, 8 } |
| Jones polynomial | [math]\displaystyle{ -q^{15}+2 q^{14}-4 q^{13}+5 q^{12}-6 q^{11}+7 q^{10}-6 q^9+5 q^8-4 q^7+3 q^6-q^5+q^4 }[/math] |
| HOMFLY-PT polynomial (db, data sources) | [math]\displaystyle{ z^8 a^{-8} +z^8 a^{-10} +7 z^6 a^{-8} +6 z^6 a^{-10} -z^6 a^{-12} +16 z^4 a^{-8} +11 z^4 a^{-10} -5 z^4 a^{-12} +13 z^2 a^{-8} +7 z^2 a^{-10} -7 z^2 a^{-12} +2 a^{-8} +2 a^{-10} -3 a^{-12} }[/math] |
| Kauffman polynomial (db, data sources) | [math]\displaystyle{ z^{10} a^{-10} +z^{10} a^{-12} +z^9 a^{-9} +3 z^9 a^{-11} +2 z^9 a^{-13} +z^8 a^{-8} -5 z^8 a^{-10} -3 z^8 a^{-12} +3 z^8 a^{-14} -5 z^7 a^{-9} -14 z^7 a^{-11} -6 z^7 a^{-13} +3 z^7 a^{-15} -7 z^6 a^{-8} +8 z^6 a^{-10} +3 z^6 a^{-12} -9 z^6 a^{-14} +3 z^6 a^{-16} +6 z^5 a^{-9} +18 z^5 a^{-11} +4 z^5 a^{-13} -5 z^5 a^{-15} +3 z^5 a^{-17} +16 z^4 a^{-8} -8 z^4 a^{-10} -9 z^4 a^{-12} +10 z^4 a^{-14} -3 z^4 a^{-16} +2 z^4 a^{-18} -8 z^3 a^{-11} -2 z^3 a^{-13} +2 z^3 a^{-15} -3 z^3 a^{-17} +z^3 a^{-19} -13 z^2 a^{-8} +7 z^2 a^{-10} +11 z^2 a^{-12} -7 z^2 a^{-14} +z^2 a^{-16} -z^2 a^{-18} -z a^{-9} +3 z a^{-11} -z a^{-15} +2 z a^{-17} -z a^{-19} +2 a^{-8} -2 a^{-10} -3 a^{-12} }[/math] |
| The A2 invariant | [math]\displaystyle{ q^{-14} +2 q^{-18} + q^{-22} +2 q^{-28} +2 q^{-32} - q^{-34} - q^{-36} - q^{-38} - q^{-40} - q^{-44} }[/math] |
| The G2 invariant | Data:K11a355/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["K11a355"];
|
In[4]:=
|
Alexander[K][t]
|
KnotTheory::loading: Loading precomputed data in PD4Knots`.
|
Out[4]=
|
[math]\displaystyle{ 2 t^4-4 t^3+6 t^2-7 t+7-7 t^{-1} +6 t^{-2} -4 t^{-3} +2 t^{-4} }[/math] |
In[5]:=
|
Conway[K][z]
|
Out[5]=
|
[math]\displaystyle{ 2 z^8+12 z^6+22 z^4+13 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{ \{1\} }[/math] |
In[7]:=
|
{KnotDet[K], KnotSignature[K]}
|
Out[7]=
|
{ 45, 8 } |
In[8]:=
|
Jones[K][q]
|
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
|
Out[8]=
|
[math]\displaystyle{ -q^{15}+2 q^{14}-4 q^{13}+5 q^{12}-6 q^{11}+7 q^{10}-6 q^9+5 q^8-4 q^7+3 q^6-q^5+q^4 }[/math] |
In[9]:=
|
HOMFLYPT[K][a, z]
|
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
|
Out[9]=
|
[math]\displaystyle{ z^8 a^{-8} +z^8 a^{-10} +7 z^6 a^{-8} +6 z^6 a^{-10} -z^6 a^{-12} +16 z^4 a^{-8} +11 z^4 a^{-10} -5 z^4 a^{-12} +13 z^2 a^{-8} +7 z^2 a^{-10} -7 z^2 a^{-12} +2 a^{-8} +2 a^{-10} -3 a^{-12} }[/math] |
In[10]:=
|
Kauffman[K][a, z]
|
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
|
Out[10]=
|
[math]\displaystyle{ z^{10} a^{-10} +z^{10} a^{-12} +z^9 a^{-9} +3 z^9 a^{-11} +2 z^9 a^{-13} +z^8 a^{-8} -5 z^8 a^{-10} -3 z^8 a^{-12} +3 z^8 a^{-14} -5 z^7 a^{-9} -14 z^7 a^{-11} -6 z^7 a^{-13} +3 z^7 a^{-15} -7 z^6 a^{-8} +8 z^6 a^{-10} +3 z^6 a^{-12} -9 z^6 a^{-14} +3 z^6 a^{-16} +6 z^5 a^{-9} +18 z^5 a^{-11} +4 z^5 a^{-13} -5 z^5 a^{-15} +3 z^5 a^{-17} +16 z^4 a^{-8} -8 z^4 a^{-10} -9 z^4 a^{-12} +10 z^4 a^{-14} -3 z^4 a^{-16} +2 z^4 a^{-18} -8 z^3 a^{-11} -2 z^3 a^{-13} +2 z^3 a^{-15} -3 z^3 a^{-17} +z^3 a^{-19} -13 z^2 a^{-8} +7 z^2 a^{-10} +11 z^2 a^{-12} -7 z^2 a^{-14} +z^2 a^{-16} -z^2 a^{-18} -z a^{-9} +3 z a^{-11} -z a^{-15} +2 z a^{-17} -z a^{-19} +2 a^{-8} -2 a^{-10} -3 a^{-12} }[/math] |
"Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {}
Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {}
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["K11a355"];
|
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^4-4 t^3+6 t^2-7 t+7-7 t^{-1} +6 t^{-2} -4 t^{-3} +2 t^{-4} }[/math], [math]\displaystyle{ -q^{15}+2 q^{14}-4 q^{13}+5 q^{12}-6 q^{11}+7 q^{10}-6 q^9+5 q^8-4 q^7+3 q^6-q^5+q^4 }[/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]=
|
{} |
Vassiliev invariants
| V2 and V3: | (13, 45) |
| 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]8 is the signature of K11a355. 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. |
|
