K11a65
From Knot Atlas
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Visit K11a65's page at Knotilus!
Visit K11a65's page at the original Knot Atlas! |
| K11a65 Quick Notes |
K11a65 Further Notes and Views
Knot presentations
| Planar diagram presentation | X4251 X8394 X16,5,17,6 X10,8,11,7 X2,9,3,10 X22,11,1,12 X20,13,21,14 X18,15,19,16 X6,17,7,18 X14,19,15,20 X12,21,13,22 |
| Gauss code | 1, -5, 2, -1, 3, -9, 4, -2, 5, -4, 6, -11, 7, -10, 8, -3, 9, -8, 10, -7, 11, -6 |
| Dowker-Thistlethwaite code | 4 8 16 10 2 22 20 18 6 14 12 |
| Conway Notation | [2522] |
Three dimensional invariants
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Four dimensional invariants
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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 -3 t^2+15 t-23+15 t^{-1} -3 t^{-2} } |
| 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 -3 z^4+3 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 | { 59, -2 } |
| 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-2+4 q^{-1} -6 q^{-2} +8 q^{-3} -9 q^{-4} +9 q^{-5} -7 q^{-6} +6 q^{-7} -4 q^{-8} +2 q^{-9} - q^{-10} } |
| 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 -a^{10}+2 z^2 a^8+a^8-z^4 a^6-z^4 a^4+z^2 a^4+a^4-z^4 a^2-z^2 a^2-a^2+z^2+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 z^7 a^{11}-5 z^5 a^{11}+7 z^3 a^{11}-3 z a^{11}+2 z^8 a^{10}-9 z^6 a^{10}+11 z^4 a^{10}-5 z^2 a^{10}+a^{10}+2 z^9 a^9-7 z^7 a^9+3 z^5 a^9+4 z^3 a^9-2 z a^9+z^{10} a^8-z^8 a^8-7 z^6 a^8+9 z^4 a^8-3 z^2 a^8+a^8+4 z^9 a^7-15 z^7 a^7+17 z^5 a^7-8 z^3 a^7+z a^7+z^{10} a^6-z^8 a^6-2 z^6 a^6+z^4 a^6+z^2 a^6+2 z^9 a^5-5 z^7 a^5+7 z^5 a^5-4 z^3 a^5+2 z^8 a^4-2 z^6 a^4+2 z^4 a^4-2 z^2 a^4+a^4+2 z^7 a^3-2 z^3 a^3+z a^3+2 z^6 a^2-3 z^2 a^2+a^2+2 z^5 a-3 z^3 a+z a+z^4-2 z^2+1} |
| The A2 invariant | Data:K11a65/QuantumInvariant/A2/1,0 |
| The G2 invariant | Data:K11a65/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]:=
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AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.
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In[3]:=
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K = Knot["K11a65"];
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In[4]:=
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Alexander[K][t]
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
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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 -3 t^2+15 t-23+15 t^{-1} -3 t^{-2} } |
In[5]:=
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Conway[K][z]
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Out[5]=
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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 -3 z^4+3 z^2+1} |
In[6]:=
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Alexander[K, 2][t]
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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.
