Template:Torus Knot Page
From Knot Atlas
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| Image:Torus Knot Page.jpg | See other torus knots
Visit [{{{KnotilusURL}}} Torus Knot Page's page] at Knotilus! Visit Torus Knot Page's page at the original Knot Atlas! |
| Edit Torus Knot Page Quick Notes
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Edit Torus Knot Page Further Notes and Views
[edit] Knot presentations
| Planar diagram presentation | Data:Torus Knot Page/PD Presentation |
| Gauss code | Data:Torus Knot Page/Gauss Code |
| Dowker-Thistlethwaite code | Data:Torus Knot Page/DT Code |
| Braid presentation | {{{braid_table}}} |
[edit] Polynomial invariants
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]:=
| 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]:=
| K = Knot["Torus Knot Page"];
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In[4]:=
| Alexander[K][t]
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
| Data:Torus Knot Page/Alexander Polynomial |
In[5]:=
| Conway[K][z]
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Out[5]=
| Data:Torus Knot Page/Conway Polynomial |
In[6]:=
| 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]=
| Data:Torus Knot Page/2nd AlexanderIdeal |
In[7]:=
| {KnotDet[K], KnotSignature[K]}
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Out[7]=
| { Data:Torus Knot Page/Determinant, Data:Torus Knot Page/Signature } |
In[8]:=
| Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
| Data:Torus Knot Page/Jones Polynomial |
In[9]:=
| HOMFLYPT[K][a, z]
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KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
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Out[9]=
| Data:Torus Knot Page/HOMFLYPT Polynomial |
In[10]:=
| Kauffman[K][a, z]
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KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
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Out[10]=
| Data:Torus Knot Page/Kauffman Polynomial |
[edit] "Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {{{{same_alexander}}}}
Same Jones Polynomial (up to mirroring, 
):
{{{{same_jones}}}}
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]:=
| 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]:=
| K = Knot["Torus Knot Page"];
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In[4]:=
| {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]=
| { Data:Torus Knot Page/Alexander Polynomial, Data:Torus Knot Page/Jones Polynomial } |
In[5]:=
| 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]=
| {{{{same_alexander}}}} |
In[6]:=
| 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]=
| {{{{same_jones}}}} |
[edit] Khovanov Homology
| The coefficients of the monomials trqj are shown, along with their alternating sums χ (fixed j, alternation over r). The squares with yellow highlighting are those on the "critical diagonals", where j−2r = s + 1 or j−2r = s−1, where s = Data:Torus Knot Page/Signature is the signature of Torus Knot Page. Nonzero entries off the critical diagonals (if any exist) are highlighted in red. | Data:Torus Knot Page/KhovanovTable |
| Integral Khovanov Homology
(db, data source) | Data:Torus Knot Page/Integral Khovanov Homology |
[edit] Computer Talk
Much of the above data can be recomputed by Mathematica using the packageKnotTheory`. See A Sample KnotTheory` Session.
[edit] 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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