Large pieces of high-curvature hyperbolic space are better crocheted than knitted, I think, unless you are willing to use double-pointed needles (which I am not). Still, I made a silly hyperbolic toddler tutu, visible on a public ravelry page...

It's fun to toss these at students and ask (a) is the curvature positive or
negative on these objects? and (b) which one has greater curvature?

I had knitted the above bits from the inside out, so to speak, by beginning with a small number of stitches and putting an increase on every stitch or every-other stitch. (That's exactly what one does when crocheting.) In summer of 2004, Ari Turner asked me what would happen if I knit them outside in, i.e. by starting with a needleful of stitches and uniformly decreasing. Here are the results:

These were done at 2:1, 3:2, and 4:3 stitch ratios, respectively. And it's much, much easier to knit outside->in than inside->out for these things...The recipe is as follows:

Cast on as many stitches as your needle will hold. (I use circular needles,
always.)

2:1 stitch ratio? *K2 togtbl* each row until you feel done. Cast off.

3:2 stitich ratio? *K1, K2togtbl* each row until you feel done. Cast off.

4:3 stitch ratio? *K2, K2togtbl* each row until you feel done. Cast off.

Taken together, Ari and I agreed they look much like Xmas ornaments.

A nice, loose **pseudosphere** (photo on public ravelry page here) may be
made in the following way:

Cast on as many stitches as your circular needle will hold. Join without
twisting to work in the round.

*K10, K2togtbl* until one stitch remains. Yes, really, that works, and yes,
really, that's all there is to it.

One cool thing that can be done with a hyperbolic octagon is folding it into
pants. Knitting instructions for baby pants can be found in *Making Mathematics with Needlework* and a
customizable version of the pattern is at the Wolfram
Demonstrations Project. An eight-colored pair of hyperbolic pants may be
seen here
(public ravelry page); really, they are a dual map to K_{8} embedded on
a 2-holed torus.

I did once make a hyperbolic toddler tutu (public ravelry page)...

The color orders in the stripes of this scarf form the
group S_{3}; think of S_{3} as acting on the triple (green,
grey, black).

More braids and some knots appear on my public ravelry pages:

triquetra
(flattened trefoil knot)

triple braid

(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