K11a51
From Knot Atlas
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Visit K11a51's page at Knotilus!
Visit K11a51's page at the original Knot Atlas! |
| K11a51 Quick Notes |
K11a51 Further Notes and Views
Knot presentations
| Planar diagram presentation | X4251 X8394 X14,5,15,6 X10,8,11,7 X2,9,3,10 X20,11,21,12 X16,13,17,14 X6,15,7,16 X22,18,1,17 X12,19,13,20 X18,22,19,21 |
| Gauss code | 1, -5, 2, -1, 3, -8, 4, -2, 5, -4, 6, -10, 7, -3, 8, -7, 9, -11, 10, -6, 11, -9 |
| Dowker-Thistlethwaite code | 4 8 14 10 2 20 16 6 22 12 18 |
| Conway Notation | [22,22,2+] |
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 t^3-9 t^2+28 t-39+28 t^{-1} -9 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^6-3 z^4+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 | { 115, -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^3-3 q^2+7 q-12+16 q^{-1} -18 q^{-2} +19 q^{-3} -16 q^{-4} +12 q^{-5} -7 q^{-6} +3 q^{-7} - q^{-8} } |
| 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^8+3 z^2 a^6+2 a^6-3 z^4 a^4-4 z^2 a^4-2 a^4+z^6 a^2+2 z^4 a^2+4 z^2 a^2+3 a^2-2 z^4-3 z^2-2+z^2 a^{-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 a^4 z^{10}+a^2 z^{10}+3 a^5 z^9+6 a^3 z^9+3 a z^9+5 a^6 z^8+9 a^4 z^8+8 a^2 z^8+4 z^8+5 a^7 z^7+7 a^5 z^7+a^3 z^7+2 a z^7+3 z^7 a^{-1} +3 a^8 z^6-2 a^6 z^6-11 a^4 z^6-13 a^2 z^6+z^6 a^{-2} -6 z^6+a^9 z^5-7 a^7 z^5-20 a^5 z^5-20 a^3 z^5-16 a z^5-8 z^5 a^{-1} -5 a^8 z^4-6 a^6 z^4-6 a^4 z^4-5 a^2 z^4-3 z^4 a^{-2} -3 z^4-2 a^9 z^3+4 a^7 z^3+16 a^5 z^3+16 a^3 z^3+13 a z^3+7 z^3 a^{-1} +3 a^8 z^2+7 a^6 z^2+10 a^4 z^2+10 a^2 z^2+3 z^2 a^{-2} +7 z^2+a^9 z-a^7 z-4 a^5 z-4 a^3 z-4 a z-2 z a^{-1} -a^8-2 a^6-2 a^4-3 a^2- a^{-2} -2} |
| The A2 invariant | Data:K11a51/QuantumInvariant/A2/1,0 |
| The G2 invariant | Data:K11a51/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["K11a51"];
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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 t^3-9 t^2+28 t-39+28 t^{-1} -9 t^{-2} + t^{-3} } |
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 z^6-3 z^4+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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{ 115, -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^3-3 q^2+7 q-12+16 q^{-1} -18 q^{-2} +19 q^{-3} -16 q^{-4} +12 q^{-5} -7 q^{-6} +3 q^{-7} - q^{-8} } |
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^8+3 z^2 a^6+2 a^6-3 z^4 a^4-4 z^2 a^4-2 a^4+z^6 a^2+2 z^4 a^2+4 z^2 a^2+3 a^2-2 z^4-3 z^2-2+z^2 a^{-2} + a^{-2} } |
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 a^4 z^{10}+a^2 z^{10}+3 a^5 z^9+6 a^3 z^9+3 a z^9+5 a^6 z^8+9 a^4 z^8+8 a^2 z^8+4 z^8+5 a^7 z^7+7 a^5 z^7+a^3 z^7+2 a z^7+3 z^7 a^{-1} +3 a^8 z^6-2 a^6 z^6-11 a^4 z^6-13 a^2 z^6+z^6 a^{-2} -6 z^6+a^9 z^5-7 a^7 z^5-20 a^5 z^5-20 a^3 z^5-16 a z^5-8 z^5 a^{-1} -5 a^8 z^4-6 a^6 z^4-6 a^4 z^4-5 a^2 z^4-3 z^4 a^{-2} -3 z^4-2 a^9 z^3+4 a^7 z^3+16 a^5 z^3+16 a^3 z^3+13 a z^3+7 z^3 a^{-1} +3 a^8 z^2+7 a^6 z^2+10 a^4 z^2+10 a^2 z^2+3 z^2 a^{-2} +7 z^2+a^9 z-a^7 z-4 a^5 z-4 a^3 z-4 a z-2 z a^{-1} -a^8-2 a^6-2 a^4-3 a^2- a^{-2} -2} |
Vassiliev invariants
| V2 and V3: | (1, -4) |
| 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 K11a51. 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, 51]] |
Out[2]= | 11 |
In[3]:= | PD[Knot[11, Alternating, 51]] |
Out[3]= | PD[X[4, 2, 5, 1], X[8, 3, 9, 4], X[14, 5, 15, 6], X[10, 8, 11, 7],X[2, 9, 3, 10], X[20, 11, 21, 12], X[16, 13, 17, 14], X[6, 15, 7, 16], X[22, 18, 1, 17], X[12, 19, 13, 20],X[18, 22, 19, 21]] |
In[4]:= | GaussCode[Knot[11, Alternating, 51]] |
Out[4]= | GaussCode[1, -5, 2, -1, 3, -8, 4, -2, 5, -4, 6, -10, 7, -3, 8, -7, 9, -11, 10, -6, 11, -9] |
In[5]:= | BR[Knot[11, Alternating, 51]] |
Out[5]= | BR[Knot[11, Alternating, 51]] |
In[6]:= | alex = Alexander[Knot[11, Alternating, 51]][t] |
Out[6]= | -3 9 28 2 3 |
In[7]:= | Conway[Knot[11, Alternating, 51]][z] |
Out[7]= | 2 4 6 1 + z - 3 z + z |
In[8]:= | Select[AllKnots[], (alex === Alexander[#][t])&] |
Out[8]= | {Knot[11, Alternating, 51]} |
In[9]:= | {KnotDet[Knot[11, Alternating, 51]], KnotSignature[Knot[11, Alternating, 51]]} |
Out[9]= | {115, -2} |
In[10]:= | J=Jones[Knot[11, Alternating, 51]][q] |
Out[10]= | -8 3 7 12 16 19 18 16 2 3 |
In[11]:= | Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&] |
Out[11]= | {Knot[11, Alternating, 3], Knot[11, Alternating, 51],
Knot[11, Alternating, 331]} |
In[12]:= | A2Invariant[Knot[11, Alternating, 51]][q] |
Out[12]= | -26 -24 2 -20 -18 4 3 -12 -10 -8 |
In[13]:= | Kauffman[Knot[11, Alternating, 51]][a, z] |
Out[13]= | -2 2 4 6 8 2 z 3 5 |
In[14]:= | {Vassiliev[2][Knot[11, Alternating, 51]], Vassiliev[3][Knot[11, Alternating, 51]]} |
Out[14]= | {0, -4} |
In[15]:= | Kh[Knot[11, Alternating, 51]][q, t] |
Out[15]= | 8 9 1 2 1 5 2 7 5 |
