K11a32
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 | X4251 X8493 X14,6,15,5 X2837 X16,10,17,9 X18,11,19,12 X6,14,7,13 X22,16,1,15 X20,17,21,18 X12,19,13,20 X10,22,11,21 |
| Gauss code | 1, -4, 2, -1, 3, -7, 4, -2, 5, -11, 6, -10, 7, -3, 8, -5, 9, -6, 10, -9, 11, -8 |
| Dowker-Thistlethwaite code | 4 8 14 2 16 18 6 22 20 12 10 |
| A Braid Representative |
| ||||||
| A Morse Link Presentation | 
|
Three dimensional invariants
|
Four dimensional invariants
|
Polynomial invariants
| Alexander polynomial | [math]\displaystyle{ 3 t^3-14 t^2+32 t-41+32 t^{-1} -14 t^{-2} +3 t^{-3} }[/math] |
| Conway polynomial | [math]\displaystyle{ 3 z^6+4 z^4+3 z^2+1 }[/math] |
| 2nd Alexander ideal (db, data sources) | [math]\displaystyle{ \{1\} }[/math] |
| Determinant and Signature | { 139, 2 } |
| Jones polynomial | [math]\displaystyle{ q^9-4 q^8+8 q^7-14 q^6+19 q^5-22 q^4+23 q^3-19 q^2+15 q-9+4 q^{-1} - q^{-2} }[/math] |
| HOMFLY-PT polynomial (db, data sources) | [math]\displaystyle{ z^6 a^{-2} +2 z^6 a^{-4} +z^4 a^{-2} +7 z^4 a^{-4} -3 z^4 a^{-6} -z^4-z^2 a^{-2} +11 z^2 a^{-4} -7 z^2 a^{-6} +z^2 a^{-8} -z^2- a^{-2} +6 a^{-4} -5 a^{-6} + a^{-8} }[/math] |
| Kauffman polynomial (db, data sources) | [math]\displaystyle{ z^{10} a^{-4} +z^{10} a^{-6} +5 z^9 a^{-3} +9 z^9 a^{-5} +4 z^9 a^{-7} +9 z^8 a^{-2} +20 z^8 a^{-4} +17 z^8 a^{-6} +6 z^8 a^{-8} +8 z^7 a^{-1} +9 z^7 a^{-3} +2 z^7 a^{-5} +5 z^7 a^{-7} +4 z^7 a^{-9} -11 z^6 a^{-2} -45 z^6 a^{-4} -43 z^6 a^{-6} -12 z^6 a^{-8} +z^6 a^{-10} +4 z^6+a z^5-12 z^5 a^{-1} -32 z^5 a^{-3} -40 z^5 a^{-5} -31 z^5 a^{-7} -10 z^5 a^{-9} +5 z^4 a^{-2} +37 z^4 a^{-4} +35 z^4 a^{-6} +6 z^4 a^{-8} -2 z^4 a^{-10} -5 z^4-a z^3+7 z^3 a^{-1} +26 z^3 a^{-3} +41 z^3 a^{-5} +31 z^3 a^{-7} +8 z^3 a^{-9} -3 z^2 a^{-2} -19 z^2 a^{-4} -16 z^2 a^{-6} -z^2 a^{-8} +z^2 a^{-10} +2 z^2-2 z a^{-1} -7 z a^{-3} -13 z a^{-5} -10 z a^{-7} -2 z a^{-9} + a^{-2} +6 a^{-4} +5 a^{-6} + a^{-8} }[/math] |
| The A2 invariant | Data:K11a32/QuantumInvariant/A2/1,0 |
| The G2 invariant | Data:K11a32/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["K11a32"];
|
In[4]:=
|
Alexander[K][t]
|
KnotTheory::loading: Loading precomputed data in PD4Knots`.
|
Out[4]=
|
[math]\displaystyle{ 3 t^3-14 t^2+32 t-41+32 t^{-1} -14 t^{-2} +3 t^{-3} }[/math] |
In[5]:=
|
Conway[K][z]
|
Out[5]=
|
[math]\displaystyle{ 3 z^6+4 z^4+3 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]=
|
{ 139, 2 } |
In[8]:=
|
Jones[K][q]
|
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
|
Out[8]=
|
[math]\displaystyle{ q^9-4 q^8+8 q^7-14 q^6+19 q^5-22 q^4+23 q^3-19 q^2+15 q-9+4 q^{-1} - q^{-2} }[/math] |
In[9]:=
|
HOMFLYPT[K][a, z]
|
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
|
Out[9]=
|
[math]\displaystyle{ z^6 a^{-2} +2 z^6 a^{-4} +z^4 a^{-2} +7 z^4 a^{-4} -3 z^4 a^{-6} -z^4-z^2 a^{-2} +11 z^2 a^{-4} -7 z^2 a^{-6} +z^2 a^{-8} -z^2- a^{-2} +6 a^{-4} -5 a^{-6} + a^{-8} }[/math] |
In[10]:=
|
Kauffman[K][a, z]
|
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
|
Out[10]=
|
[math]\displaystyle{ z^{10} a^{-4} +z^{10} a^{-6} +5 z^9 a^{-3} +9 z^9 a^{-5} +4 z^9 a^{-7} +9 z^8 a^{-2} +20 z^8 a^{-4} +17 z^8 a^{-6} +6 z^8 a^{-8} +8 z^7 a^{-1} +9 z^7 a^{-3} +2 z^7 a^{-5} +5 z^7 a^{-7} +4 z^7 a^{-9} -11 z^6 a^{-2} -45 z^6 a^{-4} -43 z^6 a^{-6} -12 z^6 a^{-8} +z^6 a^{-10} +4 z^6+a z^5-12 z^5 a^{-1} -32 z^5 a^{-3} -40 z^5 a^{-5} -31 z^5 a^{-7} -10 z^5 a^{-9} +5 z^4 a^{-2} +37 z^4 a^{-4} +35 z^4 a^{-6} +6 z^4 a^{-8} -2 z^4 a^{-10} -5 z^4-a z^3+7 z^3 a^{-1} +26 z^3 a^{-3} +41 z^3 a^{-5} +31 z^3 a^{-7} +8 z^3 a^{-9} -3 z^2 a^{-2} -19 z^2 a^{-4} -16 z^2 a^{-6} -z^2 a^{-8} +z^2 a^{-10} +2 z^2-2 z a^{-1} -7 z a^{-3} -13 z a^{-5} -10 z a^{-7} -2 z a^{-9} + a^{-2} +6 a^{-4} +5 a^{-6} + a^{-8} }[/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["K11a32"];
|
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{ 3 t^3-14 t^2+32 t-41+32 t^{-1} -14 t^{-2} +3 t^{-3} }[/math], [math]\displaystyle{ q^9-4 q^8+8 q^7-14 q^6+19 q^5-22 q^4+23 q^3-19 q^2+15 q-9+4 q^{-1} - q^{-2} }[/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: | (3, 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 K11a32. 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. |
|
