Template:Rolfsen Knot Page: Difference between revisions
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This is the Rolfsen Knot Page master template. For testing purposes, some of the data in it is filled in with data for [[8_17]]. |
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{{Knot Navigation Links|ext=gif}} |
{{Knot Navigation Links|ext=gif}} |
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See the full [[The Rolfsen Knot Table|Rolfsen Knot Table]]. |
See the full [[The Rolfsen Knot Table|Rolfsen Knot Table]]. |
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Visit [http://newweb.cecm.sfu.ca/cgi-bin/KnotPlot/KnotServer/kserver?ncomp=1&ncross={{{n}}}&id={{{k}}} {{PAGENAME}}'s page] |
Visit [http://newweb.cecm.sfu.ca/cgi-bin/KnotPlot/KnotServer/kserver?ncomp=1&ncross={{{n}}}&id={{{k}}} {{PAGENAME}}'s page] |
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at the [http://www.colab.sfu.ca/KnotPlot/KnotServer/ Knot Server] |
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([http://www.pims.math.ca/knotplot/ KnotPlot] driven, includes 3D interactive images!) |
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Visit {{Data:{{PAGENAME}}/ |
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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Visit [http://www.math.toronto.edu/~drorbn/KAtlas/Knots/{{{n}}}.{{{k}}}.html {{PAGENAME}}'s page] at the original [http://www.math.toronto.edu/~drorbn/KAtlas/index.html Knot Atlas]! |
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|{{floating edit link|{{PAGENAME}} Quick Notes}} |
|{{floating edit link|{{PAGENAME}} Quick Notes}} |
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{{#ifexist:{{PAGENAME}} Quick Notes|{{:{{PAGENAME}} Quick Notes}}|}} |
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{{floating edit link|{{PAGENAME}} Further Notes and Views}} |
{{floating edit link|{{PAGENAME}} Further Notes and Views}} |
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{{#ifexist:{{PAGENAME}} Further Notes and Views|{{:{{PAGENAME}} Further Notes and Views}}|}} |
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<div style="border: solid pink 1px"> |
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{{Knot Presentations}} |
{{Knot Presentations}} |
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{| |
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{| align=center width=99% |
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|'''[[Braid Representatives|Minimum Braid Representative]]''' |
|width=33%|'''[[Braid Representatives|Minimum Braid Representative]]''' |
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|width=33%|'''[[Arc Presentations|An Arc Presentation]]''' |
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|style="padding-left: 1em;" | {{Data:{{PAGENAME}}/BraidPlot}} |
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|[[Image:{{PAGENAME}}_ML.gif]] |
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|[[Image:{{PAGENAME}}_AP.gif]]<br>{{Data:{{PAGENAME}}/Arc_Presentation}} |
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{{edit link|Notes on presentations of {{PAGENAME}}}} |
{{edit link|Notes on presentations of {{PAGENAME}}}} |
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{{#ifexist:Notes on presentations of {{PAGENAME}}|{{:Notes on presentations of {{PAGENAME}}}}|}} |
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{{Rolfsen Knot Presentations Computer Talk}} |
{{Rolfsen Knot Presentations Computer Talk}} |
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{{4D Invariants}} |
{{4D Invariants}} |
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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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J6 = {{{coloured_jones_6}}} | |
J6 = {{{coloured_jones_6}}} | |
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J7 = {{{coloured_jones_7}}} }} |
J7 = {{{coloured_jones_7}}} }} |
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{{Computer Talk for Rolfsen Table}} |
{{Computer Talk for Rolfsen Table}} |
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Latest revision as of 00:06, 3 July 2015
This is the Rolfsen Knot Page master template. For testing purposes, some of the data in it is filled in with data for 8_17.
