K11a52
From Knot Atlas
Jump to navigationJump to search
|
|
|
|
Visit K11a52's page at Knotilus!
Visit K11a52's page at the original Knot Atlas! |
| K11a52 Quick Notes |
K11a52 Further Notes and Views
Knot presentations
| Planar diagram presentation | X4251 X8394 X14,5,15,6 X12,8,13,7 X2,9,3,10 X18,12,19,11 X22,13,1,14 X20,16,21,15 X10,18,11,17 X16,20,17,19 X6,21,7,22 |
| Gauss code | 1, -5, 2, -1, 3, -11, 4, -2, 5, -9, 6, -4, 7, -3, 8, -10, 9, -6, 10, -8, 11, -7 |
| Dowker-Thistlethwaite code | 4 8 14 12 2 18 22 20 10 16 6 |
| Conway Notation | [.22.210] |
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 -2 t^3+13 t^2-32 t+43-32 t^{-1} +13 t^{-2} -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 -2 z^6+z^4+2 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 | { 137, 0 } |
| 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 q^6-5 q^5+10 q^4-15 q^3+20 q^2-22 q+22-18 q^{-1} +13 q^{-2} -7 q^{-3} +3 q^{-4} - q^{-5} } |
| 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^6 a^{-2} -z^6+2 a^2 z^4-z^4 a^{-2} +z^4 a^{-4} -z^4-a^4 z^2+3 a^2 z^2+z^2 a^{-2} -z^2-a^4+2 a^2+2 a^{-2} - a^{-4} -1} |
| 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 2 z^{10} a^{-2} +2 z^{10}+5 a z^9+12 z^9 a^{-1} +7 z^9 a^{-3} +6 a^2 z^8+14 z^8 a^{-2} +9 z^8 a^{-4} +11 z^8+5 a^3 z^7-18 z^7 a^{-1} -8 z^7 a^{-3} +5 z^7 a^{-5} +3 a^4 z^6-5 a^2 z^6-41 z^6 a^{-2} -21 z^6 a^{-4} +z^6 a^{-6} -27 z^6+a^5 z^5-6 a^3 z^5-9 a z^5-z^5 a^{-1} -9 z^5 a^{-3} -10 z^5 a^{-5} -5 a^4 z^4-a^2 z^4+27 z^4 a^{-2} +11 z^4 a^{-4} -z^4 a^{-6} +19 z^4-2 a^5 z^3+2 a^3 z^3+10 a z^3+10 z^3 a^{-1} +7 z^3 a^{-3} +3 z^3 a^{-5} +3 a^4 z^2+4 a^2 z^2-2 z^2 a^{-2} -z^2+a^5 z-3 a z-3 z a^{-1} +z a^{-5} -a^4-2 a^2-2 a^{-2} - a^{-4} -1} |
| The A2 invariant | Data:K11a52/QuantumInvariant/A2/1,0 |
| The G2 invariant | Data:K11a52/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["K11a52"];
|
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 -2 t^3+13 t^2-32 t+43-32 t^{-1} +13 t^{-2} -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 -2 z^6+z^4+2 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]=
|
{ 137, 0 } |
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 q^6-5 q^5+10 q^4-15 q^3+20 q^2-22 q+22-18 q^{-1} +13 q^{-2} -7 q^{-3} +3 q^{-4} - q^{-5} } |
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^6 a^{-2} -z^6+2 a^2 z^4-z^4 a^{-2} +z^4 a^{-4} -z^4-a^4 z^2+3 a^2 z^2+z^2 a^{-2} -z^2-a^4+2 a^2+2 a^{-2} - a^{-4} -1} |
