K11n185
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 X18,4,19,3 X16,5,17,6 X14,8,15,7 X9,21,10,20 X4,12,5,11 X2,13,3,14 X22,16,1,15 X12,18,13,17 X19,9,20,8 X21,11,22,10 |
| Gauss code | 1, -7, 2, -6, 3, -1, 4, 10, -5, 11, 6, -9, 7, -4, 8, -3, 9, -2, -10, 5, -11, -8 |
| Dowker-Thistlethwaite code | 6 18 16 14 -20 4 2 22 12 -8 -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 -2 t^3+11 t^2-24 t+31-24 t^{-1} +11 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+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 \left\{3,t^2+1\right\}} |
| Determinant and Signature | { 105, 4 } |
| 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^{11}+5 q^{10}-10 q^9+14 q^8-18 q^7+18 q^6-16 q^5+13 q^4-7 q^3+3 q^2} |
| 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 -2 z^6 a^{-6} +3 z^4 a^{-4} -7 z^4 a^{-6} +3 z^4 a^{-8} +7 z^2 a^{-4} -9 z^2 a^{-6} +5 z^2 a^{-8} -z^2 a^{-10} +4 a^{-4} -4 a^{-6} + a^{-8} } |
| 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 3 z^9 a^{-7} +3 z^9 a^{-9} +6 z^8 a^{-6} +15 z^8 a^{-8} +9 z^8 a^{-10} +3 z^7 a^{-5} +7 z^7 a^{-7} +14 z^7 a^{-9} +10 z^7 a^{-11} -10 z^6 a^{-6} -24 z^6 a^{-8} -9 z^6 a^{-10} +5 z^6 a^{-12} -18 z^5 a^{-7} -35 z^5 a^{-9} -16 z^5 a^{-11} +z^5 a^{-13} +6 z^4 a^{-4} +16 z^4 a^{-6} +10 z^4 a^{-8} -5 z^4 a^{-10} -5 z^4 a^{-12} +z^3 a^{-5} +12 z^3 a^{-7} +16 z^3 a^{-9} +5 z^3 a^{-11} -9 z^2 a^{-4} -14 z^2 a^{-6} -3 z^2 a^{-8} +2 z^2 a^{-10} -3 z a^{-5} -3 z a^{-7} +z a^{-9} +z a^{-11} +4 a^{-4} +4 a^{-6} + a^{-8} } |
| 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 3 q^{-6} -2 q^{-8} +4 q^{-10} + q^{-12} -2 q^{-14} +4 q^{-16} -4 q^{-18} +2 q^{-20} -3 q^{-22} -2 q^{-24} +2 q^{-26} -3 q^{-28} +3 q^{-30} + q^{-32} - q^{-34} } |
| The G2 invariant | Data:K11n185/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["K11n185"];
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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+11 t^2-24 t+31-24 t^{-1} +11 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+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 \left\{3,t^2+1\right\}} |
In[7]:=
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{KnotDet[K], KnotSignature[K]}
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Out[7]=
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{ 105, 4 } |
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^{11}+5 q^{10}-10 q^9+14 q^8-18 q^7+18 q^6-16 q^5+13 q^4-7 q^3+3 q^2} |
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 -2 z^6 a^{-6} +3 z^4 a^{-4} -7 z^4 a^{-6} +3 z^4 a^{-8} +7 z^2 a^{-4} -9 z^2 a^{-6} +5 z^2 a^{-8} -z^2 a^{-10} +4 a^{-4} -4 a^{-6} + a^{-8} } |
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 3 z^9 a^{-7} +3 z^9 a^{-9} +6 z^8 a^{-6} +15 z^8 a^{-8} +9 z^8 a^{-10} +3 z^7 a^{-5} +7 z^7 a^{-7} +14 z^7 a^{-9} +10 z^7 a^{-11} -10 z^6 a^{-6} -24 z^6 a^{-8} -9 z^6 a^{-10} +5 z^6 a^{-12} -18 z^5 a^{-7} -35 z^5 a^{-9} -16 z^5 a^{-11} +z^5 a^{-13} +6 z^4 a^{-4} +16 z^4 a^{-6} +10 z^4 a^{-8} -5 z^4 a^{-10} -5 z^4 a^{-12} +z^3 a^{-5} +12 z^3 a^{-7} +16 z^3 a^{-9} +5 z^3 a^{-11} -9 z^2 a^{-4} -14 z^2 a^{-6} -3 z^2 a^{-8} +2 z^2 a^{-10} -3 z a^{-5} -3 z a^{-7} +z a^{-9} +z a^{-11} +4 a^{-4} +4 a^{-6} + a^{-8} } |
"Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {10_122,}
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["K11n185"];
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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 -2 t^3+11 t^2-24 t+31-24 t^{-1} +11 t^{-2} -2 t^{-3} } , 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^{11}+5 q^{10}-10 q^9+14 q^8-18 q^7+18 q^6-16 q^5+13 q^4-7 q^3+3 q^2} } |
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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{10_122,} |
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, 2) |
| 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=} 4 is the signature of K11n185. 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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