K11n67
From Knot Atlas
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 (Knotscape image) |
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. |
Knot presentations
| Planar diagram presentation | X4251 X8493 X14,5,15,6 X2837 X9,19,10,18 X11,17,12,16 X13,20,14,21 X6,15,7,16 X17,11,18,10 X19,1,20,22 X21,12,22,13 |
| Gauss code | 1, -4, 2, -1, 3, -8, 4, -2, -5, 9, -6, 11, -7, -3, 8, 6, -9, 5, -10, 7, -11, 10 |
| Dowker-Thistlethwaite code | 4 8 14 2 -18 -16 -20 6 -10 -22 -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 | [math]\displaystyle{ -2 t+5-2 t^{-1} }[/math] |
| Conway polynomial | [math]\displaystyle{ 1-2 z^2 }[/math] |
| 2nd Alexander ideal (db, data sources) | [math]\displaystyle{ \{1\} }[/math] |
| Determinant and Signature | { 9, 0 } |
| Jones polynomial | [math]\displaystyle{ -q^7+2 q^6-2 q^5+3 q^4-2 q^3+q^2-q+ q^{-1} - q^{-2} + q^{-3} }[/math] |
| HOMFLY-PT polynomial (db, data sources) | [math]\displaystyle{ z^4 a^{-4} -z^4+a^2 z^2-z^2 a^{-2} +3 z^2 a^{-4} -z^2 a^{-6} -4 z^2+2 a^2- a^{-2} +3 a^{-4} - a^{-6} -2 }[/math] |
| Kauffman polynomial (db, data sources) | [math]\displaystyle{ z^9 a^{-3} +z^9 a^{-5} +z^8 a^{-2} +3 z^8 a^{-4} +2 z^8 a^{-6} +a z^7-6 z^7 a^{-3} -4 z^7 a^{-5} +z^7 a^{-7} +a^2 z^6-7 z^6 a^{-2} -18 z^6 a^{-4} -11 z^6 a^{-6} +z^6-5 a z^5-z^5 a^{-1} +10 z^5 a^{-3} +z^5 a^{-5} -5 z^5 a^{-7} -5 a^2 z^4+12 z^4 a^{-2} +31 z^4 a^{-4} +17 z^4 a^{-6} -7 z^4+5 a z^3-z^3 a^{-1} -9 z^3 a^{-3} +3 z^3 a^{-5} +6 z^3 a^{-7} +6 a^2 z^2-8 z^2 a^{-2} -20 z^2 a^{-4} -9 z^2 a^{-6} +9 z^2-a z+2 z a^{-1} +4 z a^{-3} -z a^{-7} -2 a^2+ a^{-2} +3 a^{-4} + a^{-6} -2 }[/math] |
| The A2 invariant | Data:K11n67/QuantumInvariant/A2/1,0 |
| The G2 invariant | Data:K11n67/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["K11n67"];
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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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[math]\displaystyle{ -2 t+5-2 t^{-1} }[/math] |
In[5]:=
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Conway[K][z]
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Out[5]=
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[math]\displaystyle{ 1-2 z^2 }[/math] |
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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[math]\displaystyle{ \{1\} }[/math] |
In[7]:=
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{KnotDet[K], KnotSignature[K]}
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Out[7]=
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{ 9, 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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[math]\displaystyle{ -q^7+2 q^6-2 q^5+3 q^4-2 q^3+q^2-q+ q^{-1} - q^{-2} + q^{-3} }[/math] |
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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[math]\displaystyle{ z^4 a^{-4} -z^4+a^2 z^2-z^2 a^{-2} +3 z^2 a^{-4} -z^2 a^{-6} -4 z^2+2 a^2- a^{-2} +3 a^{-4} - a^{-6} -2 }[/math] |
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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[math]\displaystyle{ z^9 a^{-3} +z^9 a^{-5} +z^8 a^{-2} +3 z^8 a^{-4} +2 z^8 a^{-6} +a z^7-6 z^7 a^{-3} -4 z^7 a^{-5} +z^7 a^{-7} +a^2 z^6-7 z^6 a^{-2} -18 z^6 a^{-4} -11 z^6 a^{-6} +z^6-5 a z^5-z^5 a^{-1} +10 z^5 a^{-3} +z^5 a^{-5} -5 z^5 a^{-7} -5 a^2 z^4+12 z^4 a^{-2} +31 z^4 a^{-4} +17 z^4 a^{-6} -7 z^4+5 a z^3-z^3 a^{-1} -9 z^3 a^{-3} +3 z^3 a^{-5} +6 z^3 a^{-7} +6 a^2 z^2-8 z^2 a^{-2} -20 z^2 a^{-4} -9 z^2 a^{-6} +9 z^2-a z+2 z a^{-1} +4 z a^{-3} -z a^{-7} -2 a^2+ a^{-2} +3 a^{-4} + a^{-6} -2 }[/math] |
"Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {6_1, 9_46, K11n97, K11n139,}
Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {}
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["K11n67"];
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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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{ [math]\displaystyle{ -2 t+5-2 t^{-1} }[/math], [math]\displaystyle{ -q^7+2 q^6-2 q^5+3 q^4-2 q^3+q^2-q+ q^{-1} - q^{-2} + q^{-3} }[/math] } |
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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{6_1, 9_46, K11n97, K11n139,} |
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, 1) |
| 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 [math]\displaystyle{ t^rq^j }[/math] are shown, along with their alternating sums [math]\displaystyle{ \chi }[/math] (fixed [math]\displaystyle{ j }[/math], alternation over [math]\displaystyle{ r }[/math]). The squares with yellow highlighting are those on the "critical diagonals", where [math]\displaystyle{ j-2r=s+1 }[/math] or [math]\displaystyle{ j-2r=s-1 }[/math], where [math]\displaystyle{ s= }[/math]0 is the signature of K11n67. 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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