A Möbius band, a.k.a. Möbius strip, is a nonorientable surface (has only one side) with a boundary. The most common way to make one is by taking a strip of paper, giving one end a half-twist, and taping the ends together. This is an extrinsic-twist construction, because it starts with something untwisted and adds the twist at the end. An intrinsic-twist construction is preferable for Mobius bands, because the twist occurs as part of the construction. Luckily, we can create intrinsic-twist Möbius bands with knitting. Instructions (three different kinds) are given at the bottom of this page.
A Short History of Knitting Möbius bands:
sm's Möbius bands:
I only knit seamless intrinsic-twist Möbius
bands. It's very difficult to find the center circle, but knitting forces a
one-half-stitch glide reflection symmetry across the central circle, so it's
not impossible. I almost always use seed stitch because it's so simple and
symmetric, but occasionally (as in the upper Möbius band in the old photo to
the right) I'll use faux garter stitch or the like; it creates the illusion of
symmetry but of course there's a hidden break in the symmetry.
Recently I've been making much smaller Möbius bands. The two on the right have central circles of approximately 20 inches in circumference; in the photo below, the band on the left has a circumference of perhaps 14 inches. The smaller bands are six or fewer inches in circumference.
In order to make Möbius bands so small, I had to create a sufficiently short circular needle (after all, I refuse to use double-pointed needles). After much thinking and asking around, I happened upon some bamboo circulars sold via eBay which have hollow cables. This allowed me to shorten the bamboo points, detach one point from the cable, shorten the cable appropriately, and re-glue. Awesome. Then, of course, I had to experiment with all the different kinds of super-clearance yarn available at my local yarn store...several are shown in the left-hand picture. Note that the right-hand picture has a tiny Klein bottle as well. Yes, that's cat hair on the background pillow... and the colorful yarn is Noro Kujaku. More mini Möbius pictures are available on a public ravelry page.
In 1999, it was pointed out to me that K6 embeds in the Möbius band. K6 is the complete graph on six vertices; in other words, take 6 points and join each pair of them by an edge. This can be drawn without edge-crossings on a Möbius band. (In fact, K6 is the largest complete graph which cellularly embeds on a Möbius strip.) Over the 2001-2002 school year, I designed and knitted a Möbius-band scarf with K6 embedded in it. Here are pix of the finished product. The base yarn is handspun wool/silk, the edges are a color of Lopi no longer produced, and the vertices are some random wool yarn which felted when I dyed it purple. Yes, it was stupid to dye it loose in the washing machine; now it's lumpy. Oh, well.
The next two pictures show closeups of an area surrounding the center circle. can you find it?
That said, here are two tips.
(1) A curved arrow on the diagrams indicates that one should start a new piece of yarn.
(2) Begin by casting on 72 stitches in cotton. (One can't use a scrap-yarn-less construction because of the color changes along the central circle.) This makes it easier to pull out the cast on than if one uses wool.
There is a fancier and smaller version of this embedding called Dusk or Dawn (the link is to its public ravelry page).
(For any mathematicians reading this, note that the first two methods are basically forming an S1 and forming the strip by adding a layer to the bottom, then top, then bottom, etc. Also, while such Möbius bands are essentially seamless, they still contain glide-reflection with glide one-half stitch. This is induced by knitting into the loops between the stitches, which is necessary in order to form the intrinsic twist.)
Method 1: Knit from the inside out, using scrap yarn.
You'll need two different colors of yarn, one for the Möbius band itself and a scrap yarn for casting on. I find plain cotton yarn the easiest to work with in all cases, but especially for the scrap. You'll also need a sufficiently flexible circular needle. I think I usually use a size 7 or 8 in 24" length for this, but other sizes should work as well.
Cast on 90 stitches in your scrap yarn. Because you'll be removing the scrap yarn later, use a cast on method which easily pulls out. (My favorite is basically a crochet chain cast on; thanks to Cindy Kraus for explaining it to me. I create a slipknot loop and, using my fingers, pull another loop through it to start. Then I wrap the free end of the yarn over the needle, and (again using my fingers) pull a loop through the previous loop. Then I repeat. If that made no sense, try this description instead.) In case you're wondering why the scrap yarn is necessary at all, it's to avoid any appearance of a seam in the finished product.
This means that your Möbius band will have a central circle of 90 stitches
in circumference. If you're making a wearable Möbius band rather than just a
mathematical manipulative, you will of course want to adjust this number for
gauge and fit.
Onto the cast-on row, knit one row loosely, using the yarn-for-the-Möbius-band-itself. Leave a bit of a tail when you begin so that you have some to knit in when you're done. (No knots allowed!) Note: you may want to purl one row instead. I think that whether you should purl or knit depends on the handedness of your cast-on row and the handedness of your knitting; the point is that some ways of knitting will ensnare the scrap yarn so it won't pull out easily, so be aware of the issue.
