This page is devoted to knitted orientable surfaces, such as the torus, 2-holed torus, etc. I also have a page on nonorientable surfaces and pages on Möbius bands, projective planes, and Klein bottles.

While the surfaces on this page actually used more than one piece of yarn, 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.

Miles Reid published two patterns for a torus (ref) in 1971, but the one most like mine didn't have curvature. (His pattern with curvature is like mine, but with axes (rows/stitches) switched.)

Here's my first-ever torus:

My first-ever two-holed torus:

And, here's my first-ever 3-holed torus. I know, the execution isn't pretty.
It'll be better next time.

The next photo shows an embedding of K_{7} on the torus, created
for a conference honoring Martin Gardner. K_{7} is the complete graph
on seven vertices, in which each pair of the seven vertices is joined by an
edge. K_{7} is the largest complete graph that can be drawn on the
torus without edges crossing. This particular torus seems all edge, no face
(like all hat, no cattle) to me so I've made a larger one that looks less
edge-y, so to speak.

See photos on :

larger K_{7} on a torus.

K_{7} on a torus, all edge and no face.

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 pages for some of my other orientable surfaces:

three
surfaces from one skein of yarn

orthogonal
2-holed torus

(5,3) torus
knot embedded on a torus

(3,2) torus
knot embedded on a torus with one chirality

(3,2) torus
knot embedded on a torus with the other chirality

a
hyperbolic octagon that folds into a 2-holed torus (and also a pair of
pants)