Knot Atlas - User contributions [en]

Notes for 8 20's three dimensional invariants

2009-12-13T22:36:56Z

Rybu: ribbon diagram for 8_20

<br/>
[[8_20]] ribbon diagram from A. Kawauchi's text.
{| align=center
|[[Image:8_20.ribbon.png|thumb|337px|Ribbon diagram for [[8_20]] ]]
|}

File:8 20.ribbon.png

2009-12-13T22:33:40Z

Rybu: 8_20 ribbon diagram.

8_20 ribbon diagram.

Brunnian link

2008-03-13T02:01:52Z

Rybu: added Kanenobu reference

[[Image:Brunnian.gif|right|200px|thumb|This four-component link is a Brunnian link.]]

A '''Brunnian link''' is a nontrivial link which becomes trivial if any component is removed. In other words, cutting any loop frees all the other loops (so that there is no [[L2a1|Hopf link]] between any two loops).

The name Brunnian is after Hermann Brunn, whose 1892 article ''Über Verkettung'' included examples of such links.

The best-known and simplest possible Brunnian link is the [[L6a4|Borromean rings]], a link of three unknots. (This is also the only Brunnian link contained in tables which don't include links with 12 or more crossings.) However for every number three or above, there are an infinite number of links with the Brunnian property containing that number of loops. Here are some relatively simple three-component Brunnian links which are not the same as the Borromean rings:

<gallery>
Image:Brunnian-3-not-Borromean.gif|12-crossing link.
Image:Three-triang-18crossings-Brunnian.gif|18-crossing link.
</gallery>

T. Kanenobu has proven there exist hyperbolic Brunnian links with arbitrarily many components.

==See also==
*[["Rubberband" Brunnian Links]]
T.Kanenobu, Hyperbolic links with Brunnian properties. J. Math. Soc. Japan. Vol 38 No. 2, (1986) 295--308.

==Links==
*[http://en.wikipedia.org/wiki/Brunnian_link Wikipedia article "Brunnian link"]
*[http://www.mi.sanu.ac.yu/vismath/bor/bor1.htm "Are Borromean Links so Rare?", by Slavik Jablan]

Talk:The Multivariable Alexander Polynomial

2007-04-13T18:17:54Z

Rybu:

Regarding this comment:

:Dror doesn't understand the multivariable Alexander polynomial well enough to give simple topological reasons for the vanishing of the said polynomial for these knots.

The multivariable Alexander polynomial is zero precisely when H1 of the universal abelian cover has non-zero rank (as a module over the group-ring of covering transformations). Equivalently, if H2 of the universal abelian cover is non-trivial. In the L10n36 case, H2 is free on one generator, which is represented by a map of a genus 2 surface into the link complement. So far I haven't found a very appealing description of this surface, but it's there... -ryan budney

Talk:The Multivariable Alexander Polynomial

2007-04-13T18:12:22Z

Rybu:

Regarding this comment:

:Dror doesn't understand the multivariable Alexander polynomial well enough to give simple topological reasons for the vanishing of the said polynomial for these knots.

The multivariable Alexander polynomial is zero precisely when H1 of the universal abelian cover has non-zero rank (as a module over the group-ring of covering transformations). Equivalently, if H2 of the universal abelian cover is non-trivial. In this case, H2 is free on one generator, which is represented by a map of a genus 2 surface into the link complement. So far I haven't found a very appealing description of this surface, but it's there... -ryan budney

Talk:The Multivariable Alexander Polynomial

2007-04-13T18:11:40Z

Rybu: comment

Regarding this comment:

:Dror doesn't understand the multivariable Alexander polynomial well enough to give simple topological reasons for the vanishing of the said polynomial for these knots.

The multivariable Alexander polynomial is zero precisely when H1 of the universal abelian cover has non-zero rank (as a module over the group-ring of covering transformations). Equivalently, if H2 of the universal abelian cover is non-trivial. In this case, H2 is free on one generator, which is represented by a map of a genus 2 surface into the link complement. So far I haven't found a very appealing description of this surface, but it's there...

To Do

2007-03-31T07:40:42Z

Rybu: added some comments -- is this the right place?

* Update this To Do list.
* Make sure all KnotAtlas associated software is in either the KnotTheory or KnotAtlas svn repositories, and document how to get programs from svn. See [[:Category:Software]].
* Remove all the splicing programs.

== Knot Atlas issues ==
* Link to us from relevant pages on wikipedia and elsewhere.
* Add "comment" hooks near each of the sections on the knot pages.
* The right floating tables of contents in the manual don't layout properly in safari. I don't have access to safari, so can't do any testing. --[[User:Scott|Scott]] 05:08, 25 Sep 2005 (EDT)
**You seem to forget that both your officemates have iBooks they bring to school regularly. (as a Safari user, I can attest that this is still a problem a lot of places). [[User:Ben|Ben]] 17:47, 21 Nov 2005 (EST)

=== Pages to write ===
# Further edit [[How you can contribute]].
# Finish [[Extending/Modifying KnotTheory`]].
# Prepare [[List of Modules in KnotTheory`]].
# Document "The A2 invariant", "The G2 invariant" and "other quantum invariants".

