K11n91
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 X10,3,11,4 X12,6,13,5 X7,16,8,17 X9,18,10,19 X2,11,3,12 X20,13,21,14 X15,8,16,9 X17,6,18,7 X22,19,1,20 X14,21,15,22 |
| Gauss code | 1, -6, 2, -1, 3, 9, -4, 8, -5, -2, 6, -3, 7, -11, -8, 4, -9, 5, 10, -7, 11, -10 |
| Dowker-Thistlethwaite code | 4 10 12 -16 -18 2 20 -8 -6 22 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{ -t^2+8 t-13+8 t^{-1} - t^{-2} }[/math] |
| Conway polynomial | [math]\displaystyle{ -z^4+4 z^2+1 }[/math] |
| 2nd Alexander ideal (db, data sources) | [math]\displaystyle{ \left\{2,t^2+t+1\right\} }[/math] |
| Determinant and Signature | { 31, -2 } |
| Jones polynomial | [math]\displaystyle{ q^{-1} -2 q^{-2} +4 q^{-3} -4 q^{-4} +5 q^{-5} -5 q^{-6} +4 q^{-7} -3 q^{-8} +2 q^{-9} - q^{-10} }[/math] |
| HOMFLY-PT polynomial (db, data sources) | [math]\displaystyle{ -a^{10}+2 z^2 a^8+2 a^8-z^4 a^6-2 z^2 a^6-3 a^6+3 z^2 a^4+3 a^4+z^2 a^2 }[/math] |
| Kauffman polynomial (db, data sources) | [math]\displaystyle{ z^7 a^{11}-5 z^5 a^{11}+7 z^3 a^{11}-3 z a^{11}+2 z^8 a^{10}-10 z^6 a^{10}+14 z^4 a^{10}-6 z^2 a^{10}+a^{10}+z^9 a^9-2 z^7 a^9-6 z^5 a^9+11 z^3 a^9-3 z a^9+4 z^8 a^8-18 z^6 a^8+23 z^4 a^8-11 z^2 a^8+2 a^8+z^9 a^7-2 z^7 a^7-2 z^5 a^7+2 z^3 a^7-z a^7+2 z^8 a^6-8 z^6 a^6+13 z^4 a^6-12 z^2 a^6+3 a^6+z^7 a^5-z^5 a^5-z a^5+4 z^4 a^4-6 z^2 a^4+3 a^4+2 z^3 a^3+z^2 a^2 }[/math] |
| The A2 invariant | Data:K11n91/QuantumInvariant/A2/1,0 |
| The G2 invariant | Data:K11n91/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["K11n91"];
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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^2+8 t-13+8 t^{-1} - t^{-2} }[/math] |
In[5]:=
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Conway[K][z]
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Out[5]=
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[math]\displaystyle{ -z^4+4 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\{2,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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{ 31, -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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[math]\displaystyle{ q^{-1} -2 q^{-2} +4 q^{-3} -4 q^{-4} +5 q^{-5} -5 q^{-6} +4 q^{-7} -3 q^{-8} +2 q^{-9} - q^{-10} }[/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{ -a^{10}+2 z^2 a^8+2 a^8-z^4 a^6-2 z^2 a^6-3 a^6+3 z^2 a^4+3 a^4+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^7 a^{11}-5 z^5 a^{11}+7 z^3 a^{11}-3 z a^{11}+2 z^8 a^{10}-10 z^6 a^{10}+14 z^4 a^{10}-6 z^2 a^{10}+a^{10}+z^9 a^9-2 z^7 a^9-6 z^5 a^9+11 z^3 a^9-3 z a^9+4 z^8 a^8-18 z^6 a^8+23 z^4 a^8-11 z^2 a^8+2 a^8+z^9 a^7-2 z^7 a^7-2 z^5 a^7+2 z^3 a^7-z a^7+2 z^8 a^6-8 z^6 a^6+13 z^4 a^6-12 z^2 a^6+3 a^6+z^7 a^5-z^5 a^5-z a^5+4 z^4 a^4-6 z^2 a^4+3 a^4+2 z^3 a^3+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["K11n91"];
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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^2+8 t-13+8 t^{-1} - t^{-2} }[/math], [math]\displaystyle{ q^{-1} -2 q^{-2} +4 q^{-3} -4 q^{-4} +5 q^{-5} -5 q^{-6} +4 q^{-7} -3 q^{-8} +2 q^{-9} - q^{-10} }[/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: | (4, -9) |
| 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]-2 is the signature of K11n91. 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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