K11n3
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 X8394 X10,6,11,5 X7,15,8,14 X2,9,3,10 X11,16,12,17 X13,20,14,21 X15,7,16,6 X17,22,18,1 X19,12,20,13 X21,18,22,19 |
| Gauss code | 1, -5, 2, -1, 3, 8, -4, -2, 5, -3, -6, 10, -7, 4, -8, 6, -9, 11, -10, 7, -11, 9 |
| Dowker-Thistlethwaite code | 4 8 10 -14 2 -16 -20 -6 -22 -12 -18 |
| A Braid Representative |
| ||||||
| A Morse Link Presentation | 
|
Three dimensional invariants
|
Four dimensional invariants
|
Polynomial invariants
| Alexander polynomial | [math]\displaystyle{ -3 t^2+11 t-15+11 t^{-1} -3 t^{-2} }[/math] |
| Conway polynomial | [math]\displaystyle{ -3 z^4-z^2+1 }[/math] |
| 2nd Alexander ideal (db, data sources) | [math]\displaystyle{ \{1\} }[/math] |
| Determinant and Signature | { 43, -2 } |
| Jones polynomial | [math]\displaystyle{ q^3-2 q^2+4 q-6+7 q^{-1} -7 q^{-2} +7 q^{-3} -5 q^{-4} +3 q^{-5} - q^{-6} }[/math] |
| HOMFLY-PT polynomial (db, data sources) | [math]\displaystyle{ -a^6+3 z^2 a^4+3 a^4-2 z^4 a^2-4 z^2 a^2-2 a^2-z^4-z^2+z^2 a^{-2} + a^{-2} }[/math] |
| Kauffman polynomial (db, data sources) | [math]\displaystyle{ a^3 z^9+a z^9+2 a^4 z^8+4 a^2 z^8+2 z^8+a^5 z^7-a^3 z^7+2 z^7 a^{-1} -7 a^4 z^6-13 a^2 z^6+z^6 a^{-2} -5 z^6-a^5 z^5-a^3 z^5-7 a z^5-7 z^5 a^{-1} +3 a^6 z^4+15 a^4 z^4+16 a^2 z^4-4 z^4 a^{-2} +a^7 z^3+4 a^5 z^3+4 a^3 z^3+7 a z^3+6 z^3 a^{-1} -3 a^6 z^2-11 a^4 z^2-10 a^2 z^2+4 z^2 a^{-2} +2 z^2-a^7 z-2 a^5 z-2 a^3 z-3 a z-2 z a^{-1} +a^6+3 a^4+2 a^2- a^{-2} }[/math] |
| The A2 invariant | Data:K11n3/QuantumInvariant/A2/1,0 |
| The G2 invariant | Data:K11n3/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["K11n3"];
|
In[4]:=
|
Alexander[K][t]
|
KnotTheory::loading: Loading precomputed data in PD4Knots`.
|
Out[4]=
|
[math]\displaystyle{ -3 t^2+11 t-15+11 t^{-1} -3 t^{-2} }[/math] |
In[5]:=
|
Conway[K][z]
|
Out[5]=
|
[math]\displaystyle{ -3 z^4-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]=
|
{ 43, -2 } |
In[8]:=
|
Jones[K][q]
|
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
|
Out[8]=
|
[math]\displaystyle{ q^3-2 q^2+4 q-6+7 q^{-1} -7 q^{-2} +7 q^{-3} -5 q^{-4} +3 q^{-5} - q^{-6} }[/math] |
In[9]:=
|
HOMFLYPT[K][a, z]
|
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
|
Out[9]=
|
[math]\displaystyle{ -a^6+3 z^2 a^4+3 a^4-2 z^4 a^2-4 z^2 a^2-2 a^2-z^4-z^2+z^2 a^{-2} + a^{-2} }[/math] |
In[10]:=
|
Kauffman[K][a, z]
|
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
|
Out[10]=
|
[math]\displaystyle{ a^3 z^9+a z^9+2 a^4 z^8+4 a^2 z^8+2 z^8+a^5 z^7-a^3 z^7+2 z^7 a^{-1} -7 a^4 z^6-13 a^2 z^6+z^6 a^{-2} -5 z^6-a^5 z^5-a^3 z^5-7 a z^5-7 z^5 a^{-1} +3 a^6 z^4+15 a^4 z^4+16 a^2 z^4-4 z^4 a^{-2} +a^7 z^3+4 a^5 z^3+4 a^3 z^3+7 a z^3+6 z^3 a^{-1} -3 a^6 z^2-11 a^4 z^2-10 a^2 z^2+4 z^2 a^{-2} +2 z^2-a^7 z-2 a^5 z-2 a^3 z-3 a z-2 z a^{-1} +a^6+3 a^4+2 a^2- a^{-2} }[/math] |
"Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {10_7, K11a59,}
Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {9_22,}
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["K11n3"];
|
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^2+11 t-15+11 t^{-1} -3 t^{-2} }[/math], [math]\displaystyle{ q^3-2 q^2+4 q-6+7 q^{-1} -7 q^{-2} +7 q^{-3} -5 q^{-4} +3 q^{-5} - q^{-6} }[/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]=
|
{10_7, K11a59,} |
In[6]:=
|
DeleteCases[
Select[
AllKnots[],
(J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) &
],
K
]
|
KnotTheory::loading: Loading precomputed data in Jones4Knots11`.
|
Out[6]=
|
{9_22,} |
Vassiliev invariants
| V2 and V3: | (-1, -1) |
| 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 K11n3. 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. |
|
