Knitting with cotton is very different than knitting with wool. The yarn itself does not stretch at all, still the knitted piece can be very stretchy depending on the stitch pattern used.
I tried knitting a bit of cotton lace and made these two shrugs. Different yarn and colour shade, both 100 per cent cotton.
The left one is knitted using the easiest pattern of a simple rectangle, for the one on the right I made a paper pattern by cutting an old sweater. This one is knitted in one piece as well and side seams sewn together.
The close-up of the lace patterns: