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.
I will post explanations and instructions for knitting these surfaces after publishing a paper on those topics; it will be a detailed version of the talk I gave at the 2006 Joint Mathematics Meetings.