T(11,4): Difference between revisions
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{{Torus Knot Page| |
{{Torus Knot Page| |
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m = 11 | |
m = 11 | |
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Latest revision as of 11:37, 31 August 2005
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.jpg)
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See other torus knots |
| Edit T(11,4) Quick Notes
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Edit T(11,4) Further Notes and Views
Knot presentations
| Planar diagram presentation | X3,53,4,52 X20,54,21,53 X37,55,38,54 X21,5,22,4 X38,6,39,5 X55,7,56,6 X39,23,40,22 X56,24,57,23 X7,25,8,24 X57,41,58,40 X8,42,9,41 X25,43,26,42 X9,59,10,58 X26,60,27,59 X43,61,44,60 X27,11,28,10 X44,12,45,11 X61,13,62,12 X45,29,46,28 X62,30,63,29 X13,31,14,30 X63,47,64,46 X14,48,15,47 X31,49,32,48 X15,65,16,64 X32,66,33,65 X49,1,50,66 X33,17,34,16 X50,18,51,17 X1,19,2,18 X51,35,52,34 X2,36,3,35 X19,37,20,36 |
| Gauss code | -30, -32, -1, 4, 5, 6, -9, -11, -13, 16, 17, 18, -21, -23, -25, 28, 29, 30, -33, -2, -4, 7, 8, 9, -12, -14, -16, 19, 20, 21, -24, -26, -28, 31, 32, 33, -3, -5, -7, 10, 11, 12, -15, -17, -19, 22, 23, 24, -27, -29, -31, 1, 2, 3, -6, -8, -10, 13, 14, 15, -18, -20, -22, 25, 26, 27 |
| Dowker-Thistlethwaite code | 18 52 -38 24 58 -44 30 64 -50 36 4 -56 42 10 -62 48 16 -2 54 22 -8 60 28 -14 66 34 -20 6 40 -26 12 46 -32 |
| Braid presentation |
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Polynomial invariants
| Alexander polynomial | [math]\displaystyle{ t^{15}-t^{14}+t^{11}-t^{10}+t^7-t^6+t^4-t^2+1- t^{-2} + t^{-4} - t^{-6} + t^{-7} - t^{-10} + t^{-11} - t^{-14} + t^{-15} }[/math] |
| Conway polynomial | [math]\displaystyle{ z^{30}+29 z^{28}+377 z^{26}+2900 z^{24}+14675 z^{22}+51380 z^{20}+127470 z^{18}+225760 z^{16}+283951 z^{14}+249288 z^{12}+148070 z^{10}+56577 z^8+12825 z^6+1510 z^4+75 z^2+1 }[/math] |
| 2nd Alexander ideal (db, data sources) | [math]\displaystyle{ \{1\} }[/math] |
| Determinant and Signature | { 11, 22 } |
| Jones polynomial | [math]\displaystyle{ -q^{28}-q^{26}+q^{25}-q^{24}+q^{23}-q^{22}+q^{21}-q^{20}+q^{19}+q^{17}+q^{15} }[/math] |
| HOMFLY-PT polynomial (db, data sources) | Data:T(11,4)/HOMFLYPT Polynomial |
| Kauffman polynomial (db, data sources) | Data:T(11,4)/Kauffman Polynomial |
| The A2 invariant | Data:T(11,4)/QuantumInvariant/A2/1,0 |
| The G2 invariant | Data:T(11,4)/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(11,4)"];
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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^{15}-t^{14}+t^{11}-t^{10}+t^7-t^6+t^4-t^2+1- t^{-2} + t^{-4} - t^{-6} + t^{-7} - t^{-10} + t^{-11} - t^{-14} + t^{-15} }[/math] |
In[5]:=
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Conway[K][z]
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Out[5]=
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[math]\displaystyle{ z^{30}+29 z^{28}+377 z^{26}+2900 z^{24}+14675 z^{22}+51380 z^{20}+127470 z^{18}+225760 z^{16}+283951 z^{14}+249288 z^{12}+148070 z^{10}+56577 z^8+12825 z^6+1510 z^4+75 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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{ 11, 22 } |
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^{28}-q^{26}+q^{25}-q^{24}+q^{23}-q^{22}+q^{21}-q^{20}+q^{19}+q^{17}+q^{15} }[/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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Data:T(11,4)/HOMFLYPT Polynomial |
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(11,4)/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(11,4)"];
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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^{15}-t^{14}+t^{11}-t^{10}+t^7-t^6+t^4-t^2+1- t^{-2} + t^{-4} - t^{-6} + t^{-7} - t^{-10} + t^{-11} - t^{-14} + t^{-15} }[/math], [math]\displaystyle{ -q^{28}-q^{26}+q^{25}-q^{24}+q^{23}-q^{22}+q^{21}-q^{20}+q^{19}+q^{17}+q^{15} }[/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: | (75, 550) |
| 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]22 is the signature of T(11,4). 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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