K11a288
From Knot Atlas
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 (Knotscape image) |
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. |
Knot presentations
| Planar diagram presentation | X6271 X10,3,11,4 X16,6,17,5 X18,7,19,8 X20,10,21,9 X4,11,5,12 X8,14,9,13 X2,16,3,15 X22,17,1,18 X14,19,15,20 X12,22,13,21 |
| Gauss code | 1, -8, 2, -6, 3, -1, 4, -7, 5, -2, 6, -11, 7, -10, 8, -3, 9, -4, 10, -5, 11, -9 |
| Dowker-Thistlethwaite code | 6 10 16 18 20 4 8 2 22 14 12 |
| A Braid Representative |
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| A Morse Link Presentation | 
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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^4-7 t^3+23 t^2-44 t+55-44 t^{-1} +23 t^{-2} -7 t^{-3} + t^{-4} } |
| 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^8+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 | { 205, 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+12 q^4-21 q^3+29 q^2-33 q+34-29 q^{-1} +22 q^{-2} -13 q^{-3} +5 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^8-a^2 z^6-2 z^6 a^{-2} +4 z^6-2 a^2 z^4-5 z^4 a^{-2} +z^4 a^{-4} +7 z^4-2 a^2 z^2-4 z^2 a^{-2} +z^2 a^{-4} +6 z^2-a^2- a^{-2} +3} |
| 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 5 z^{10} a^{-2} +5 z^{10}+16 a z^9+28 z^9 a^{-1} +12 z^9 a^{-3} +20 a^2 z^8+21 z^8 a^{-2} +11 z^8 a^{-4} +30 z^8+13 a^3 z^7-12 a z^7-45 z^7 a^{-1} -15 z^7 a^{-3} +5 z^7 a^{-5} +5 a^4 z^6-31 a^2 z^6-65 z^6 a^{-2} -21 z^6 a^{-4} +z^6 a^{-6} -79 z^6+a^5 z^5-14 a^3 z^5-13 a z^5+6 z^5 a^{-1} -4 z^5 a^{-3} -8 z^5 a^{-5} -2 a^4 z^4+19 a^2 z^4+47 z^4 a^{-2} +14 z^4 a^{-4} -z^4 a^{-6} +53 z^4+6 a^3 z^3+10 a z^3+8 z^3 a^{-1} +8 z^3 a^{-3} +4 z^3 a^{-5} -6 a^2 z^2-12 z^2 a^{-2} -3 z^2 a^{-4} -15 z^2-a^3 z-2 a z-2 z a^{-1} -z a^{-3} +a^2+ a^{-2} +3} |
| The A2 invariant | 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^{14}+3 q^{12}-5 q^{10}+3 q^8+q^6-5 q^4+8 q^2-5+6 q^{-2} - q^{-4} -2 q^{-6} +5 q^{-8} -6 q^{-10} +3 q^{-12} -2 q^{-16} + q^{-18} } |
| The G2 invariant | Data:K11a288/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["K11a288"];
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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^4-7 t^3+23 t^2-44 t+55-44 t^{-1} +23 t^{-2} -7 t^{-3} + t^{-4} } |
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^8+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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{ 205, 0 } |
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^6-5 q^5+12 q^4-21 q^3+29 q^2-33 q+34-29 q^{-1} +22 q^{-2} -13 q^{-3} +5 q^{-4} - q^{-5} } |
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^8-a^2 z^6-2 z^6 a^{-2} +4 z^6-2 a^2 z^4-5 z^4 a^{-2} +z^4 a^{-4} +7 z^4-2 a^2 z^2-4 z^2 a^{-2} +z^2 a^{-4} +6 z^2-a^2- a^{-2} +3} |
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 5 z^{10} a^{-2} +5 z^{10}+16 a z^9+28 z^9 a^{-1} +12 z^9 a^{-3} +20 a^2 z^8+21 z^8 a^{-2} +11 z^8 a^{-4} +30 z^8+13 a^3 z^7-12 a z^7-45 z^7 a^{-1} -15 z^7 a^{-3} +5 z^7 a^{-5} +5 a^4 z^6-31 a^2 z^6-65 z^6 a^{-2} -21 z^6 a^{-4} +z^6 a^{-6} -79 z^6+a^5 z^5-14 a^3 z^5-13 a z^5+6 z^5 a^{-1} -4 z^5 a^{-3} -8 z^5 a^{-5} -2 a^4 z^4+19 a^2 z^4+47 z^4 a^{-2} +14 z^4 a^{-4} -z^4 a^{-6} +53 z^4+6 a^3 z^3+10 a z^3+8 z^3 a^{-1} +8 z^3 a^{-3} +4 z^3 a^{-5} -6 a^2 z^2-12 z^2 a^{-2} -3 z^2 a^{-4} -15 z^2-a^3 z-2 a z-2 z a^{-1} -z a^{-3} +a^2+ a^{-2} +3} |
"Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {}
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]:=
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AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
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In[3]:=
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K = Knot["K11a288"];
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In[4]:=
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{A = Alexander[K][t], J = Jones[K][q]}
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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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^4-7 t^3+23 t^2-44 t+55-44 t^{-1} +23 t^{-2} -7 t^{-3} + t^{-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 q^6-5 q^5+12 q^4-21 q^3+29 q^2-33 q+34-29 q^{-1} +22 q^{-2} -13 q^{-3} +5 q^{-4} - q^{-5} } } |
In[5]:=
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DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]
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KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.
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KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.
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Out[5]=
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{} |
In[6]:=
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DeleteCases[
Select[
AllKnots[],
(J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) &
],
K
]
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KnotTheory::loading: Loading precomputed data in Jones4Knots11`.
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Out[6]=
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{} |
Vassiliev invariants
| V2 and V3: | (1, 0) |
| 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=} 0 is the signature of K11a288. 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.
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. |
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