Seifert algorithm

Apply the oriented smoothing to every crossing of an oriented knot diagram, fill the resulting Seifert circles by disjoint disks at suitable heights, and reconnect them by twisted bands at the original crossings. The boundary is the original knot. If the connected surface has $s$ disks and $c$ bands, its Euler characteristic is $s-c$ and its genus is $(1-s+c)/2$.

A Seifert circle is a component of the crossing-free diagram obtained by oriented smoothing at every crossing. These circles bound the disks used in the Seifert algorithm.