Topological Knitting: featuring the HRSFA Klein Bottle Hat!

Apologies: this post has been very long in coming but I’ve finally gotten around to writing it up.   I did a bunch of knitting last fall and winter, partly spurred on by the HRSFA yarn that the awesome hobbitknitter gave me last summer.  (And general thanks to hobbitknitter for encouragement, yarn, loan of needles when I went in for jury duty, etc.)  At the beginning of last fall it occurred to me that a Klein bottle would be a befittingly awesome thing to turn HRSFA yarn into.

But before we get to that: my Mobius strip, which was sort of a warm-up project for the hat.

There are a number of ways of knitting a Mobius strip:  the one I used was the “inside-out” method, basically the one explained here.   The idea behind this is the following: if you cut a Mobius strip in half you get a single strip with two full twists in it.  Now, knitting a strip with two full twists is easy — you just put two twists in the row of stitches before you join the ends in circular knitting.  So the hard part here is “undoing the cut” by grafting (sewing with yarn) the unfinished edge of the double-twisted self to itself — the first time I tried to make a Mobius strip I had the wrong number of twists in my original strip, and the strip ended up getting internal twists when I tried to join it up.  (Some people thing grafting is cheating, but I think it’s cool.  Also it was a fun challenge, and I had to learn how to do it for the Klein bottle anyway.)

I knitted the strip in k1p1 to make both sides look the same (I needed an odd number of stitiches around for the grafting, so the strip came out as seed stitch rather than rib stitch).  As it turns out, k1p1 is not perfectly symmetric with respect to turning it upside down, and the grafting produced a half-stitch offset, so things didn’t exactly match up where I joined, but I managed to join things in a way that made the seam as unnoticeable as possible.  (I also think seed stitch is just generally pretty good at hiding grafting seams.)

OK: now for the Klein Bottle:

The construction of this was a lot simpler topologically: I just knitted a tube with a slit in it, fed the narrow end of the tube through the slit, and joined it to the other end of the tube. The only tricky thing here was doing all the increases and decreases in k2p2 rib, which I think I managed fairly nicely.

More pictures of the hat in various configurations can be found in my Knitting album on Google here along with the Mobius pictures and a couple pictures of my hyperbolic scrunchie (which didn’t turn out as well as the others; I might post on hyperbolic knitting later if future projects turn out better). Here’s one last one (this is a bit of an in-joke among recent Fusion High Ladies; read the alt-text or click through and read the caption on the link):