K11a31
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 X14,6,15,5 X2837 X16,10,17,9 X18,11,19,12 X6,14,7,13 X22,16,1,15 X20,18,21,17 X10,19,11,20 X12,22,13,21 |
| Gauss code | 1, -4, 2, -1, 3, -7, 4, -2, 5, -10, 6, -11, 7, -3, 8, -5, 9, -6, 10, -9, 11, -8 |
| Dowker-Thistlethwaite code | 4 8 14 2 16 18 6 22 20 10 12 |
| 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+14 t^2-28 t+35-28 t^{-1} +14 t^{-2} -3 t^{-3} }[/math] |
| Conway polynomial | [math]\displaystyle{ -3 z^6-4 z^4+z^2+1 }[/math] |
| 2nd Alexander ideal (db, data sources) | [math]\displaystyle{ \{1\} }[/math] |
| Determinant and Signature | { 125, 4 } |
| Jones polynomial | [math]\displaystyle{ -q^{11}+4 q^{10}-8 q^9+13 q^8-18 q^7+20 q^6-20 q^5+17 q^4-12 q^3+8 q^2-3 q+1 }[/math] |
| HOMFLY-PT polynomial (db, data sources) | [math]\displaystyle{ -z^6 a^{-4} -2 z^6 a^{-6} +z^4 a^{-2} -z^4 a^{-4} -7 z^4 a^{-6} +3 z^4 a^{-8} +2 z^2 a^{-2} +3 z^2 a^{-4} -10 z^2 a^{-6} +7 z^2 a^{-8} -z^2 a^{-10} + a^{-2} +3 a^{-4} -6 a^{-6} +4 a^{-8} - a^{-10} }[/math] |
| Kauffman polynomial (db, data sources) | [math]\displaystyle{ z^{10} a^{-6} +z^{10} a^{-8} +4 z^9 a^{-5} +8 z^9 a^{-7} +4 z^9 a^{-9} +5 z^8 a^{-4} +13 z^8 a^{-6} +15 z^8 a^{-8} +7 z^8 a^{-10} +3 z^7 a^{-3} -2 z^7 a^{-5} -4 z^7 a^{-7} +8 z^7 a^{-9} +7 z^7 a^{-11} +z^6 a^{-2} -11 z^6 a^{-4} -37 z^6 a^{-6} -35 z^6 a^{-8} -6 z^6 a^{-10} +4 z^6 a^{-12} -7 z^5 a^{-3} -10 z^5 a^{-5} -20 z^5 a^{-7} -29 z^5 a^{-9} -11 z^5 a^{-11} +z^5 a^{-13} -3 z^4 a^{-2} +8 z^4 a^{-4} +38 z^4 a^{-6} +30 z^4 a^{-8} -3 z^4 a^{-10} -6 z^4 a^{-12} +4 z^3 a^{-3} +10 z^3 a^{-5} +25 z^3 a^{-7} +26 z^3 a^{-9} +6 z^3 a^{-11} -z^3 a^{-13} +3 z^2 a^{-2} -6 z^2 a^{-4} -22 z^2 a^{-6} -13 z^2 a^{-8} +2 z^2 a^{-10} +2 z^2 a^{-12} -4 z a^{-5} -9 z a^{-7} -7 z a^{-9} -2 z a^{-11} - a^{-2} +3 a^{-4} +6 a^{-6} +4 a^{-8} + a^{-10} }[/math] |
| The A2 invariant | Data:K11a31/QuantumInvariant/A2/1,0 |
| The G2 invariant | Data:K11a31/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["K11a31"];
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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+14 t^2-28 t+35-28 t^{-1} +14 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^6-4 z^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{ \{1\} }[/math] |
In[7]:=
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{KnotDet[K], KnotSignature[K]}
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Out[7]=
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{ 125, 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{ -q^{11}+4 q^{10}-8 q^9+13 q^8-18 q^7+20 q^6-20 q^5+17 q^4-12 q^3+8 q^2-3 q+1 }[/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^6 a^{-4} -2 z^6 a^{-6} +z^4 a^{-2} -z^4 a^{-4} -7 z^4 a^{-6} +3 z^4 a^{-8} +2 z^2 a^{-2} +3 z^2 a^{-4} -10 z^2 a^{-6} +7 z^2 a^{-8} -z^2 a^{-10} + a^{-2} +3 a^{-4} -6 a^{-6} +4 a^{-8} - a^{-10} }[/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^{-6} +z^{10} a^{-8} +4 z^9 a^{-5} +8 z^9 a^{-7} +4 z^9 a^{-9} +5 z^8 a^{-4} +13 z^8 a^{-6} +15 z^8 a^{-8} +7 z^8 a^{-10} +3 z^7 a^{-3} -2 z^7 a^{-5} -4 z^7 a^{-7} +8 z^7 a^{-9} +7 z^7 a^{-11} +z^6 a^{-2} -11 z^6 a^{-4} -37 z^6 a^{-6} -35 z^6 a^{-8} -6 z^6 a^{-10} +4 z^6 a^{-12} -7 z^5 a^{-3} -10 z^5 a^{-5} -20 z^5 a^{-7} -29 z^5 a^{-9} -11 z^5 a^{-11} +z^5 a^{-13} -3 z^4 a^{-2} +8 z^4 a^{-4} +38 z^4 a^{-6} +30 z^4 a^{-8} -3 z^4 a^{-10} -6 z^4 a^{-12} +4 z^3 a^{-3} +10 z^3 a^{-5} +25 z^3 a^{-7} +26 z^3 a^{-9} +6 z^3 a^{-11} -z^3 a^{-13} +3 z^2 a^{-2} -6 z^2 a^{-4} -22 z^2 a^{-6} -13 z^2 a^{-8} +2 z^2 a^{-10} +2 z^2 a^{-12} -4 z a^{-5} -9 z a^{-7} -7 z a^{-9} -2 z a^{-11} - a^{-2} +3 a^{-4} +6 a^{-6} +4 a^{-8} + a^{-10} }[/math] |
"Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {K11a317,}
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["K11a31"];
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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+14 t^2-28 t+35-28 t^{-1} +14 t^{-2} -3 t^{-3} }[/math], [math]\displaystyle{ -q^{11}+4 q^{10}-8 q^9+13 q^8-18 q^7+20 q^6-20 q^5+17 q^4-12 q^3+8 q^2-3 q+1 }[/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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{K11a317,} |
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: | (1, 1) |
| 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 K11a31. 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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