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.
Here's my first-ever torus, followed by my first-ever two-holed torus:
Miles Reid published two patterns for a torus (ref) in1971, but the one most like mine didn't have curvature. (His pattern with curvature is like mine, but with axes (rows/stitches) switched.)
And, here's my first-ever 3-holed torus. I know, the execution isn't pretty. It'll be better next time.
The photo below 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 which 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'm making a larger one which will look less edge-y, so to speak. There are more photos of this torus on its public ravelry page.
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 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