K11n77
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 X5,15,6,14 X2837 X20,10,21,9 X11,17,12,16 X13,19,14,18 X15,7,16,6 X17,13,18,12 X22,20,1,19 X10,22,11,21 |
| Gauss code | 1, -4, 2, -1, -3, 8, 4, -2, 5, -11, -6, 9, -7, 3, -8, 6, -9, 7, 10, -5, 11, -10 |
| Dowker-Thistlethwaite code | 4 8 -14 2 20 -16 -18 -6 -12 22 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 | [math]\displaystyle{ t^4-t^3-2 t^2+8 t-11+8 t^{-1} -2 t^{-2} - t^{-3} + t^{-4} }[/math] |
| Conway polynomial | [math]\displaystyle{ z^8+7 z^6+12 z^4+7 z^2+1 }[/math] |
| 2nd Alexander ideal (db, data sources) | [math]\displaystyle{ \left\{t^2-t+1\right\} }[/math] |
| Determinant and Signature | { 27, 6 } |
| Jones polynomial | [math]\displaystyle{ -q^{14}+3 q^{13}-4 q^{12}+5 q^{11}-6 q^{10}+4 q^9-4 q^8+2 q^7+q^6+q^4 }[/math] |
| HOMFLY-PT polynomial (db, data sources) | [math]\displaystyle{ z^8 a^{-8} +8 z^6 a^{-8} -z^6 a^{-10} +21 z^4 a^{-8} -9 z^4 a^{-10} +23 z^2 a^{-8} -20 z^2 a^{-10} +4 z^2 a^{-12} +9 a^{-8} -13 a^{-10} +6 a^{-12} - a^{-14} }[/math] |
| Kauffman polynomial (db, data sources) | [math]\displaystyle{ z^8 a^{-8} +z^8 a^{-10} +z^8 a^{-12} +z^8 a^{-14} +z^7 a^{-9} +2 z^7 a^{-11} +4 z^7 a^{-13} +3 z^7 a^{-15} -8 z^6 a^{-8} -10 z^6 a^{-10} -4 z^6 a^{-12} +z^6 a^{-14} +3 z^6 a^{-16} -9 z^5 a^{-9} -15 z^5 a^{-11} -15 z^5 a^{-13} -8 z^5 a^{-15} +z^5 a^{-17} +21 z^4 a^{-8} +29 z^4 a^{-10} +8 z^4 a^{-12} -8 z^4 a^{-14} -8 z^4 a^{-16} +20 z^3 a^{-9} +36 z^3 a^{-11} +23 z^3 a^{-13} +5 z^3 a^{-15} -2 z^3 a^{-17} -23 z^2 a^{-8} -31 z^2 a^{-10} -6 z^2 a^{-12} +5 z^2 a^{-14} +3 z^2 a^{-16} -13 z a^{-9} -22 z a^{-11} -12 z a^{-13} -3 z a^{-15} +9 a^{-8} +13 a^{-10} +6 a^{-12} + a^{-14} }[/math] |
| The A2 invariant | [math]\displaystyle{ q^{-14} + q^{-16} +2 q^{-18} +4 q^{-20} +2 q^{-22} + q^{-24} -5 q^{-28} -3 q^{-30} -4 q^{-32} +2 q^{-36} + q^{-38} +3 q^{-40} - q^{-42} - q^{-44} }[/math] |
| The G2 invariant | Data:K11n77/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["K11n77"];
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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{ t^4-t^3-2 t^2+8 t-11+8 t^{-1} -2 t^{-2} - t^{-3} + t^{-4} }[/math] |
In[5]:=
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Conway[K][z]
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Out[5]=
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[math]\displaystyle{ z^8+7 z^6+12 z^4+7 z^2+1 }[/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{ \left\{t^2-t+1\right\} }[/math] |
In[7]:=
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{KnotDet[K], KnotSignature[K]}
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Out[7]=
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{ 27, 6 } |
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^{14}+3 q^{13}-4 q^{12}+5 q^{11}-6 q^{10}+4 q^9-4 q^8+2 q^7+q^6+q^4 }[/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^8 a^{-8} +8 z^6 a^{-8} -z^6 a^{-10} +21 z^4 a^{-8} -9 z^4 a^{-10} +23 z^2 a^{-8} -20 z^2 a^{-10} +4 z^2 a^{-12} +9 a^{-8} -13 a^{-10} +6 a^{-12} - a^{-14} }[/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^8 a^{-8} +z^8 a^{-10} +z^8 a^{-12} +z^8 a^{-14} +z^7 a^{-9} +2 z^7 a^{-11} +4 z^7 a^{-13} +3 z^7 a^{-15} -8 z^6 a^{-8} -10 z^6 a^{-10} -4 z^6 a^{-12} +z^6 a^{-14} +3 z^6 a^{-16} -9 z^5 a^{-9} -15 z^5 a^{-11} -15 z^5 a^{-13} -8 z^5 a^{-15} +z^5 a^{-17} +21 z^4 a^{-8} +29 z^4 a^{-10} +8 z^4 a^{-12} -8 z^4 a^{-14} -8 z^4 a^{-16} +20 z^3 a^{-9} +36 z^3 a^{-11} +23 z^3 a^{-13} +5 z^3 a^{-15} -2 z^3 a^{-17} -23 z^2 a^{-8} -31 z^2 a^{-10} -6 z^2 a^{-12} +5 z^2 a^{-14} +3 z^2 a^{-16} -13 z a^{-9} -22 z a^{-11} -12 z a^{-13} -3 z a^{-15} +9 a^{-8} +13 a^{-10} +6 a^{-12} + a^{-14} }[/math] |
"Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {}
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["K11n77"];
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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{ t^4-t^3-2 t^2+8 t-11+8 t^{-1} -2 t^{-2} - t^{-3} + t^{-4} }[/math], [math]\displaystyle{ -q^{14}+3 q^{13}-4 q^{12}+5 q^{11}-6 q^{10}+4 q^9-4 q^8+2 q^7+q^6+q^4 }[/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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{} |
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: | (7, 16) |
| 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]6 is the signature of K11n77. 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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