I've written up (probably rather confusing) instructions for Seifert surfaces (two ways of crocheting them and one way of knitting them) and for crocheting crosscaps. The grid for the homology scarf is here. I'll probably get around to writing up instructions for the rest; if you're curious and don't want to wait, though (or if you find any of the instructions confusing), just send me an email (

I have a few more photos of things I've made at my Flickr page.

On to the pretty pictures! Let's start with the Seifert surfaces. These are orientable surfaces (essentially, you can't get from one side to the other without crossing an edge, as you can with the Mobius strip) with some knot or link as its boundary. A good introduction to them, with lots of pictures, is this one. It's kind of surprising that they exist for any knot or link --- and in fact, there's a rather easy algorithm for finding them. The nice thing is that the algorithm gives the surface in terms of a bunch of discs and twisted strips connecting them, both of which are easy to crochet! Putting it all together, it's not too hard to come up with a method that will let you crochet any Seifert surface. Here's the Seifert surface of a trefoil knot (I've run wire along the edge so it keeps its shape):

But we can also do links! Here's a Seifert surface of the Borromean Rings, three rings linked together in such a way that if any one is removed, the other two come apart as well. This was quite hard to make; all of the twists have to go in the right direction and it's quite difficult to keep track of them at the beginning (in one of my earlier attempts I got one wrong, giving me some less-exciting three-component link). Again, I've run wire through it so it keeps its shape.

Moving along, here's a Klein Bottle. I also have a figure-8 immersion of a Klein bottle lying around somewhere, but I'm not sure where (it's much easier to crochet, actually). I'm not too fond of this one; the seams are too obvious (the blurriness of the photo actually makes it look a bit better than it is!)

Here's another, this time done with a kind of checkerboard pattern so you can see inside (at least, you can when you're looking at it in "real life"; the pictures don't show it well at all). The downside is that it's really floppy; it was hard to get it to stand up and pose for these pictures!

We can also make a (sphere with a) crosscap, one way of shoving a projective plane into three-dimensional space.

Here's how it looks before stitching it together. We can see two discs that pass through each other along a line.

Brent, one of the other grad students, said that I should crochet an Alexander Horned Sphere. I chuckled for a second, and then thought about it and realized it should be possible. Here's my first attempt. I was getting a feel for it as I was doing it, so the sizes of the various parts aren't very consistent; still, I think it came out reasonably well for a first try.

I'm going to try doing another, this time with the "horns" coming out at greater angles (so that the resulting horned sphere would look a bit more like the picture I linked and less like a squid), and knit instead of crocheted.

I also recently finished the Long Exact Scarf of Homology. It's nothing too special (just a double-knit scarf), but the design is the Long Exact Sequence of Homology for a space X and a subspace A containing another subspace B.

I've only recently started to knit much, and so I've been trying to knit some of the things that I'd gotten good at crocheting. Here's my first attempt at the Seifert surface of a trefoil (in a previous post I'd posted pictures of such a surface, but crocheted). When I crochet it I start with a "skeleton" in the middle and crochet along the one edge; to knit it, though, I did the top and bottom halves separately and joined them in the middle. One nice thing about knitting is that the "right" and "wrong" sides look very different (while in crochet, at least with the stitch I was using, you have to look somewhat carefully to tell the difference); as a result it's more obvious that the surface is orientable. Anyway, here's the result:

The method I used to make this should generalize to any Seifert surface, so hopefully later I'll be able to knit Seifert surfaces for things like the Borromean rings.

And, finally, a crosscap. This was one of the first things I knit with double-pointed needles, and it didn't come out that great (though stuffing it with something would probably make it look better; right now it's flat and just looks like a disc). Still, it shows that it can be done!