## Nonorientable Knitted Surfaces

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. Below 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.

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

four
surfaces from one skein of yarn

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