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Out[6]=
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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]:=
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{KnotDet[K], KnotSignature[K]}
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Out[7]=
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{ 59, -2 } |
In[8]:=
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Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
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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-2+4 q^{-1} -6 q^{-2} +8 q^{-3} -9 q^{-4} +9 q^{-5} -7 q^{-6} +6 q^{-7} -4 q^{-8} +2 q^{-9} - q^{-10} } |
In[9]:=
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HOMFLYPT[K][a, z]
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KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
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Out[9]=
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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 -a^{10}+2 z^2 a^8+a^8-z^4 a^6-z^4 a^4+z^2 a^4+a^4-z^4 a^2-z^2 a^2-a^2+z^2+1} |
In[10]:=
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Kauffman[K][a, z]
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KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
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Out[10]=
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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^7 a^{11}-5 z^5 a^{11}+7 z^3 a^{11}-3 z a^{11}+2 z^8 a^{10}-9 z^6 a^{10}+11 z^4 a^{10}-5 z^2 a^{10}+a^{10}+2 z^9 a^9-7 z^7 a^9+3 z^5 a^9+4 z^3 a^9-2 z a^9+z^{10} a^8-z^8 a^8-7 z^6 a^8+9 z^4 a^8-3 z^2 a^8+a^8+4 z^9 a^7-15 z^7 a^7+17 z^5 a^7-8 z^3 a^7+z a^7+z^{10} a^6-z^8 a^6-2 z^6 a^6+z^4 a^6+z^2 a^6+2 z^9 a^5-5 z^7 a^5+7 z^5 a^5-4 z^3 a^5+2 z^8 a^4-2 z^6 a^4+2 z^4 a^4-2 z^2 a^4+a^4+2 z^7 a^3-2 z^3 a^3+z a^3+2 z^6 a^2-3 z^2 a^2+a^2+2 z^5 a-3 z^3 a+z a+z^4-2 z^2+1} |
Vassiliev invariants
| V2 and V3: | (3, -9) |
| V2,1 through V6,9: |
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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=} -2 is the signature of K11a65. Nonzero entries off the critical diagonals (if any exist) are highlighted in red. |
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| Integral Khovanov Homology
(db, data source) |
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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, 65]] |
Out[2]= | 11 |
In[3]:= | PD[Knot[11, Alternating, 65]] |
Out[3]= | PD[X[4, 2, 5, 1], X[8, 3, 9, 4], X[16, 5, 17, 6], X[10, 8, 11, 7],X[2, 9, 3, 10], X[22, 11, 1, 12], X[20, 13, 21, 14], X[18, 15, 19, 16], X[6, 17, 7, 18], X[14, 19, 15, 20],X[12, 21, 13, 22]] |
In[4]:= | GaussCode[Knot[11, Alternating, 65]] |
Out[4]= | GaussCode[1, -5, 2, -1, 3, -9, 4, -2, 5, -4, 6, -11, 7, -10, 8, -3, 9, -8, 10, -7, 11, -6] |
In[5]:= | BR[Knot[11, Alternating, 65]] |
Out[5]= | BR[Knot[11, Alternating, 65]] |
In[6]:= | alex = Alexander[Knot[11, Alternating, 65]][t] |
Out[6]= | 3 15 2 |
In[7]:= | Conway[Knot[11, Alternating, 65]][z] |
Out[7]= | 2 4 1 + 3 z - 3 z |
In[8]:= | Select[AllKnots[], (alex === Alexander[#][t])&] |
Out[8]= | {Knot[11, Alternating, 65]} |
In[9]:= | {KnotDet[Knot[11, Alternating, 65]], KnotSignature[Knot[11, Alternating, 65]]} |
Out[9]= | {59, -2} |
In[10]:= | J=Jones[Knot[11, Alternating, 65]][q] |
Out[10]= | -10 2 4 6 7 9 9 8 6 4 |
In[11]:= | Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&] |
Out[11]= | {Knot[11, Alternating, 65]} |
In[12]:= | A2Invariant[Knot[11, Alternating, 65]][q] |
Out[12]= | -32 -30 -28 -26 2 -20 -18 -16 -12 -10 |
In[13]:= | Kauffman[Knot[11, Alternating, 65]][a, z] |
Out[13]= | 2 4 8 10 3 7 9 11 2 |
In[14]:= | {Vassiliev[2][Knot[11, Alternating, 65]], Vassiliev[3][Knot[11, Alternating, 65]]} |
Out[14]= | {0, -9} |
In[15]:= | Kh[Knot[11, Alternating, 65]][q, t] |
Out[15]= | 2 3 1 1 1 3 1 3 3 |