=== More complicated projects ===
# Identify torus knots.
## should we have an 'identify this knot' function in [[KnotTheory`]]? How to implement it?
## one way to do this would be to: 1) triangulate the knot/link complement then 2) use the Rubinstein algorithm to find the connect-sum decomposition 3) use the Jaco-Rubinstein algorithm to compute the JSJ-decomposition of the prime parts and identify the Seifert-fibred bits and 4) use SnapPea to identify the hyperbolic parts. (1) is implemented in several places such as Orb and Knotscape, (2) is implemented in the software called Regina, by Ben Burton (do a Google search for "Regina sourceforge" (3) is not yet implemented and (4) has been implemented for a long time now, in SnapPea's isometry checker. Perhaps you would like to use a less big hammer for this problem, though? -ryan
# Clarify orientation issues, especially for links.
## As in, if you reverse the orientation of a particular strand, do you get the same oriented link? The above JSJ schema gives a systematic way to do this.
# Extend tables beyond 11 crossings.
# Add pages on rational knots and other classes of "special" knots.
# Add a page for "sporadic" special knots.

=== Data to include ===
# Add Tristram-Levine signatures.
# Chiral? Invertible?
## More generally, list the full symmetry group of the knot (isotopy classes of diffeomorphisms of the pair (S^3,K) and describe whether or not it can be realized as a subgroup of Diff(S^3,K), the fixed point sets, how it acts on the orientations of S^3 and K respectively, etc... If you're going to start including symmetry data, why not do it all? Again the JSJ-schema I outlined above is one way to approach this. -ryan
# Indicate which knots are the components of given links.
# Add something about hyperbolic invariants.
# "Similar Links" using [[The Multivariable Alexander Polynomial]]; add the The Multivariable Alexander Polynomial to the simulated Mathematica sessions.

==Things to calculate==
# Compute ''HOMFLYPT'' for torus knots -- this has been added to the [[Upload_Queues]].
# compute Reshetikhin-Turaev invariants for everything in sight.

==KnotTheory`==
# Put the MIT license notice in all the source files. (You can consider this note as MIT licensing for now. If you're one of the contributors, and you didn't complain when we proposed MIT licensing for everything, you can still complain now!)
# Write some functions to generate knot families, taking advantage of ConwayNotation. Pretzel knots, etc.
# Unite <code>DrawPD</code> and <code>BraidPlot</code> under Draw.
# Remove <code>Simplify</code> from ColouredJones.
# Custom knot drawings! Output in ps, TeX, etc.
# Browsable source code for <code>KnotTheory`</code>.
# Fix the <tt>In</tt>/<tt>Out</tt> numbering in [[Graphical Input]].

==katlas software==
* Find out if it's possible to eliminate the need for &action=purge when transcluded pages change?
* Tell people about the modifications to [[ImagePage.php]]
* Fix/improve the "disappearing transclusion" tag <nowiki><ifpageexists></nowiki>.
* Replace the "bracket" splicing model by a "templated" one.

==texvc==
# can texvc use dvi2bitmap instead of convert? might be much faster? [http://heim.ifi.uio.no/~simek/document/document.ps]
## alternatively see [http://meta.wikimedia.org/wiki/Problems_with_texvc]
## what about a cron job to randomly delete entries from mw_math, and the corresponding png's?
# Apply patches to texvc, for making transparent, font-sizes png in math mode. [http://meta.wikimedia.org/wiki/Help_talk:Formula]

==Other things==
# Add the Universal Vassiliev Invariant.
# Something about knot colouring.
# Add Ozsvath-Szabo stuff.
# Make all sources public.
# Add Dowker codes for links.
# Identify the knot at http://commons.wikimedia.org/wiki/Image:Gateknot.jpg (see also [[User talk:Drorbn]]).

==Papers to consider extracting data from==
# Extract data from Stanford's paper [http://www.maths.warwick.ac.uk/gt/GTMon4/paper23.abs.html].
# Extract data from Livingston's paper [http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-1.abs.html].
# Extract data from Rasmussen's paper, {{arXiv|math.GT/0402131}}. (I think this has been done - I imported s-invariant data from KnotInfo. --[[User:Scott|Scott]] 02:06, 26 May 2006 (EDT))
# Extract data from Hikami's paper, {{arXiv|math.GT/0403224}}.
# Extract data from Stoimenow's paper, {{arXiv|math.GT/0405076}}.
# Extract data from De Wit and Links, {{arXiv|math.GT/0501224}}, and from De Wit, Ishii and Links, {{arXiv|math.GT/0405403}}.
# Extract data from Cha-Livingston's paper, {{arXiv|math.GT/0503125}}.
# Extract data from Hirasawa-Teragaito's paper, {{arXiv|math.GT/0504446}}.
# Extract data from Cimasoni-Florens' paper, {{arXiv|math.GT/0505185}}.
# Extract data from Ashton-Cantarella-Piatek-Rawdon's paper, {{arXiv|math.DG/0508248}}.
# Boundary slope data. There's Dunfield's paper, and associated program, {{arXiv|math.GT/9901120}}, and also a paper by Hoste and Shanahan {{arXiv|math.GT/0505442}}, which contains data for many two component links.
# Extract data from Baldwin-Gillam's paper, {{arXiv|math.GT/0610167}}.

[[Special:Whatlinkshere&target=Template:Todo-later|Pages with the 'todo-later' template]]
[[Special:Whatlinkshere&target=Template:Todo|Pages with the 'todo' template]]