K11a234
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 X14,4,15,3 X16,6,17,5 X18,8,19,7 X20,10,21,9 X22,12,1,11 X2,14,3,13 X12,16,13,15 X6,18,7,17 X8,20,9,19 X10,22,11,21 |
| Gauss code | 1, -7, 2, -1, 3, -9, 4, -10, 5, -11, 6, -8, 7, -2, 8, -3, 9, -4, 10, -5, 11, -6 |
| Dowker-Thistlethwaite code | 4 14 16 18 20 22 2 12 6 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 | [math]\displaystyle{ 2 t^4-4 t^3+5 t^2-5 t+5-5 t^{-1} +5 t^{-2} -4 t^{-3} +2 t^{-4} }[/math] |
| Conway polynomial | [math]\displaystyle{ 2 z^8+12 z^6+21 z^4+11 z^2+1 }[/math] |
| 2nd Alexander ideal (db, data sources) | [math]\displaystyle{ \{1\} }[/math] |
| Determinant and Signature | { 37, 8 } |
| Jones polynomial | [math]\displaystyle{ -q^{15}+2 q^{14}-3 q^{13}+4 q^{12}-5 q^{11}+5 q^{10}-5 q^9+4 q^8-3 q^7+3 q^6-q^5+q^4 }[/math] |
| HOMFLY-PT polynomial (db, data sources) | [math]\displaystyle{ z^8 a^{-8} +z^8 a^{-10} +7 z^6 a^{-8} +6 z^6 a^{-10} -z^6 a^{-12} +16 z^4 a^{-8} +10 z^4 a^{-10} -5 z^4 a^{-12} +14 z^2 a^{-8} +3 z^2 a^{-10} -6 z^2 a^{-12} +4 a^{-8} -2 a^{-10} - a^{-12} }[/math] |
| Kauffman polynomial (db, data sources) | [math]\displaystyle{ z^{10} a^{-10} +z^{10} a^{-12} +z^9 a^{-9} +3 z^9 a^{-11} +2 z^9 a^{-13} +z^8 a^{-8} -5 z^8 a^{-10} -4 z^8 a^{-12} +2 z^8 a^{-14} -5 z^7 a^{-9} -16 z^7 a^{-11} -9 z^7 a^{-13} +2 z^7 a^{-15} -7 z^6 a^{-8} +6 z^6 a^{-10} +5 z^6 a^{-12} -6 z^6 a^{-14} +2 z^6 a^{-16} +5 z^5 a^{-9} +25 z^5 a^{-11} +14 z^5 a^{-13} -4 z^5 a^{-15} +2 z^5 a^{-17} +16 z^4 a^{-8} -7 z^4 a^{-12} +5 z^4 a^{-14} -2 z^4 a^{-16} +2 z^4 a^{-18} +3 z^3 a^{-9} -13 z^3 a^{-11} -11 z^3 a^{-13} +3 z^3 a^{-15} -z^3 a^{-17} +z^3 a^{-19} -14 z^2 a^{-8} -3 z^2 a^{-10} +6 z^2 a^{-12} -2 z^2 a^{-14} +z^2 a^{-16} -2 z^2 a^{-18} -3 z a^{-9} +z a^{-11} +2 z a^{-13} -z a^{-15} -z a^{-19} +4 a^{-8} +2 a^{-10} - a^{-12} }[/math] |
| The A2 invariant | [math]\displaystyle{ q^{-14} +2 q^{-18} + q^{-20} +2 q^{-22} + q^{-24} -2 q^{-30} - q^{-34} - q^{-44} }[/math] |
| The G2 invariant | Data:K11a234/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["K11a234"];
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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^4-4 t^3+5 t^2-5 t+5-5 t^{-1} +5 t^{-2} -4 t^{-3} +2 t^{-4} }[/math] |
In[5]:=
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Conway[K][z]
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Out[5]=
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[math]\displaystyle{ 2 z^8+12 z^6+21 z^4+11 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{ \{1\} }[/math] |
In[7]:=
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{KnotDet[K], KnotSignature[K]}
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Out[7]=
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{ 37, 8 } |
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^{15}+2 q^{14}-3 q^{13}+4 q^{12}-5 q^{11}+5 q^{10}-5 q^9+4 q^8-3 q^7+3 q^6-q^5+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} +z^8 a^{-10} +7 z^6 a^{-8} +6 z^6 a^{-10} -z^6 a^{-12} +16 z^4 a^{-8} +10 z^4 a^{-10} -5 z^4 a^{-12} +14 z^2 a^{-8} +3 z^2 a^{-10} -6 z^2 a^{-12} +4 a^{-8} -2 a^{-10} - a^{-12} }[/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^{10} a^{-10} +z^{10} a^{-12} +z^9 a^{-9} +3 z^9 a^{-11} +2 z^9 a^{-13} +z^8 a^{-8} -5 z^8 a^{-10} -4 z^8 a^{-12} +2 z^8 a^{-14} -5 z^7 a^{-9} -16 z^7 a^{-11} -9 z^7 a^{-13} +2 z^7 a^{-15} -7 z^6 a^{-8} +6 z^6 a^{-10} +5 z^6 a^{-12} -6 z^6 a^{-14} +2 z^6 a^{-16} +5 z^5 a^{-9} +25 z^5 a^{-11} +14 z^5 a^{-13} -4 z^5 a^{-15} +2 z^5 a^{-17} +16 z^4 a^{-8} -7 z^4 a^{-12} +5 z^4 a^{-14} -2 z^4 a^{-16} +2 z^4 a^{-18} +3 z^3 a^{-9} -13 z^3 a^{-11} -11 z^3 a^{-13} +3 z^3 a^{-15} -z^3 a^{-17} +z^3 a^{-19} -14 z^2 a^{-8} -3 z^2 a^{-10} +6 z^2 a^{-12} -2 z^2 a^{-14} +z^2 a^{-16} -2 z^2 a^{-18} -3 z a^{-9} +z a^{-11} +2 z a^{-13} -z a^{-15} -z a^{-19} +4 a^{-8} +2 a^{-10} - a^{-12} }[/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["K11a234"];
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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^4-4 t^3+5 t^2-5 t+5-5 t^{-1} +5 t^{-2} -4 t^{-3} +2 t^{-4} }[/math], [math]\displaystyle{ -q^{15}+2 q^{14}-3 q^{13}+4 q^{12}-5 q^{11}+5 q^{10}-5 q^9+4 q^8-3 q^7+3 q^6-q^5+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: | (11, 35) |
| 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]8 is the signature of K11a234. 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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