K11a67
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 X8394 X16,5,17,6 X12,8,13,7 X2,9,3,10 X18,12,19,11 X22,13,1,14 X20,15,21,16 X10,18,11,17 X6,19,7,20 X14,21,15,22 |
| Gauss code | 1, -5, 2, -1, 3, -10, 4, -2, 5, -9, 6, -4, 7, -11, 8, -3, 9, -6, 10, -8, 11, -7 |
| Dowker-Thistlethwaite code | 4 8 16 12 2 18 22 20 10 6 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{ -2 t^3+12 t^2-29 t+39-29 t^{-1} +12 t^{-2} -2 t^{-3} }[/math] |
| Conway polynomial | [math]\displaystyle{ -2 z^6+z^2+1 }[/math] |
| 2nd Alexander ideal (db, data sources) | [math]\displaystyle{ \{1\} }[/math] |
| Determinant and Signature | { 125, 0 } |
| Jones polynomial | [math]\displaystyle{ q^4-4 q^3+9 q^2-14 q+18-20 q^{-1} +20 q^{-2} -16 q^{-3} +12 q^{-4} -7 q^{-5} +3 q^{-6} - q^{-7} }[/math] |
| HOMFLY-PT polynomial (db, data sources) | [math]\displaystyle{ -z^2 a^6-a^6+2 z^4 a^4+3 z^2 a^4+a^4-z^6 a^2-z^4 a^2+z^2 a^2+2 a^2-z^6-2 z^4-3 z^2-2+z^4 a^{-2} +z^2 a^{-2} + a^{-2} }[/math] |
| Kauffman polynomial (db, data sources) | [math]\displaystyle{ 2 a^4 z^{10}+2 a^2 z^{10}+4 a^5 z^9+11 a^3 z^9+7 a z^9+3 a^6 z^8+4 a^4 z^8+13 a^2 z^8+12 z^8+a^7 z^7-11 a^5 z^7-25 a^3 z^7+13 z^7 a^{-1} -11 a^6 z^6-27 a^4 z^6-40 a^2 z^6+9 z^6 a^{-2} -15 z^6-4 a^7 z^5+5 a^5 z^5+7 a^3 z^5-24 a z^5-18 z^5 a^{-1} +4 z^5 a^{-3} +13 a^6 z^4+29 a^4 z^4+24 a^2 z^4-8 z^4 a^{-2} +z^4 a^{-4} -z^4+5 a^7 z^3+5 a^5 z^3+8 a^3 z^3+18 a z^3+9 z^3 a^{-1} -z^3 a^{-3} -6 a^6 z^2-9 a^4 z^2+3 z^2 a^{-2} +6 z^2-2 a^7 z-3 a^5 z-3 a^3 z-4 a z-2 z a^{-1} +a^6+a^4-2 a^2- a^{-2} -2 }[/math] |
| The A2 invariant | Data:K11a67/QuantumInvariant/A2/1,0 |
| The G2 invariant | Data:K11a67/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["K11a67"];
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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^3+12 t^2-29 t+39-29 t^{-1} +12 t^{-2} -2 t^{-3} }[/math] |
In[5]:=
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Conway[K][z]
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Out[5]=
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[math]\displaystyle{ -2 z^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{ \{1\} }[/math] |
In[7]:=
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{KnotDet[K], KnotSignature[K]}
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Out[7]=
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{ 125, 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^4-4 q^3+9 q^2-14 q+18-20 q^{-1} +20 q^{-2} -16 q^{-3} +12 q^{-4} -7 q^{-5} +3 q^{-6} - q^{-7} }[/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^6-a^6+2 z^4 a^4+3 z^2 a^4+a^4-z^6 a^2-z^4 a^2+z^2 a^2+2 a^2-z^6-2 z^4-3 z^2-2+z^4 a^{-2} +z^2 a^{-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{ 2 a^4 z^{10}+2 a^2 z^{10}+4 a^5 z^9+11 a^3 z^9+7 a z^9+3 a^6 z^8+4 a^4 z^8+13 a^2 z^8+12 z^8+a^7 z^7-11 a^5 z^7-25 a^3 z^7+13 z^7 a^{-1} -11 a^6 z^6-27 a^4 z^6-40 a^2 z^6+9 z^6 a^{-2} -15 z^6-4 a^7 z^5+5 a^5 z^5+7 a^3 z^5-24 a z^5-18 z^5 a^{-1} +4 z^5 a^{-3} +13 a^6 z^4+29 a^4 z^4+24 a^2 z^4-8 z^4 a^{-2} +z^4 a^{-4} -z^4+5 a^7 z^3+5 a^5 z^3+8 a^3 z^3+18 a z^3+9 z^3 a^{-1} -z^3 a^{-3} -6 a^6 z^2-9 a^4 z^2+3 z^2 a^{-2} +6 z^2-2 a^7 z-3 a^5 z-3 a^3 z-4 a z-2 z a^{-1} +a^6+a^4-2 a^2- a^{-2} -2 }[/math] |
"Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {K11a104, K11a168,}
Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {K11a317,}
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["K11a67"];
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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^3+12 t^2-29 t+39-29 t^{-1} +12 t^{-2} -2 t^{-3} }[/math], [math]\displaystyle{ q^4-4 q^3+9 q^2-14 q+18-20 q^{-1} +20 q^{-2} -16 q^{-3} +12 q^{-4} -7 q^{-5} +3 q^{-6} - q^{-7} }[/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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{K11a104, K11a168,} |
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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{K11a317,} |
Vassiliev invariants
| V2 and V3: | (1, -3) |
| 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 K11a67. 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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