# 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 link traverses the meridian cycle of a torus p times and the longitudinal cycle q times; when p and q are coprime, the result is a knot, and when not (ha!) the result is a gcd(p,q)-component link with each component a (p/gcd(p,q), p/gcd(p,q)) torus knot.

Here we have (in increasing order of complexity) a (4,2) torus link, a (6,4) torus link, and an (8,6) torus link. Each is knitted so that both the knotting and the linking are intrinsic to the construction (rather than induced afterwards via grafting). They were made as proof-of-concept for the methodology for knitting torus knots and links that the artist introduced at the 2014 JMM.

An embedding of a graph on a surface shows the graph without edges crossing. Instantiated here is the smallest possible snark graph embedded polyhedrally on the Klein bottle (proved in 2012 by the artist). It is properly edge-colored.

Constructing a knitting pattern for this object was arduous. To avoid obvious seams, a symmetric and uniform texture was needed; surface texture dictated that adjacent stitches interlock colors; and, for an edge to be clearly visible it had to be at least two rows long/stitches wide. Additionally, the graph needed to be distributed as evenly as possible across the surface, and many iterations were needed to produce an acceptable embedding. Construction was also arduous; 125 ends needed to be woven in.