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 2 z^{10} a^{-2} +2 z^{10}+5 a z^9+12 z^9 a^{-1} +7 z^9 a^{-3} +6 a^2 z^8+14 z^8 a^{-2} +9 z^8 a^{-4} +11 z^8+5 a^3 z^7-18 z^7 a^{-1} -8 z^7 a^{-3} +5 z^7 a^{-5} +3 a^4 z^6-5 a^2 z^6-41 z^6 a^{-2} -21 z^6 a^{-4} +z^6 a^{-6} -27 z^6+a^5 z^5-6 a^3 z^5-9 a z^5-z^5 a^{-1} -9 z^5 a^{-3} -10 z^5 a^{-5} -5 a^4 z^4-a^2 z^4+27 z^4 a^{-2} +11 z^4 a^{-4} -z^4 a^{-6} +19 z^4-2 a^5 z^3+2 a^3 z^3+10 a z^3+10 z^3 a^{-1} +7 z^3 a^{-3} +3 z^3 a^{-5} +3 a^4 z^2+4 a^2 z^2-2 z^2 a^{-2} -z^2+a^5 z-3 a z-3 z a^{-1} +z a^{-5} -a^4-2 a^2-2 a^{-2} - a^{-4} -1} |
Vassiliev invariants
| V2 and V3: | (2, 0) |
| 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 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^rq^j} are shown, along with their alternating sums 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 \chi} (fixed 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} , alternation over 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 r} ). 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=} 0 is the signature of K11a52. 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.
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 \textrm{Include}(\textrm{ColouredJonesM.mhtml})}
In[1]:= |
<< KnotTheory` |
Loading KnotTheory` (version of August 17, 2005, 14:44:34)... | |
In[2]:= | Crossings[Knot[11, Alternating, 52]] |
Out[2]= | 11 |
In[3]:= | PD[Knot[11, Alternating, 52]] |
Out[3]= | PD[X[4, 2, 5, 1], X[8, 3, 9, 4], X[14, 5, 15, 6], X[12, 8, 13, 7],X[2, 9, 3, 10], X[18, 12, 19, 11], X[22, 13, 1, 14], X[20, 16, 21, 15], X[10, 18, 11, 17], X[16, 20, 17, 19],X[6, 21, 7, 22]] |
In[4]:= | GaussCode[Knot[11, Alternating, 52]] |
Out[4]= | GaussCode[1, -5, 2, -1, 3, -11, 4, -2, 5, -9, 6, -4, 7, -3, 8, -10, 9, -6, 10, -8, 11, -7] |
In[5]:= | BR[Knot[11, Alternating, 52]] |
Out[5]= | BR[Knot[11, Alternating, 52]] |
In[6]:= | alex = Alexander[Knot[11, Alternating, 52]][t] |
Out[6]= | 2 13 32 2 3 |
In[7]:= | Conway[Knot[11, Alternating, 52]][z] |
Out[7]= | 2 4 6 1 + 2 z + z - 2 z |
In[8]:= | Select[AllKnots[], (alex === Alexander[#][t])&] |
Out[8]= | {Knot[11, Alternating, 52]} |
In[9]:= | {KnotDet[Knot[11, Alternating, 52]], KnotSignature[Knot[11, Alternating, 52]]} |
Out[9]= | {137, 0} |
In[10]:= | J=Jones[Knot[11, Alternating, 52]][q] |
Out[10]= | -5 3 7 13 18 2 3 4 5 6 |
In[11]:= | Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&] |
Out[11]= | {Knot[11, Alternating, 52]} |
In[12]:= | A2Invariant[Knot[11, Alternating, 52]][q] |
Out[12]= | -16 -12 3 3 2 3 4 2 4 6 8 |
In[13]:= | Kauffman[Knot[11, Alternating, 52]][a, z] |
Out[13]= | 2-4 2 2 4 z 3 z 5 2 2 z |
In[14]:= | {Vassiliev[2][Knot[11, Alternating, 52]], Vassiliev[3][Knot[11, Alternating, 52]]} |
Out[14]= | {0, 0} |
In[15]:= | Kh[Knot[11, Alternating, 52]][q, t] |
Out[15]= | 12 1 2 1 5 2 8 5 |
