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, probably in a book.

There are public pages for some of my other nonorientable surfaces:

four
surfaces from one skein of yarn

connected sum of 3 projective planes (P#P#P)

a striped P#P#P