K11a41
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 X14,5,15,6 X2837 X18,10,19,9 X20,12,21,11 X16,13,17,14 X6,15,7,16 X22,18,1,17 X12,20,13,19 X10,22,11,21 |
| Gauss code | 1, -4, 2, -1, 3, -8, 4, -2, 5, -11, 6, -10, 7, -3, 8, -7, 9, -5, 10, -6, 11, -9 |
| Dowker-Thistlethwaite code | 4 8 14 2 18 20 16 6 22 12 10 |
| A Braid Representative | {{{braid_table}}} |
| 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^3-11 t^2+27 t-35+27 t^{-1} -11 t^{-2} +2 t^{-3} }[/math] |
| Conway polynomial | [math]\displaystyle{ 2 z^6+z^4+z^2+1 }[/math] |
| 2nd Alexander ideal (db, data sources) | [math]\displaystyle{ \{1\} }[/math] |
| Determinant and Signature | { 115, 2 } |
| Jones polynomial | [math]\displaystyle{ q^9-3 q^8+6 q^7-11 q^6+15 q^5-18 q^4+19 q^3-16 q^2+13 q-8+4 q^{-1} - q^{-2} }[/math] |
| HOMFLY-PT polynomial (db, data sources) | [math]\displaystyle{ z^6 a^{-2} +z^6 a^{-4} +2 z^4 a^{-2} +2 z^4 a^{-4} -2 z^4 a^{-6} -z^4+2 z^2 a^{-2} +3 z^2 a^{-4} -4 z^2 a^{-6} +z^2 a^{-8} -z^2+ a^{-2} +2 a^{-4} -3 a^{-6} + a^{-8} }[/math] |
| Kauffman polynomial (db, data sources) | [math]\displaystyle{ z^{10} a^{-4} +z^{10} a^{-6} +4 z^9 a^{-3} +7 z^9 a^{-5} +3 z^9 a^{-7} +7 z^8 a^{-2} +12 z^8 a^{-4} +9 z^8 a^{-6} +4 z^8 a^{-8} +7 z^7 a^{-1} +4 z^7 a^{-3} -6 z^7 a^{-5} +3 z^7 a^{-9} -8 z^6 a^{-2} -29 z^6 a^{-4} -27 z^6 a^{-6} -9 z^6 a^{-8} +z^6 a^{-10} +4 z^6+a z^5-11 z^5 a^{-1} -18 z^5 a^{-3} -12 z^5 a^{-5} -15 z^5 a^{-7} -9 z^5 a^{-9} +26 z^4 a^{-4} +28 z^4 a^{-6} +5 z^4 a^{-8} -3 z^4 a^{-10} -6 z^4-a z^3+4 z^3 a^{-1} +14 z^3 a^{-3} +20 z^3 a^{-5} +19 z^3 a^{-7} +8 z^3 a^{-9} +z^2 a^{-2} -10 z^2 a^{-4} -13 z^2 a^{-6} -2 z^2 a^{-8} +2 z^2 a^{-10} +2 z^2-z a^{-1} -3 z a^{-3} -7 z a^{-5} -8 z a^{-7} -3 z a^{-9} - a^{-2} +2 a^{-4} +3 a^{-6} + a^{-8} }[/math] |
| The A2 invariant | Data:K11a41/QuantumInvariant/A2/1,0 |
| The G2 invariant | Data:K11a41/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["K11a41"];
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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^3-11 t^2+27 t-35+27 t^{-1} -11 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^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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{ 115, 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^9-3 q^8+6 q^7-11 q^6+15 q^5-18 q^4+19 q^3-16 q^2+13 q-8+4 q^{-1} - q^{-2} }[/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^{-2} +z^6 a^{-4} +2 z^4 a^{-2} +2 z^4 a^{-4} -2 z^4 a^{-6} -z^4+2 z^2 a^{-2} +3 z^2 a^{-4} -4 z^2 a^{-6} +z^2 a^{-8} -z^2+ a^{-2} +2 a^{-4} -3 a^{-6} + a^{-8} }[/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^{-4} +z^{10} a^{-6} +4 z^9 a^{-3} +7 z^9 a^{-5} +3 z^9 a^{-7} +7 z^8 a^{-2} +12 z^8 a^{-4} +9 z^8 a^{-6} +4 z^8 a^{-8} +7 z^7 a^{-1} +4 z^7 a^{-3} -6 z^7 a^{-5} +3 z^7 a^{-9} -8 z^6 a^{-2} -29 z^6 a^{-4} -27 z^6 a^{-6} -9 z^6 a^{-8} +z^6 a^{-10} +4 z^6+a z^5-11 z^5 a^{-1} -18 z^5 a^{-3} -12 z^5 a^{-5} -15 z^5 a^{-7} -9 z^5 a^{-9} +26 z^4 a^{-4} +28 z^4 a^{-6} +5 z^4 a^{-8} -3 z^4 a^{-10} -6 z^4-a z^3+4 z^3 a^{-1} +14 z^3 a^{-3} +20 z^3 a^{-5} +19 z^3 a^{-7} +8 z^3 a^{-9} +z^2 a^{-2} -10 z^2 a^{-4} -13 z^2 a^{-6} -2 z^2 a^{-8} +2 z^2 a^{-10} +2 z^2-z a^{-1} -3 z a^{-3} -7 z a^{-5} -8 z a^{-7} -3 z a^{-9} - a^{-2} +2 a^{-4} +3 a^{-6} + a^{-8} }[/math] |
"Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {10_121, K11a183, K11a198, K11a331,}
Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {K11a183,}
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["K11a41"];
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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^3-11 t^2+27 t-35+27 t^{-1} -11 t^{-2} +2 t^{-3} }[/math], [math]\displaystyle{ q^9-3 q^8+6 q^7-11 q^6+15 q^5-18 q^4+19 q^3-16 q^2+13 q-8+4 q^{-1} - q^{-2} }[/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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{10_121, K11a183, K11a198, K11a331,} |
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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{K11a183,} |
Vassiliev invariants
| V2 and V3: | (1, 0) |
| 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 K11a41. 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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