K11n121
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 X10,3,11,4 X5,16,6,17 X7,12,8,13 X18,9,19,10 X2,11,3,12 X13,20,14,21 X15,6,16,7 X22,18,1,17 X8,19,9,20 X21,14,22,15 |
| Gauss code | 1, -6, 2, -1, -3, 8, -4, -10, 5, -2, 6, 4, -7, 11, -8, 3, 9, -5, 10, 7, -11, -9 |
| Dowker-Thistlethwaite code | 4 10 -16 -12 18 2 -20 -6 22 8 -14 |
| A Braid Representative |
| ||||||
| A Morse Link Presentation | 
|
Three dimensional invariants
|
Four dimensional invariants
|
Polynomial invariants
| Alexander polynomial | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -t^3+6 t^2-10 t+11-10 t^{-1} +6 t^{-2} - t^{-3} } |
| Conway polynomial | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -z^6+5 z^2+1} |
| 2nd Alexander ideal (db, data sources) | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1\}} |
| Determinant and Signature | { 45, -4 } |
| Jones polynomial | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1-3 q^{-1} +5 q^{-2} -6 q^{-3} +8 q^{-4} -7 q^{-5} +7 q^{-6} -5 q^{-7} +2 q^{-8} - q^{-9} } |
| HOMFLY-PT polynomial (db, data sources) | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -z^2 a^8-2 a^8+2 z^4 a^6+5 z^2 a^6+2 a^6-z^6 a^4-3 z^4 a^4-z^2 a^4+a^4+z^4 a^2+2 z^2 a^2} |
| Kauffman polynomial (db, data sources) | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^3 a^{11}-z a^{11}+2 z^4 a^{10}-z^2 a^{10}+4 z^5 a^9-5 z^3 a^9+4 z a^9+z^8 a^8-z^6 a^8+2 z^4 a^8+2 z^2 a^8-2 a^8+z^9 a^7-2 z^7 a^7+6 z^5 a^7-10 z^3 a^7+6 z a^7+4 z^8 a^6-11 z^6 a^6+9 z^4 a^6-2 z^2 a^6-2 a^6+z^9 a^5+z^7 a^5-8 z^5 a^5+3 z^3 a^5+z a^5+3 z^8 a^4-9 z^6 a^4+6 z^4 a^4-3 z^2 a^4+a^4+3 z^7 a^3-10 z^5 a^3+7 z^3 a^3+z^6 a^2-3 z^4 a^2+2 z^2 a^2} |
| The A2 invariant | Data:K11n121/QuantumInvariant/A2/1,0 |
| The G2 invariant | Data:K11n121/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["K11n121"];
|
In[4]:=
|
Alexander[K][t]
|
KnotTheory::loading: Loading precomputed data in PD4Knots`.
|
Out[4]=
|
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -t^3+6 t^2-10 t+11-10 t^{-1} +6 t^{-2} - t^{-3} } |
In[5]:=
|
Conway[K][z]
|
Out[5]=
|
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -z^6+5 z^2+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]=
|
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1\}} |
In[7]:=
|
{KnotDet[K], KnotSignature[K]}
|
Out[7]=
|
{ 45, -4 } |
In[8]:=
|
Jones[K][q]
|
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
|
Out[8]=
|
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1-3 q^{-1} +5 q^{-2} -6 q^{-3} +8 q^{-4} -7 q^{-5} +7 q^{-6} -5 q^{-7} +2 q^{-8} - q^{-9} } |
In[9]:=
|
HOMFLYPT[K][a, z]
|
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
|
Out[9]=
|
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -z^2 a^8-2 a^8+2 z^4 a^6+5 z^2 a^6+2 a^6-z^6 a^4-3 z^4 a^4-z^2 a^4+a^4+z^4 a^2+2 z^2 a^2} |
In[10]:=
|
Kauffman[K][a, z]
|
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
|
Out[10]=
|
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^3 a^{11}-z a^{11}+2 z^4 a^{10}-z^2 a^{10}+4 z^5 a^9-5 z^3 a^9+4 z a^9+z^8 a^8-z^6 a^8+2 z^4 a^8+2 z^2 a^8-2 a^8+z^9 a^7-2 z^7 a^7+6 z^5 a^7-10 z^3 a^7+6 z a^7+4 z^8 a^6-11 z^6 a^6+9 z^4 a^6-2 z^2 a^6-2 a^6+z^9 a^5+z^7 a^5-8 z^5 a^5+3 z^3 a^5+z a^5+3 z^8 a^4-9 z^6 a^4+6 z^4 a^4-3 z^2 a^4+a^4+3 z^7 a^3-10 z^5 a^3+7 z^3 a^3+z^6 a^2-3 z^4 a^2+2 z^2 a^2} |
"Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {K11n14,}
Same Jones Polynomial (up to mirroring, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q\leftrightarrow q^{-1}} ): {}
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["K11n121"];
|
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]=
|
{ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -t^3+6 t^2-10 t+11-10 t^{-1} +6 t^{-2} - t^{-3} } , } |
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]=
|
{K11n14,} |
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: | (5, -11) |
| 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 are shown, along with their alternating sums (fixed , alternation over ). The squares with yellow highlighting are those on the "critical diagonals", where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s+1} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s-1} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s=} -4 is the signature of K11n121. 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. |
|
