Illustrating the Genus of a Seifert Surface

A Seifert surface of a knot ∗ is an orientable surface whose boundary coincides with that knot. These surfaces are named after Herbert Seifert, who gave an algorithm to construct such a surface from a diagram describing the knot in 1934. The steps in this algorithm are relatively easy to understand, this does not hold for the geometric shape, which does not become immediately apparent from the resulting schematic diagrams of the surface. Texts on knot theory almost always only contain these schematic drawings, from which it is hard to understand their structure and properties. Seifert also introduced the notion of the genus of a knot as the minimal genus of all Seifert surfaces of this knot. In this way the genus of a knot or link can be defined via Seifert surfaces. The aim of this project is to illustrate the genus of a Seifert surface by gradually transforming the surface to a normal-form that clearly shows the genus.