Basically, it's a sock for a jam jar (or in this particular case, a pesto jar). I started with a flat round disc and then knitted further without increasing to turn it into a tube.
I made this at the suggestion of someone who was collecting knitted things for a photoshoot - I don't think it's exactly what she had in mind, the reference pics were of very plain, tweedy jug-shaped things. I'm thinking of doing a few more: a black mohair one with horizontal welts and a fawn fuzzy one with cables all over.
Here's how to do a flat round disc to start off with:
Using 4 dpns, cast on 8 sts and distribute over 3 needles, with 2 sts on the 1st needle and 3 sts each on the 2nd and 3rd needles. Join into a round and knit all sts through the back loop to keep the center flat.
1st round: Knit into front and back of each st (called inc 1)
2nd round: Knit to end.
3rd rnd: As first.
4th and 5th rnds: K to end
6th rnd: *Inc 1, k2, inc 1, rep from * to end
7th and 8th rnds: K to end
9th rnd: *Inc 1, K4, inc 1, rep from * to end
10th and 11th rnds: K to end
12th rnd: *Inc 1, K6, inc 1, rep from * to end.
13th and 14th rnds: K to end
... you get the idea. Keep measuring the knitting against the bottom of the jar - I only did about 8 rounds, using 2.5 mm needles and DK cotton. (It's a small jar)
Then knit straight in the pattern of your choice. Ribbed patterns like the one I used need to be able to pull in, so the disc needs to be a little bigger than the bottom of the jar. At this point you can easily switch to a circular and magic loop it if that's easier.
The stitch pattern:
You need a multiple of 6 sts - I think I had to k2tog twice to get the right amount of stitches
rnd 1 - 6: *P2, K4* rep from * to * to end
rnd 7 - 12: *K3, P2, K1* rep from * to * to end