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(KnotPlot image) |
See the full Rolfsen Knot Table. Visit Rolfsen Knot Page's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit Rolfsen Knot Page at Knotilus! |
Knot presentations
Minimum Braid Representative | A Morse Link Presentation | An Arc Presentation | |||
Length is 8, width is 3, Braid index is 3 |
[{7, 10}, {9, 2}, {10, 4}, {3, 5}, {4, 8}, {6, 9}, {5, 1}, {2, 7}, {1, 6}, {8, 3}] |
[edit Notes on presentations of Rolfsen Knot Page]
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[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
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In[3]:=
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K = Knot["Rolfsen Knot Page"];
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In[4]:=
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PD[K]
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
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Data:Rolfsen Knot Page/PD Presentation |
In[5]:=
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GaussCode[K]
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Out[5]=
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Data:Rolfsen Knot Page/Gauss Code |
In[6]:=
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DTCode[K]
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Out[6]=
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Data:Rolfsen Knot Page/DT Code |
(The path below may be different on your system)
In[7]:=
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AppendTo[$Path, "C:/bin/LinKnot/"];
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In[8]:=
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ConwayNotation[K]
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Out[8]=
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Data:Rolfsen Knot Page/Conway Notation |
In[9]:=
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br = BR[K]
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KnotTheory::credits: The minimum braids representing the knots with up to 10 crossings were provided by Thomas Gittings. See arXiv:math.GT/0401051.
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Out[9]=
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Data:Rolfsen Knot Page/BraidWord |
In[10]:=
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{First[br], Crossings[br], BraidIndex[K]}
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KnotTheory::credits: The braid index data known to KnotTheory` is taken from Charles Livingston's http://www.indiana.edu/~knotinfo/.
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KnotTheory::loading: Loading precomputed data in IndianaData`.
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Out[10]=
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{ 3, 8, 3 } |
In[11]:=
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Show[BraidPlot[br]]
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Out[11]=
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-Graphics- |
In[12]:=
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Show[DrawMorseLink[K]]
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KnotTheory::credits: "MorseLink was added to KnotTheory` by Siddarth Sankaran at the University of Toronto in the summer of 2005."
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KnotTheory::credits: "DrawMorseLink was written by Siddarth Sankaran at the University of Toronto in the summer of 2005."
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Out[12]=
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-Graphics- |
In[13]:=
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ap = ArcPresentation[K]
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Out[13]=
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ArcPresentation[{7, 10}, {9, 2}, {10, 4}, {3, 5}, {4, 8}, {6, 9}, {5, 1}, {2, 7}, {1, 6}, {8, 3}] |
In[14]:=
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Draw[ap]
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Out[14]=
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-Graphics- |
Three dimensional invariants
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[edit Notes for Rolfsen Knot Page's three dimensional invariants] |
Four dimensional invariants
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[edit Notes for Rolfsen Knot Page's four dimensional invariants] |
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["Rolfsen 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:Rolfsen Knot Page/Alexander Polynomial |
In[5]:=
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Conway[K][z]
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Out[5]=
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Data:Rolfsen 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:Rolfsen 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:Rolfsen Knot Page/Determinant, Data:Rolfsen 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:Rolfsen 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:Rolfsen 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:Rolfsen 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["Rolfsen 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:Rolfsen Knot Page/Alexander Polynomial, Data:Rolfsen 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:Rolfsen Knot Page/V 2, Data:Rolfsen 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:Rolfsen Knot Page/Signature is the signature of Rolfsen Knot Page. Nonzero entries off the critical diagonals (if any exist) are highlighted in red. | Data:Rolfsen Knot Page/KhovanovTable |
Integral Khovanov Homology
(db, data source) |
Data:Rolfsen Knot Page/Integral Khovanov Homology |
The Coloured Jones Polynomials
2 | {{{coloured_jones_2}}} |
3 | {{{coloured_jones_3}}} |
4 | {{{coloured_jones_4}}} |
5 | {{{coloured_jones_5}}} |
6 | {{{coloured_jones_6}}} |
7 | {{{coloured_jones_7}}} |
Computer Talk
Much of the above data can be recomputed by Mathematica using the package KnotTheory`
. See A Sample KnotTheory` Session, or any of the Computer Talk sections above.
Modifying This Page
Read me first: Modifying Knot Pages
See/edit the Rolfsen Knot Page master template (intermediate). See/edit the Rolfsen_Splice_Base (expert). Back to the top. |
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