Seifert surface in crochet

As I mentioned in a comment to the post on the 3-4-5, my latest project in crochet is a so-called Seifert surface, a surface with a knot as its edge.
I came up with the idea from a picture of the Seifert surface of a trefoil I found here. A Moebius strip is another example of a Seifert surface, although its surface is not really a knot – but the ‘un-knot’, a loop with a twist.

For the moebius strips I made before, I started in the middle, with a loop with a half twist. After that first loop, every next line of stitches goes round twice, on both sides of the center string.

Moebius inside out

But now I thought you could also start on the outside, with the edge of the moebius strip (it has only one edge, of course) and work your way in. The edge is just a loop, no knots, but to make it work you need to give it two complete twists. (Remember that when you cut a moebius strip in half over the center line, you get a loop with two turns in it.) After closing the edge with the turns, work your way in until the width of the strip is half what you want the moebius strip to be, then end with the center loop connecting the frontside with the back.

That was not too hard, and actually great fun (I know… I’m probably a crochet-nerd). Next I wanted to do the trefoil. First I used the picture from the blogpost as an example. That worked (although I made a mistake and twisted one of the bands the wrong way), but I didn’t really like the result. The circles on top and bottom are there for no good reason and just obscure the structure.

seifert surface for a trefoil

So I tried the same method I used for the Moebius strip and started from the edge. The edge is a trefoil, the simplest knot you can think of, and it needs three twists. (I had to use some paper strips to figure out the right combination of the orientation of the twists and the knot.) It would have been clever to count the stitches of the first loop, the edge, and make sure they’re a multiple of six, but I didn’t. So one of the arms is 17 stitches long, and the others 18. It did work out pretty well with the different colors being ‘localised’ more or less.

Anyway, it was quite a puzzle, but in the end it worked out. The pictures prove it.

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  1. Machiel

    Very nice! Do you think you are ready for this structure? ;-)
    See: http://www.newscientist.com/article/dn11410-mathematicians-finally-map-248dimension-structure.html

  2. Hmmm, maybe I’ll have to practise a bit more, first.

  3. crochet secrets

    Absolutely wonderful! Lovely pictures, too!

    (Absolutely wonderful… spam. Lovely spam, too)

  4. Pingback: Kijk, cijfers! « Qulog 2.0

  5. Nick

    Actually, the moebius strip is not a Seifert Surface since it is non-orientable. A Seifert Surface for the unknot might be a disk. Similarly, the moebius strip with 3 twists is not a Seifert Surface either.

  6. Oh, thanks, I didn’t know that – although it says so right in the second paragraph of the wikipedia article I linked: shows how well I read the thing…

    But, well, crocheting a disk isn’t really a challenge, is it? :-)

  7. RaiulBaztepo

    Hello!
    Very Interesting post! Thank you for such interesting resource!
    PS: Sorry for my bad english, I’v just started to learn this language ;)
    See you!
    Your, Raiul Baztepo

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