Seifert surface

Remarks

The genus $g(K)$ of a knot $K$ is the least genus of a Seifert surface for $K$. Example: the unknot $\set{(x, y, 0) : x^2 + y^2 = 1}$ in $\R^3 \subset S^3$ bounds the flat disk $\set{(x, y, 0) : x^2 + y^2 \le 1}$, which is compact, connected, orientable and has the unknot as boundary; so the unknot has genus $0$. Conversely, a knot of genus $0$ bounds a disk and is the unknot. Non-example: a band with one half-twist, a Möbius band, has the unknot as boundary, but it is not orientable, so it is not a Seifert surface. A band with three half-twists is a Möbius band whose boundary is the trefoil, so it fails for the same reason; the trefoil has genus $1$ instead. Seifert's algorithm builds a Seifert surface from any diagram. One smooths each crossing according to the orientation, fills the resulting circles with disks, and joins the disks by twisted bands at the crossings.