Saturday, January 30, 2010

Seifert surfaces

Hi folks! My next mathematical project comes from a suggestion from local knot-theorist Julia. The idea is to knit the Seifert surface of a trefoil knot. This will be very similar to a project ( of Matthew Wright's.

So, what is a Seifert surface? There's a theorem in knot theory that given any ('tame') knot, there is an orientable surface whose boundary is this knot. I think there's a good constructive proof of this, which also allows you to find out what this surface is. This surface is the Seifert surface of the knot.
If you want to play with them, there's a very cool program, "Seifert View", which will draw them for you (it's free, but sadly only runs on windows).

So, how to knit this? I think this should be pretty straightforward -- I'll start from the middle of one side, increase outwards to form a disc, then split into three to form the 'struts', before rejoining and decreasing to form the top disc. The magic is that before rejoining the 'struts' at the top, I'll flip them over to give the twist. When I'm done, I'll crochet around the edges to highlight the boundary knot. This way has the advantage of being seamless, though possibly at the expense of making the joins less neat.
I'm also considering threading some wire through this boundary, as Matthew Wright suggests, to give the shape rigidity, and then see if I can hang it as a decoration.

I'm planning to write up a pattern for this one, partly because it should be good and quick and partly because I need to get more practice with the LaTeX pattern formatting package I wrote. Since it's so similar to Matthew Wright's work, though, I'll have to ask him before I go publishing it.
I'm not sure how well this approach would generalise to other knots, I may have to ask Julia about that.

So, let's see how it goes -- should be good and quick, since it will be quite small.


smbelcas said...

Because a Seifert surface is a surface (albeit with boundary and sometimes lots of them) you could certainly knit a general Seifert surface by classifying it as a surface and then using my algorithm for knitting surfaces to do it.

Not that this is the easiest or best way, mind you... just that it can be done.

Jhadur said...

Ooh, I haven't come across this algorithm -- do you have a link for it?*
I was thinking that it would be fairly straightforward to make this work for any Seifert surface embedded as a series of discs connected with twisted strips, and it'd not just give the topology but a sensible embedding too. And I *think* that's the form you get from the proof of their existence, but I'm not too sure on that.

* - Puns \o/

smbelcas said...

It's in a paper in the Journal of Mathematics and the Arts, 3(2) June 2009, 67–83. Not freely available online, but if your institution has Taylor&Francis online access you can get to it. Or you can email me *grin*.