T(9,4)
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See other torus knots |
| Edit T(9,4) Quick Notes
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Edit T(9,4) Further Notes and Views
Knot presentations
| Planar diagram presentation | X11,25,12,24 X52,26,53,25 X39,27,40,26 X53,13,54,12 X40,14,41,13 X27,15,28,14 X41,1,42,54 X28,2,29,1 X15,3,16,2 X29,43,30,42 X16,44,17,43 X3,45,4,44 X17,31,18,30 X4,32,5,31 X45,33,46,32 X5,19,6,18 X46,20,47,19 X33,21,34,20 X47,7,48,6 X34,8,35,7 X21,9,22,8 X35,49,36,48 X22,50,23,49 X9,51,10,50 X23,37,24,36 X10,38,11,37 X51,39,52,38 |
| Gauss code | 8, 9, -12, -14, -16, 19, 20, 21, -24, -26, -1, 4, 5, 6, -9, -11, -13, 16, 17, 18, -21, -23, -25, 1, 2, 3, -6, -8, -10, 13, 14, 15, -18, -20, -22, 25, 26, 27, -3, -5, -7, 10, 11, 12, -15, -17, -19, 22, 23, 24, -27, -2, -4, 7 |
| Dowker-Thistlethwaite code | 28 -44 -18 34 -50 -24 40 -2 -30 46 -8 -36 52 -14 -42 4 -20 -48 10 -26 -54 16 -32 -6 22 -38 -12 |
| Braid presentation |
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Polynomial invariants
| Alexander polynomial | [math]\displaystyle{ t^{12}-t^{11}+t^8-t^7+t^4-t^2+1- t^{-2} + t^{-4} - t^{-7} + t^{-8} - t^{-11} + t^{-12} }[/math] |
| Conway polynomial | [math]\displaystyle{ z^{24}+23 z^{22}+230 z^{20}+1311 z^{18}+4693 z^{16}+10963 z^{14}+16834 z^{12}+16720 z^{10}+10318 z^8+3675 z^6+665 z^4+50 z^2+1 }[/math] |
| 2nd Alexander ideal (db, data sources) | [math]\displaystyle{ \{1\} }[/math] |
| Determinant and Signature | { 9, 16 } |
| Jones polynomial | [math]\displaystyle{ -q^{23}-q^{21}+q^{20}-q^{19}+q^{18}-q^{17}+q^{16}+q^{14}+q^{12} }[/math] |
| HOMFLY-PT polynomial (db, data sources) | Data:T(9,4)/HOMFLYPT Polynomial |
| Kauffman polynomial (db, data sources) | Data:T(9,4)/Kauffman Polynomial |
| The A2 invariant | Data:T(9,4)/QuantumInvariant/A2/1,0 |
| The G2 invariant | Data:T(9,4)/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(9,4)"];
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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^{12}-t^{11}+t^8-t^7+t^4-t^2+1- t^{-2} + t^{-4} - t^{-7} + t^{-8} - t^{-11} + t^{-12} }[/math] |
In[5]:=
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Conway[K][z]
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Out[5]=
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[math]\displaystyle{ z^{24}+23 z^{22}+230 z^{20}+1311 z^{18}+4693 z^{16}+10963 z^{14}+16834 z^{12}+16720 z^{10}+10318 z^8+3675 z^6+665 z^4+50 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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{ 9, 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^{23}-q^{21}+q^{20}-q^{19}+q^{18}-q^{17}+q^{16}+q^{14}+q^{12} }[/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(9,4)/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(9,4)/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(9,4)"];
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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^{12}-t^{11}+t^8-t^7+t^4-t^2+1- t^{-2} + t^{-4} - t^{-7} + t^{-8} - t^{-11} + t^{-12} }[/math], [math]\displaystyle{ -q^{23}-q^{21}+q^{20}-q^{19}+q^{18}-q^{17}+q^{16}+q^{14}+q^{12} }[/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: | (50, 300) |
| 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(9,4). 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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