K11a321
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 X12,3,13,4 X16,5,17,6 X22,8,1,7 X18,9,19,10 X20,11,21,12 X2,13,3,14 X8,15,9,16 X4,17,5,18 X10,19,11,20 X14,21,15,22 |
| Gauss code | 1, -7, 2, -9, 3, -1, 4, -8, 5, -10, 6, -2, 7, -11, 8, -3, 9, -5, 10, -6, 11, -4 |
| Dowker-Thistlethwaite code | 6 12 16 22 18 20 2 8 4 10 14 |
| 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{ -3 t^3+15 t^2-27 t+31-27 t^{-1} +15 t^{-2} -3 t^{-3} }[/math] |
| Conway polynomial | [math]\displaystyle{ -3 z^6-3 z^4+6 z^2+1 }[/math] |
| 2nd Alexander ideal (db, data sources) | [math]\displaystyle{ \{11,t+1\} }[/math] |
| Determinant and Signature | { 121, -4 } |
| Jones polynomial | [math]\displaystyle{ 1-3 q^{-1} +7 q^{-2} -12 q^{-3} +17 q^{-4} -19 q^{-5} +20 q^{-6} -17 q^{-7} +13 q^{-8} -8 q^{-9} +3 q^{-10} - q^{-11} }[/math] |
| HOMFLY-PT polynomial (db, data sources) | [math]\displaystyle{ -z^2 a^{10}-2 a^{10}+3 z^4 a^8+7 z^2 a^8+3 a^8-2 z^6 a^6-6 z^4 a^6-5 z^2 a^6-2 a^6-z^6 a^4-z^4 a^4+3 z^2 a^4+2 a^4+z^4 a^2+2 z^2 a^2 }[/math] |
| Kauffman polynomial (db, data sources) | [math]\displaystyle{ z^5 a^{13}-2 z^3 a^{13}+z a^{13}+3 z^6 a^{12}-4 z^4 a^{12}+z^2 a^{12}+6 z^7 a^{11}-10 z^5 a^{11}+8 z^3 a^{11}-4 z a^{11}+7 z^8 a^{10}-10 z^6 a^{10}+7 z^4 a^{10}-5 z^2 a^{10}+2 a^{10}+5 z^9 a^9-z^7 a^9-10 z^5 a^9+12 z^3 a^9-4 z a^9+2 z^{10} a^8+7 z^8 a^8-18 z^6 a^8+16 z^4 a^8-8 z^2 a^8+3 a^8+10 z^9 a^7-21 z^7 a^7+20 z^5 a^7-11 z^3 a^7+2 z a^7+2 z^{10} a^6+5 z^8 a^6-19 z^6 a^6+20 z^4 a^6-11 z^2 a^6+2 a^6+5 z^9 a^5-11 z^7 a^5+11 z^5 a^5-8 z^3 a^5+z a^5+5 z^8 a^4-13 z^6 a^4+12 z^4 a^4-7 z^2 a^4+2 a^4+3 z^7 a^3-8 z^5 a^3+5 z^3 a^3+z^6 a^2-3 z^4 a^2+2 z^2 a^2 }[/math] |
| The A2 invariant | Data:K11a321/QuantumInvariant/A2/1,0 |
| The G2 invariant | Data:K11a321/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["K11a321"];
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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{ -3 t^3+15 t^2-27 t+31-27 t^{-1} +15 t^{-2} -3 t^{-3} }[/math] |
In[5]:=
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Conway[K][z]
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Out[5]=
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[math]\displaystyle{ -3 z^6-3 z^4+6 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{ \{11,t+1\} }[/math] |
In[7]:=
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{KnotDet[K], KnotSignature[K]}
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Out[7]=
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{ 121, -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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[math]\displaystyle{ 1-3 q^{-1} +7 q^{-2} -12 q^{-3} +17 q^{-4} -19 q^{-5} +20 q^{-6} -17 q^{-7} +13 q^{-8} -8 q^{-9} +3 q^{-10} - q^{-11} }[/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^2 a^{10}-2 a^{10}+3 z^4 a^8+7 z^2 a^8+3 a^8-2 z^6 a^6-6 z^4 a^6-5 z^2 a^6-2 a^6-z^6 a^4-z^4 a^4+3 z^2 a^4+2 a^4+z^4 a^2+2 z^2 a^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^5 a^{13}-2 z^3 a^{13}+z a^{13}+3 z^6 a^{12}-4 z^4 a^{12}+z^2 a^{12}+6 z^7 a^{11}-10 z^5 a^{11}+8 z^3 a^{11}-4 z a^{11}+7 z^8 a^{10}-10 z^6 a^{10}+7 z^4 a^{10}-5 z^2 a^{10}+2 a^{10}+5 z^9 a^9-z^7 a^9-10 z^5 a^9+12 z^3 a^9-4 z a^9+2 z^{10} a^8+7 z^8 a^8-18 z^6 a^8+16 z^4 a^8-8 z^2 a^8+3 a^8+10 z^9 a^7-21 z^7 a^7+20 z^5 a^7-11 z^3 a^7+2 z a^7+2 z^{10} a^6+5 z^8 a^6-19 z^6 a^6+20 z^4 a^6-11 z^2 a^6+2 a^6+5 z^9 a^5-11 z^7 a^5+11 z^5 a^5-8 z^3 a^5+z a^5+5 z^8 a^4-13 z^6 a^4+12 z^4 a^4-7 z^2 a^4+2 a^4+3 z^7 a^3-8 z^5 a^3+5 z^3 a^3+z^6 a^2-3 z^4 a^2+2 z^2 a^2 }[/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["K11a321"];
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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{ -3 t^3+15 t^2-27 t+31-27 t^{-1} +15 t^{-2} -3 t^{-3} }[/math], [math]\displaystyle{ 1-3 q^{-1} +7 q^{-2} -12 q^{-3} +17 q^{-4} -19 q^{-5} +20 q^{-6} -17 q^{-7} +13 q^{-8} -8 q^{-9} +3 q^{-10} - q^{-11} }[/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: | (6, -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]-4 is the signature of K11a321. 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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