Now, if you were doing ordinary circular knitting, you'd continue by stretching the other end of your row to the other tip of the needle, and knitting onto that. However, you're going to introduce the intrinsic twist by instead knitting into the loops between the stitches of the row you just knit. In order to do this, don't stretch the other end of the row. Leave it where it is, and bring the tip of the needle to it. Then rotate the-other-end-of-the-row a bit so that you can access the loops between your stitches. Because these loops are not at the tip of a needle, you cannot do the ordinary (insert needle into loop)-(wrap yarn 'round needle)-(pull new loop through old)-(slide old loop off needle). Instead, you'll just do the first three of these operations, leaving your old loops still on the skinny part of the circlar needle... but instead of just knitting or just purling, you must *k1p1*. This has the effect of casting on an additional 89 stitches so that each 'row' has 179 stitches. (Note that if you cast on n stitches originally, knitting into the loops between will add n-1 stitches, giving an odd number of stitches total. This ensures that your *k1p1* will become seed stitch rather than ribbing, and that your seam will be invisible.) Bizarre though this may seem, it's consistent with the fact that a Möbius Band has only one edge, so it will appear that you'll be knitting twice 'round the strip to traverse the edge once. Also, this will probably hurt your fingers or at least be rather uncomfortable, so don't be shocked when that happens.
When you've finished knitting into the loops-between-the-stitches, your
needle will be loop-de-looped. Now you can just do *k1p1* forever, or rather,
until you feel like you're done. For a cute li'l strip, five 'rows' should be
fine. For a scarf, you'll want more like twenty.
When you're done, just cast off as usual. There are three tasks that remain: knit in the end of the yarn, get rid of the scrap yarn, and knit in the beginning of the yarn. When knitting in the beginning of the yarn, look carefully at your stitches so that you don't create a hole or piece of seam.
This is a modified/expanded version of Maria Iano-Fletcher's translation of Miles Reid's pattern.
Method 2: Knit from the inside out, no scrap yarn needed.
Once you're used to it, this is the fastest method. However, the yarn forming the central circle doesn't look quite as spiffy as when scrap yarn is used, because this method induces a little bit of additional torque on the yarn. (At least, it does when I do it. This could be a peculiarity of my less-than-orthodox left-handed knitting style.) Also, lots of people seem to have trouble figuring out how to do this, so be forewarned that it does require some thinking.
You'll need some yarn and a circular needle which can coil twice 'round a circle (the way they're conventionally used requires that they coil only once 'round a circle). Make a slip knot and slip it over one point of your needle. Grab the point in your right hand and hold the knot so it doesn't slide away. Bend the needle so that the other point is (a) pointing to the left (b) in front of the point with the slipknot (c) in front of the yarn. Now, using the cable instead of a piece of scrap yarn, do stranded cast-on. (Online instructions can be found on YouTube by Cat Bordhi---note that to be mathematically correct, you'll need to omit her last yo in order to have an odd number of stitches---or on Eunny Jang's blog. Printed instructions for stranded cast-on can be found in June Hemmons Hiatt's The Principles of Knitting, p. 138; in Mary Thomas's Knitting Book, p. 66, it's called invisible cast-on.) The number of stitches you cast on will be the number in the central circle of the Möbius band, which is roughly half the number of stitches on the boundary of the Möbius band. Pull the yarn to the front, so that it's coming toward you from behind the cable (not the point of the needle). This will ensure that you end up with an odd number of stitches, which is necessary for the whole seed stitch thing to work out correctly.
What you do next depends on your handedness. If you're left-handed, like me, you'll turn the whole business around and knit into the slip knot. If you're right-handed, you'll purl into the slip knot. After that, continue in seed stitch (*k1p1*), always knitting/purling by slipping the needle into the side of the stitch closest to you. I know, that sounds silly to say---that's how we always knit/purl---but you might feel like you're putting the needle in from a funny direction or in a funny piece of the stitch. And you might actually be. Just alter your usual stitch in whatever way you have to, so that none of your stitches twist. (If that confuses you, I recommend reading The Principles of Knitting, pages 23-24 and 32-34, or Anna Zilboorg's Knitting for Anarchists, pages 14-22.)
Then keep going until you've reached the desired width. Bind off in pattern and weave in the end. Now go back, untie the slip knot (which may take some doing) and weave in that end in pattern as well, being careful not to leave a hole. If you don't remove the slip knot, it will be very easy to see where you started.
Method 3: Knit from the outside in
Catherine Kehr sent me this method. I find it more difficult to do because (a) the contortions required just before the final graft are not easy and (b) trying to decide exactly which stitches should match up for the graft is hard, and an error will create a hole. In other words, I have to think too much using Catherine's method *grin*. The text below is mostly Catherine's, but slightly edited by me.
First, make a Möbius strip out of paper. If you're ambitious, color the edge so you can keep track of it later. Slice it in half down the midline. You will see that you now have one big ring with two full 360 twists in it instead of one small ring with a 180. It's now 720 because you took two 180s (the two halves of the Möbius strip) and pulled them into a ring by adding another full twist (this isn't the most scientific explanation, but it's essentially correct). Now try to put the Möbius strip back together, because you'll need to do this operation on your knitting.
Using yarn this time, and a circular needle, cast on 500 sts (or whatever twice your preferred finished length would be). Add two full twists and then join the knitting into a (twisted) circle. Knit to half your preferred width. Now twist it into the form of a Möbius strip and graft.
Catherine says: "For those of you who think this might be 'cheating' -- this technique is as legitimate a one-edged construction as inside-out (but with less harangue and more invisibility). Grafting the midline is no different that the grafting technique used when knitting inside out (aka invisble or provisional cast-on)."
There are some more-detailed instructions (including photos) for this construction method at the girl from auntie, but note that the sample has a seam so as to illustrate where to graft, and isn't made from a symmetric stitch.