K11a264
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 X8394 X16,6,17,5 X14,7,15,8 X4,9,5,10 X18,12,19,11 X20,14,21,13 X2,16,3,15 X22,17,1,18 X12,20,13,19 X10,22,11,21 |
| Gauss code | 1, -8, 2, -5, 3, -1, 4, -2, 5, -11, 6, -10, 7, -4, 8, -3, 9, -6, 10, -7, 11, -9 |
| Dowker-Thistlethwaite code | 6 8 16 14 4 18 20 2 22 12 10 |
| 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+6 t^3-16 t^2+28 t-33+28 t^{-1} -16 t^{-2} +6 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^8-2 z^6+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 | { 135, 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^8+4 q^7-9 q^6+14 q^5-19 q^4+22 q^3-21 q^2+19 q-13+8 q^{-1} -4 q^{-2} + q^{-3} } |
| 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} -5 z^6 a^{-2} +2 z^6 a^{-4} +z^6-9 z^4 a^{-2} +7 z^4 a^{-4} -z^4 a^{-6} +3 z^4-5 z^2 a^{-2} +7 z^2 a^{-4} -2 z^2 a^{-6} +2 z^2+ a^{-2} + a^{-4} - a^{-6} } |
| 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} a^{-4} +6 z^9 a^{-1} +12 z^9 a^{-3} +6 z^9 a^{-5} +11 z^8 a^{-2} +13 z^8 a^{-4} +9 z^8 a^{-6} +7 z^8+4 a z^7-10 z^7 a^{-1} -22 z^7 a^{-3} +8 z^7 a^{-7} +a^2 z^6-39 z^6 a^{-2} -36 z^6 a^{-4} -13 z^6 a^{-6} +4 z^6 a^{-8} -19 z^6-10 a z^5+10 z^5 a^{-3} -14 z^5 a^{-5} -13 z^5 a^{-7} +z^5 a^{-9} -2 a^2 z^4+40 z^4 a^{-2} +34 z^4 a^{-4} +7 z^4 a^{-6} -5 z^4 a^{-8} +16 z^4+5 a z^3+4 z^3 a^{-1} +2 z^3 a^{-3} +11 z^3 a^{-5} +7 z^3 a^{-7} -z^3 a^{-9} -13 z^2 a^{-2} -13 z^2 a^{-4} -4 z^2 a^{-6} +z^2 a^{-8} -5 z^2-2 z a^{-5} -2 z a^{-7} - a^{-2} + a^{-4} + a^{-6} } |
| 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^8-2 q^6+2 q^4-2 q^2-1+4 q^{-2} -3 q^{-4} +6 q^{-6} - q^{-8} + q^{-10} + q^{-12} -4 q^{-14} +3 q^{-16} -2 q^{-18} + q^{-22} - q^{-24} } |
| The G2 invariant | Data:K11a264/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["K11a264"];
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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+6 t^3-16 t^2+28 t-33+28 t^{-1} -16 t^{-2} +6 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^8-2 z^6+2 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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{ 135, 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^8+4 q^7-9 q^6+14 q^5-19 q^4+22 q^3-21 q^2+19 q-13+8 q^{-1} -4 q^{-2} + q^{-3} } |
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} -5 z^6 a^{-2} +2 z^6 a^{-4} +z^6-9 z^4 a^{-2} +7 z^4 a^{-4} -z^4 a^{-6} +3 z^4-5 z^2 a^{-2} +7 z^2 a^{-4} -2 z^2 a^{-6} +2 z^2+ a^{-2} + a^{-4} - a^{-6} } |
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 2 z^{10} a^{-2} +2 z^{10} a^{-4} +6 z^9 a^{-1} +12 z^9 a^{-3} +6 z^9 a^{-5} +11 z^8 a^{-2} +13 z^8 a^{-4} +9 z^8 a^{-6} +7 z^8+4 a z^7-10 z^7 a^{-1} -22 z^7 a^{-3} +8 z^7 a^{-7} +a^2 z^6-39 z^6 a^{-2} -36 z^6 a^{-4} -13 z^6 a^{-6} +4 z^6 a^{-8} -19 z^6-10 a z^5+10 z^5 a^{-3} -14 z^5 a^{-5} -13 z^5 a^{-7} +z^5 a^{-9} -2 a^2 z^4+40 z^4 a^{-2} +34 z^4 a^{-4} +7 z^4 a^{-6} -5 z^4 a^{-8} +16 z^4+5 a z^3+4 z^3 a^{-1} +2 z^3 a^{-3} +11 z^3 a^{-5} +7 z^3 a^{-7} -z^3 a^{-9} -13 z^2 a^{-2} -13 z^2 a^{-4} -4 z^2 a^{-6} +z^2 a^{-8} -5 z^2-2 z a^{-5} -2 z a^{-7} - a^{-2} + a^{-4} + a^{-6} } |
"Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {K11a157, K11a305,}
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["K11a264"];
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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+6 t^3-16 t^2+28 t-33+28 t^{-1} -16 t^{-2} +6 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^8+4 q^7-9 q^6+14 q^5-19 q^4+22 q^3-21 q^2+19 q-13+8 q^{-1} -4 q^{-2} + q^{-3} } } |
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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{K11a157, K11a305,} |
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: | (2, 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 K11a264. 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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