# sarah-marie belcastro

I am a mathematician who knits as well as a knitter who does mathematics. It has always seemed natural to me to combine mathematics and knitting, whether that results in knitting a model of a mathematical object or in using mathematics to design a garment. Indeed, over my mathematical life both of these types of combinations have occurred. Most of the mathematical models I have created are only of aesthetic value and have no real function; it is rare that I am able to adapt a mathematical object for use as a garment. (It is perhaps too much to hope that I could regularly combine artistry and function in addition to knitting and mathematics.)

A (p,q) torus knot traverses the meridian cycle of a torus p times and the longitudinal cycle q times. Here are three instantiations of a (5,3) torus knot:

(a, middle) The knot embedded on a torus. A (p,q) torus knot may be drawn on a standard flat torus as a line of slope q/p. The challenge is to design a thickened line with constant slope on a curved surface.

(b, top) The knot projection knitted with a neighborhood of the embedding torus. The knitting proceeds meridianwise, as opposed to the embedded knot, which is knitted longitudinally. Here, one must form the knitting needle into a (5,3) torus knot prior to working rounds.

(c, bottom) The knot projection knitted into a cowl. The result looks like a skinny knotted torus.

This Klein bottle was knitted from an intrinsic-twist Mobius band with the boundary self-identified. A Klein bottle can be viewed as the connected sum of two projective planes; here, the stripes highlight the two circles that generate the fundamental groups of the individual projective planes. In some positions, this coloring of the Klein bottle resembles an ouroboros (a snake eating its own tail). The design is more than 10 years old; I recently realized that I had no high-quality example of it (only worn classroom models) and thus created one. Dream in Color veil-dyed yarn was chosen to add a color depth to the seed-stitch texture. Images of this piece graced the cover of the March-April 2013 issue of American Scientist.