K11a54
From Knot Atlas
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Visit K11a54's page at Knotilus!
Visit K11a54's page at the original Knot Atlas! |
| K11a54 Quick Notes |
K11a54 Further Notes and Views
Knot presentations
| Planar diagram presentation | X4251 X8394 X14,6,15,5 X16,7,17,8 X2,9,3,10 X18,11,19,12 X20,13,21,14 X22,16,1,15 X12,17,13,18 X10,19,11,20 X6,21,7,22 |
| Gauss code | 1, -5, 2, -1, 3, -11, 4, -2, 5, -10, 6, -9, 7, -3, 8, -4, 9, -6, 10, -7, 11, -8 |
| Dowker-Thistlethwaite code | 4 8 14 16 2 18 20 22 12 10 6 |
| Conway Notation | [.31.210] |
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 2 t^3-13 t^2+33 t-43+33 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-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 | { 139, -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+5 q-10+16 q^{-1} -20 q^{-2} +23 q^{-3} -22 q^{-4} +18 q^{-5} -13 q^{-6} +7 q^{-7} -3 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+a^8-2 z^4 a^6-3 z^2 a^6-2 a^6+z^6 a^4+z^4 a^4+z^2 a^4+a^4+z^6 a^2+z^4 a^2-z^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 z^6 a^{10}-3 z^4 a^{10}+2 z^2 a^{10}+3 z^7 a^9-8 z^5 a^9+6 z^3 a^9-2 z a^9+5 z^8 a^8-12 z^6 a^8+10 z^4 a^8-5 z^2 a^8+a^8+5 z^9 a^7-8 z^7 a^7+2 z^5 a^7+3 z^3 a^7-2 z a^7+2 z^{10} a^6+9 z^8 a^6-31 z^6 a^6+34 z^4 a^6-14 z^2 a^6+2 a^6+12 z^9 a^5-21 z^7 a^5+11 z^5 a^5-z^3 a^5+z a^5+2 z^{10} a^4+15 z^8 a^4-37 z^6 a^4+28 z^4 a^4-8 z^2 a^4+a^4+7 z^9 a^3-14 z^5 a^3+5 z^3 a^3+z a^3+11 z^8 a^2-14 z^6 a^2+2 z^4 a^2-z^2 a^2+10 z^7 a-14 z^5 a+3 z^3 a+5 z^6-5 z^4+1+z^5 a^{-1} } |
| The A2 invariant | Data:K11a54/QuantumInvariant/A2/1,0 |
| The G2 invariant | Data:K11a54/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["K11a54"];
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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 2 t^3-13 t^2+33 t-43+33 t^{-1} -13 t^{-2} +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 2 z^6-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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{ 139, -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+5 q-10+16 q^{-1} -20 q^{-2} +23 q^{-3} -22 q^{-4} +18 q^{-5} -13 q^{-6} +7 q^{-7} -3 q^{-8} + q^{-9} } |
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 z^2 a^8+a^8-2 z^4 a^6-3 z^2 a^6-2 a^6+z^6 a^4+z^4 a^4+z^2 a^4+a^4+z^6 a^2+z^4 a^2-z^4+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^6 a^{10}-3 z^4 a^{10}+2 z^2 a^{10}+3 z^7 a^9-8 z^5 a^9+6 z^3 a^9-2 z a^9+5 z^8 a^8-12 z^6 a^8+10 z^4 a^8-5 z^2 a^8+a^8+5 z^9 a^7-8 z^7 a^7+2 z^5 a^7+3 z^3 a^7-2 z a^7+2 z^{10} a^6+9 z^8 a^6-31 z^6 a^6+34 z^4 a^6-14 z^2 a^6+2 a^6+12 z^9 a^5-21 z^7 a^5+11 z^5 a^5-z^3 a^5+z a^5+2 z^{10} a^4+15 z^8 a^4-37 z^6 a^4+28 z^4 a^4-8 z^2 a^4+a^4+7 z^9 a^3-14 z^5 a^3+5 z^3 a^3+z a^3+11 z^8 a^2-14 z^6 a^2+2 z^4 a^2-z^2 a^2+10 z^7 a-14 z^5 a+3 z^3 a+5 z^6-5 z^4+1+z^5 a^{-1} } |
Vassiliev invariants
| V2 and V3: | (-1, 3) |
| 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 K11a54. 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, 54]] |
Out[2]= | 11 |
In[3]:= | PD[Knot[11, Alternating, 54]] |
Out[3]= | PD[X[4, 2, 5, 1], X[8, 3, 9, 4], X[14, 6, 15, 5], X[16, 7, 17, 8],X[2, 9, 3, 10], X[18, 11, 19, 12], X[20, 13, 21, 14], X[22, 16, 1, 15], X[12, 17, 13, 18], X[10, 19, 11, 20],X[6, 21, 7, 22]] |
In[4]:= | GaussCode[Knot[11, Alternating, 54]] |
Out[4]= | GaussCode[1, -5, 2, -1, 3, -11, 4, -2, 5, -10, 6, -9, 7, -3, 8, -4, 9, -6, 10, -7, 11, -8] |
In[5]:= | BR[Knot[11, Alternating, 54]] |
Out[5]= | BR[Knot[11, Alternating, 54]] |
In[6]:= | alex = Alexander[Knot[11, Alternating, 54]][t] |
Out[6]= | 2 13 33 2 3 |
In[7]:= | Conway[Knot[11, Alternating, 54]][z] |
Out[7]= | 2 4 6 1 - z - z + 2 z |
In[8]:= | Select[AllKnots[], (alex === Alexander[#][t])&] |
Out[8]= | {Knot[11, Alternating, 54], Knot[11, Alternating, 172]} |
In[9]:= | {KnotDet[Knot[11, Alternating, 54]], KnotSignature[Knot[11, Alternating, 54]]} |
Out[9]= | {139, -2} |
In[10]:= | J=Jones[Knot[11, Alternating, 54]][q] |
Out[10]= | -9 3 7 13 18 22 23 20 16 2 |
In[11]:= | Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&] |
Out[11]= | {Knot[11, Alternating, 54]} |
In[12]:= | A2Invariant[Knot[11, Alternating, 54]][q] |
Out[12]= | -28 -24 3 3 2 3 4 3 -10 3 4 |
In[13]:= | Kauffman[Knot[11, Alternating, 54]][a, z] |
Out[13]= | 4 6 8 3 5 7 9 2 2 4 2 |
In[14]:= | {Vassiliev[2][Knot[11, Alternating, 54]], Vassiliev[3][Knot[11, Alternating, 54]]} |
Out[14]= | {0, 3} |
In[15]:= | Kh[Knot[11, Alternating, 54]][q, t] |
Out[15]= | 7 10 1 2 1 5 2 8 |
