Have you encountered moebius knitting? It's an arcane mathemagical concept translated into deceptively easy twisted knitting. (A moebius strip has only one side and one edge – it's explained here if you aren't familiar with the idea. Grab a piece of paper and sticky tape and go check it out. Then come back for the yarny version.)
You can make a moebius cowl by knitting a strip lengthwise, then giving it a half-twist before grafting it closed. What's way more fun though is using Cat Bordhi's technique,* from A Treasury of Magical Knitting, to knit a moebius from the inside out. A unique, but fast and easy, cast-on** sets you up to work in the round in a way that feels normal... but actually extends both up and down at the same time, ending with a cast-off edge that wraps right around the top and bottom of your cowl. Magically.
Why bother? Well, because it's really, really fun. But also: it looks great. The half-twist drapes beautifully around your neck and shoulders. If you're using yarn with long colour shifts, those colours will appear perfectly symmetrically, which can be helpful to ensure aesthetic balance. And if you're not sure how far your yarn will get you, a moebius cowl can minimise the amount of planning required; you're done when you run out of yarn, and your stitch patterns will be perfectly balanced.
1. Start with a slipknot on your lefthand needle tip. Pull the tip through so that the slipknot sits on the cable. Hold your righthand needle tip and the slipknot together between the thumb and middle finger of your right hand. Support the cable with your left thumb and middle finger, and tension the yarn with the index finger.
2. Dip the righthand tip under the cable, up over the yarn, and bring it forward. One stitch made.
3. Hook needle behind yarn above the cable. Another stitch made.
Repeat steps 2-3 to make as many stitches as required. Don’t work too tight!
Each time you make a stitch on the righthand needle tip, a loop also appears underneath the cable. You will work into these in the first round, but don’t include them in your cast-on count. I find it extremely helpful to place stitch markers every 20 stitches or so while casting on – so, if told to cast on 120 stitches, you simply stop at the sixth marker. This not only makes counting easier, it saves confusion as to which stitches are above the cable and which below! (If you counted all the stitches both above and below – both of which will be knitted into – you'd find you have 240 stitches on the needle.)
4. Spread the stitches along the cable as you go, allowing the cable to form a double loop. Check that the cable is crossed exactly once when you join to start working in the round – creating the 90° twist essential for a moebius loop.
5. When you start knitting your first round, you will find every other stitch is mounted the wrong way around. Simply knit into the back of these stitches.
6. After knitting the full number of cast-on stitches (120, in our example), you will find yourself directly above the start-of-round marker, which is sitting on the cable below your needle tips. Continue following the instructions for this first round until you arrive back at the stitch marker, this time on your needle and ready to be slipped.
You will see that the stitches above your cast-on (shown here as knit stitches) will form the "right side" of your pattern, with the "wrong side" appearing below. So choose your stitches with care: reversible patterns such as those based on garter or knit/purl combinations, or welted patterns, work best.
My moebius design Purzelbaum uses three related garter-stitch lace patterns to show off a wonderful graduated yarn – just change stitches whenever the colour changes, and lo, a perfectly balanced, fully reversible cowl. Try it out!
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* Bordhi is an actual genius who has two books as well as a Craftsy class on this technique. Check them out!
** There are a number of other ways to cast on for a moebius. For instance, Iris Schreier uses a modified long-tail cast-on; or you can cast on using any method you like and then pick up stitches between those cast-on stitches to create your double loop. I find Bordhi's method still the neatest and certainly the fastest; I just need to take care to work loosely enough that the stitches can slide easily right around the double loop.