K11a311
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 X12,4,13,3 X14,6,15,5 X20,8,21,7 X18,10,19,9 X4,12,5,11 X2,14,3,13 X22,15,1,16 X10,18,11,17 X8,20,9,19 X16,21,17,22 |
| Gauss code | 1, -7, 2, -6, 3, -1, 4, -10, 5, -9, 6, -2, 7, -3, 8, -11, 9, -5, 10, -4, 11, -8 |
| Dowker-Thistlethwaite code | 6 12 14 20 18 4 2 22 10 8 16 |
| 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 -4 t^2+20 t-31+20 t^{-1} -4 t^{-2} } |
| 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 -4 z^4+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 | { 79, 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^{10}+2 q^9-4 q^8+7 q^7-10 q^6+12 q^5-12 q^4+12 q^3-9 q^2+6 q-3+ q^{-1} } |
| 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^4 a^{-2} -2 z^4 a^{-4} -z^4 a^{-6} +z^2 a^{-2} -z^2 a^{-4} +z^2 a^{-6} +2 z^2 a^{-8} +z^2+ a^{-2} + a^{-8} - a^{-10} } |
| 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^{10} a^{-6} +z^{10} a^{-8} +3 z^9 a^{-5} +5 z^9 a^{-7} +2 z^9 a^{-9} +5 z^8 a^{-4} +3 z^8 a^{-6} +2 z^8 a^{-10} +6 z^7 a^{-3} -3 z^7 a^{-5} -17 z^7 a^{-7} -7 z^7 a^{-9} +z^7 a^{-11} +5 z^6 a^{-2} -7 z^6 a^{-4} -11 z^6 a^{-6} -8 z^6 a^{-8} -9 z^6 a^{-10} +3 z^5 a^{-1} -9 z^5 a^{-3} +z^5 a^{-5} +25 z^5 a^{-7} +7 z^5 a^{-9} -5 z^5 a^{-11} -6 z^4 a^{-2} +3 z^4 a^{-4} +9 z^4 a^{-6} +11 z^4 a^{-8} +12 z^4 a^{-10} +z^4-3 z^3 a^{-1} +6 z^3 a^{-3} -3 z^3 a^{-5} -23 z^3 a^{-7} -4 z^3 a^{-9} +7 z^3 a^{-11} +3 z^2 a^{-2} +2 z^2 a^{-4} -4 z^2 a^{-6} -7 z^2 a^{-8} -5 z^2 a^{-10} -z^2+2 z a^{-5} +6 z a^{-7} +2 z a^{-9} -2 z a^{-11} - a^{-2} + a^{-8} + a^{-10} } |
| The A2 invariant | Data:K11a311/QuantumInvariant/A2/1,0 |
| The G2 invariant | Data:K11a311/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["K11a311"];
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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 -4 t^2+20 t-31+20 t^{-1} -4 t^{-2} } |
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 -4 z^4+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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{ 79, 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^{10}+2 q^9-4 q^8+7 q^7-10 q^6+12 q^5-12 q^4+12 q^3-9 q^2+6 q-3+ q^{-1} } |
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^4 a^{-2} -2 z^4 a^{-4} -z^4 a^{-6} +z^2 a^{-2} -z^2 a^{-4} +z^2 a^{-6} +2 z^2 a^{-8} +z^2+ a^{-2} + a^{-8} - a^{-10} } |
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^{10} a^{-6} +z^{10} a^{-8} +3 z^9 a^{-5} +5 z^9 a^{-7} +2 z^9 a^{-9} +5 z^8 a^{-4} +3 z^8 a^{-6} +2 z^8 a^{-10} +6 z^7 a^{-3} -3 z^7 a^{-5} -17 z^7 a^{-7} -7 z^7 a^{-9} +z^7 a^{-11} +5 z^6 a^{-2} -7 z^6 a^{-4} -11 z^6 a^{-6} -8 z^6 a^{-8} -9 z^6 a^{-10} +3 z^5 a^{-1} -9 z^5 a^{-3} +z^5 a^{-5} +25 z^5 a^{-7} +7 z^5 a^{-9} -5 z^5 a^{-11} -6 z^4 a^{-2} +3 z^4 a^{-4} +9 z^4 a^{-6} +11 z^4 a^{-8} +12 z^4 a^{-10} +z^4-3 z^3 a^{-1} +6 z^3 a^{-3} -3 z^3 a^{-5} -23 z^3 a^{-7} -4 z^3 a^{-9} +7 z^3 a^{-11} +3 z^2 a^{-2} +2 z^2 a^{-4} -4 z^2 a^{-6} -7 z^2 a^{-8} -5 z^2 a^{-10} -z^2+2 z a^{-5} +6 z a^{-7} +2 z a^{-9} -2 z a^{-11} - a^{-2} + a^{-8} + a^{-10} } |
"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["K11a311"];
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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 -4 t^2+20 t-31+20 t^{-1} -4 t^{-2} } , 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^{10}+2 q^9-4 q^8+7 q^7-10 q^6+12 q^5-12 q^4+12 q^3-9 q^2+6 q-3+ q^{-1} } } |
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: | (4, 10) |
| 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 K11a311. 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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