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 (from Making Mathematics with Needlework) didn't have curvature. (His pattern with curvature is like mine, but with axes (rows/stitches) switched.)
Here's my first-ever torus:
(I have a pattern for an embellished version available.)
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 K7 on the torus, created
for a conference honoring Martin Gardner. K7 is the complete graph
on seven vertices, in which each pair of the seven vertices is joined by an
edge. K7 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 K7 on a torus.
K7 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, probably in a book.
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)