T(11,3)
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.jpg)
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See other torus knots |
| Edit T(11,3) Quick Notes
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Edit T(11,3) Further Notes and Views
Knot presentations
| Planar diagram presentation | X7,37,8,36 X22,38,23,37 X23,9,24,8 X38,10,39,9 X39,25,40,24 X10,26,11,25 X11,41,12,40 X26,42,27,41 X27,13,28,12 X42,14,43,13 X43,29,44,28 X14,30,15,29 X15,1,16,44 X30,2,31,1 X31,17,32,16 X2,18,3,17 X3,33,4,32 X18,34,19,33 X19,5,20,4 X34,6,35,5 X35,21,36,20 X6,22,7,21 |
| Gauss code | 14, -16, -17, 19, 20, -22, -1, 3, 4, -6, -7, 9, 10, -12, -13, 15, 16, -18, -19, 21, 22, -2, -3, 5, 6, -8, -9, 11, 12, -14, -15, 17, 18, -20, -21, 1, 2, -4, -5, 7, 8, -10, -11, 13 |
| Dowker-Thistlethwaite code | 30 -32 34 -36 38 -40 42 -44 2 -4 6 -8 10 -12 14 -16 18 -20 22 -24 26 -28 |
| Braid presentation |
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Polynomial invariants
| Alexander polynomial | [math]\displaystyle{ t^{10}-t^9+t^7-t^6+t^4-t^3+t-1+ t^{-1} - t^{-3} + t^{-4} - t^{-6} + t^{-7} - t^{-9} + t^{-10} }[/math] |
| Conway polynomial | [math]\displaystyle{ z^{20}+19 z^{18}+152 z^{16}+666 z^{14}+1742 z^{12}+2782 z^{10}+2665 z^8+1443 z^6+390 z^4+40 z^2+1 }[/math] |
| 2nd Alexander ideal (db, data sources) | [math]\displaystyle{ \{1\} }[/math] |
| Determinant and Signature | { 1, 16 } |
| Jones polynomial | [math]\displaystyle{ -q^{22}+q^{12}+q^{10} }[/math] |
| HOMFLY-PT polynomial (db, data sources) | [math]\displaystyle{ z^{20} a^{-20} +20 z^{18} a^{-20} -z^{18} a^{-22} +171 z^{16} a^{-20} -19 z^{16} a^{-22} +817 z^{14} a^{-20} -152 z^{14} a^{-22} +z^{14} a^{-24} +2394 z^{12} a^{-20} -666 z^{12} a^{-22} +14 z^{12} a^{-24} +4446 z^{10} a^{-20} -1742 z^{10} a^{-22} +78 z^{10} a^{-24} +5226 z^8 a^{-20} -2782 z^8 a^{-22} +221 z^8 a^{-24} +3770 z^6 a^{-20} -2665 z^6 a^{-22} +338 z^6 a^{-24} +1560 z^4 a^{-20} -1443 z^4 a^{-22} +273 z^4 a^{-24} +325 z^2 a^{-20} -390 z^2 a^{-22} +105 z^2 a^{-24} +26 a^{-20} -40 a^{-22} +15 a^{-24} }[/math] |
| Kauffman polynomial (db, data sources) | Data:T(11,3)/Kauffman Polynomial |
| The A2 invariant | Data:T(11,3)/QuantumInvariant/A2/1,0 |
| The G2 invariant | Data:T(11,3)/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["T(11,3)"];
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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^{10}-t^9+t^7-t^6+t^4-t^3+t-1+ t^{-1} - t^{-3} + t^{-4} - t^{-6} + t^{-7} - t^{-9} + t^{-10} }[/math] |
In[5]:=
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Conway[K][z]
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Out[5]=
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[math]\displaystyle{ z^{20}+19 z^{18}+152 z^{16}+666 z^{14}+1742 z^{12}+2782 z^{10}+2665 z^8+1443 z^6+390 z^4+40 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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{ 1, 16 } |
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^{22}+q^{12}+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{ z^{20} a^{-20} +20 z^{18} a^{-20} -z^{18} a^{-22} +171 z^{16} a^{-20} -19 z^{16} a^{-22} +817 z^{14} a^{-20} -152 z^{14} a^{-22} +z^{14} a^{-24} +2394 z^{12} a^{-20} -666 z^{12} a^{-22} +14 z^{12} a^{-24} +4446 z^{10} a^{-20} -1742 z^{10} a^{-22} +78 z^{10} a^{-24} +5226 z^8 a^{-20} -2782 z^8 a^{-22} +221 z^8 a^{-24} +3770 z^6 a^{-20} -2665 z^6 a^{-22} +338 z^6 a^{-24} +1560 z^4 a^{-20} -1443 z^4 a^{-22} +273 z^4 a^{-24} +325 z^2 a^{-20} -390 z^2 a^{-22} +105 z^2 a^{-24} +26 a^{-20} -40 a^{-22} +15 a^{-24} }[/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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Data:T(11,3)/Kauffman Polynomial |
"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["T(11,3)"];
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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^{10}-t^9+t^7-t^6+t^4-t^3+t-1+ t^{-1} - t^{-3} + t^{-4} - t^{-6} + t^{-7} - t^{-9} + t^{-10} }[/math], [math]\displaystyle{ -q^{22}+q^{12}+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: | (40, 220) |
| 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]16 is the signature of T(11,3). 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 Torus Knot Page master template (intermediate). See/edit the Torus Knot_Splice_Base (expert). Back to the top. |
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