Template:Torus Knot Page: Difference between revisions
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| valign="top" rowspan=2 | [[Image:{{PAGENAME}}.jpg]] |
| valign="top" rowspan=2 | [[Image:{{PAGENAME}}.jpg]] |
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|See [[36 Torus Knots|other torus knots]] |
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|Visit [{{{KnotilusURL}}} {{PAGENAME}}'s page] at [http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/html/start.html Knotilus]! |
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Visit [http:// |
Visit [http://knotilus.math.uwo.ca/draw.php?knot={{urlencode:{{Data:{{PAGENAME}}/Gauss Code}}}} {{PAGENAME}}] at [http://knotilus.math.uwo.ca/ Knotilus]! |
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| valign="top" | [{{SERVER}}{{localurl:{{PAGENAME}} Quick Notes|action=edit}} |
| valign="top" | [{{SERVER}}{{localurl:{{PAGENAME}} Quick Notes|action=edit}} Edit {{PAGENAME}} Quick Notes] |
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{{#ifexist:{{PAGENAME}} Quick Notes|{{:{{PAGENAME}} Quick Notes}}|}} |
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[{{SERVER}}{{localurl:{{PAGENAME}} Further Notes and Views|action=edit}} |
[{{SERVER}}{{localurl:{{PAGENAME}} Further Notes and Views|action=edit}} Edit {{PAGENAME}} Further Notes and Views] |
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{{#ifexist:{{PAGENAME}} Further Notes and Views|{{:{{PAGENAME}} Further Notes and Views}}|}} |
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===Knot presentations=== |
===Knot presentations=== |
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|'''[[DT (Dowker-Thistlethwaite) Codes|Dowker-Thistlethwaite code]]''' |
|'''[[DT (Dowker-Thistlethwaite) Codes|Dowker-Thistlethwaite code]]''' |
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|style="padding-left: 1em;" | {{Data:{{PAGENAME}}/DT Code}} |
|style="padding-left: 1em;" | {{Data:{{PAGENAME}}/DT Code}} |
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{| |
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|'''[[Braid Representatives|Braid presentation]]''' |
|'''[[Braid Representatives|Braid presentation]]''' |
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|style="padding-left: |
|style="padding-left: 3em" | {{{braid_table}}} |
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{{Polynomial Invariants}} |
{{Polynomial Invariants}} |
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{{Similar Knots| |
{{Similar Knots| |
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same_alexander = {{{same_alexander}}} | |
same_alexander = {{{same_alexander}}} | |
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same_jones = {{{same_jones}}} }} |
same_jones = {{{same_jones}}} }} |
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{{Vassiliev Invariants}} |
{{Vassiliev Invariants}} |
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{{Khovanov Homology|table={{{khovanov_table}}}}} |
{{Khovanov Homology|table={{{khovanov_table}}}}} |
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{{Computer Talk Header}} |
{{Computer Talk Header}} |
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{{{computer_talk}}} |
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=== Modifying This Page === |
=== Modifying This Page === |
Latest revision as of 01:36, 3 July 2015
[[Image:Data:Torus Knot Page/Previous Knot.jpg|80px|link=Data:Torus Knot Page/Previous Knot]] |
[[Image:Data:Torus Knot Page/Next Knot.jpg|80px|link=Data:Torus Knot Page/Next Knot]] |
File:Torus Knot Page.jpg | See other torus knots
Visit Torus Knot Page at Knotilus! |
Edit Torus Knot Page Quick Notes
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Edit Torus Knot Page Further Notes and Views
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}}} |
Polynomial invariants
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["Torus Knot Page"];
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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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Data:Torus Knot Page/Alexander Polynomial |
In[5]:=
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Conway[K][z]
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Out[5]=
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Data:Torus Knot Page/Conway Polynomial |
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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Data:Torus Knot Page/2nd AlexanderIdeal |
In[7]:=
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{KnotDet[K], KnotSignature[K]}
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Out[7]=
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{ Data:Torus Knot Page/Determinant, Data:Torus Knot Page/Signature } |
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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Data:Torus Knot Page/Jones Polynomial |
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:Torus Knot Page/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:Torus Knot Page/Kauffman Polynomial |
"Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {{{{same_alexander}}}}
Same Jones Polynomial (up to mirroring, ): {{{{same_jones}}}}
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["Torus Knot Page"];
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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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{ Data:Torus Knot Page/Alexander Polynomial, Data:Torus Knot Page/Jones Polynomial } |
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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{{{{same_alexander}}}} |
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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{{{{same_jones}}}} |
Vassiliev invariants
V2 and V3: | (Data:Torus Knot Page/V 2, Data:Torus Knot Page/V 3) |
V2,1 through V6,9: |
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 are shown, along with their alternating sums (fixed , alternation over ). The squares with yellow highlighting are those on the "critical diagonals", where or , where 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 |
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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