Seifert Surface

A Seifert surface is a connected, orientable surface in three dimensional space whose boundary is exactly a given knot or link, providing a way to study a one dimensional curve through the two dimensional surface it bounds. The surfaces are named after the German mathematician Herbert Seifert, who in a 1934 paper gave a systematic algorithm, now called Seifert's algorithm, for constructing such a surface directly from any diagram of a knot or link by resolving each crossing in a standard way and then attaching twisted bands. A given knot has many possible Seifert surfaces, but the smallest possible genus among all of them, called the genus of the knot, is an invariant that does not depend on which diagram or construction was used, making it a genuine and useful measure of a knot's topological complexity. Seifert surfaces remain a basic tool throughout knot theory and in the broader study of three dimensional and four dimensional topology.