This page is devoted to knitted nonorientable surfaces of higher genus than two. For Möbius bands, projective planes, and Klein bottles, see my other pages (as linked). I also have a page on orientable surfaces.
While the surfaces on this page used more than one piece of yarn (in order to accomplish the color changes), each could have been knitted from a single strand. Well, okay, the ball of yarn would have had to be topological---I might have needed to shrink it in order to shove it through some stitches every now and then---but mathematically speaking, each could have been knitted from a single strand.
Every nonorientable surface is equivalent to the connected sum of some number of projective planes; that number is called the genus of the surface. A projective plane (P) has genus 1, a Klein bottle (P#P) has genus 2, and this...
has genus 3. It's knitted somewhat sloppily (hey, it was my first attempt) with the colored bits corresponding to the three homotopy generators.
Here is a genus-4 surface
(photo taken by Bob Vallin at the 2006 JMM). Again, colored bits correspond
to homotopy generators.
Here is a nonorientable surface of genus 5... each color corresponds to a
different projective plane.
A paper describing general methods for knitting these surfaces appears in the Journal of Mathematics and the Arts. (Citation: Every Topological Surface Can Be Knit: A Proof, Journal of Mathematics and the Arts, 3(2) June 2009, 67–83.) Knitting patterns will eventually appear... somewhere.